research article incomplete -fibonacci and -lucas...
TRANSCRIPT
Hindawi Publishing CorporationChinese Journal of MathematicsVolume 2013 Article ID 107145 7 pageshttpdxdoiorg1011552013107145
Research ArticleIncomplete 119896-Fibonacci and 119896-Lucas Numbers
Joseacute L Ramiacuterez
Instituto de Matematicas y sus Aplicaciones Universidad Sergio Arboleda Colombia
Correspondence should be addressed to Jose L Ramırez joselramirezimausergioarboledaeduco
Received 10 July 2013 Accepted 11 September 2013
Academic Editors B Li and W van der Hoek
Copyright copy 2013 Jose L Ramırez This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
We define the incomplete k-Fibonacci and k-Lucas numbers we study the recurrence relations and some properties of thesenumbers
1 Introduction
Fibonacci numbers and their generalizations have manyinteresting properties and applications to almost every fieldof science and art (eg see [1]) Fibonacci numbers 119865
119899are
defined by the recurrence relation
1198650= 0 119865
1= 1 119865
119899+1= 119865119899+ 119865119899minus1 119899 ⩾ 1 (1)
There exist a lot of properties about Fibonacci numbersIn particular there is a beautiful combinatorial identity toFibonacci numbers [1]
119865119899=
lfloor(119899minus1)2rfloor
sum
119894=0
(119899 minus 119894 minus 1
119894) (2)
From (2) Filipponi [2] introduced the incomplete Fibonaccinumbers119865
119899(119904) and the incomplete Lucas numbers119871
119899(119904)They
are defined by
119865119899(119904) =
119904
sum
119895=0
(119899 minus 1 minus 119895
119895) (119899 = 1 2 3 0 le 119904 le lfloor
119899 minus 1
2rfloor)
119871119899(119904) =
119904
sum
119895=0
119899
119899 minus 119895(119899 minus 119895
119895) (119899 = 1 2 3 0 le 119904 le lfloor
119899
2rfloor)
(3)
Further in [3] generating functions of the incompleteFibonacci and Lucas numbers are determined In [4] Djord-jevic gave the incomplete generalized Fibonacci and Lucasnumbers In [5] Djordjevic and Srivastava defined incom-plete generalized Jacobsthal and Jacobsthal-Lucas numbers
In [6] the authors define the incomplete Fibonacci and Lucas119901-numbers Also the authors define the incomplete bivariateFibonacci and Lucas 119901-polynomials in [7]
On the other hand many kinds of generalizations ofFibonacci numbers have been presented in the literature Inparticular a generalization is the 119896-Fibonacci Numbers
For any positive real number 119896 the 119896-Fibonacci sequencesay 119865
119896119899119899isinN is defined recurrently by
1198651198960= 0 119865
1198961= 1 119865
119896119899+1= 119896119865119896119899+ 119865119896119899minus1
119899 ⩾ 1
(4)
In [8] 119896-Fibonacci numbers were found by studyingthe recursive application of two geometrical transformationsused in the four-triangle longest-edge (4TLE) partitionThese numbers have been studied in several papers see [8ndash14]
For any positive real number 119896 the 119896-Lucas sequence say119871119896119899119899isinN
is defined recurrently by
1198711198960= 2 119871
1198961= 119896 119871
119896119899+1= 119896119871119896119899+ 119871119896119899minus1
(5)
If 119896 = 1 we have the classical Lucas numbers Moreover119871119896119899= 119865119896119899minus1
+ 119865119896119899+1
119899 ⩾ 1 see [15]In [12] the explicit formula to 119896-Fibonacci numbers is
119865119896119899=
lfloor(119899minus1)2rfloor
sum
119894=0
(119899 minus 119894 minus 1
119894) 119896119899minus2119894minus1
(6)
and the explicit formula of 119896-Lucas numbers is
119871119896119899(119909) =
lfloor1198992rfloor
sum
119894=0
119899
119899 minus 119894(119899 minus 119894
119894) 119896119899minus2i (7)
2 Chinese Journal of Mathematics
From (6) and (7) we introduce the incomplete 119896-Fibonacci and 119896-Lucas numbers and we obtain new recurrentrelations new identities and their generating functions
2 The Incomplete 119896-Fibonacci Numbers
Definition 1 The incomplete 119896-Fibonacci numbers aredefined by
119865119897
119896119899=
119897
sum
119894=0
(119899 minus 1 minus 119894
119894) 119896119899minus2119894minus1
0 le 119897 le lfloor119899 minus 1
2rfloor (8)
In Table 1 some values of incomplete 119896-Fibonacci num-bers are provided
We note that
119865lfloor(119899minus1)2rfloor
1119899= 119865119899 (9)
For 119896 = 1 we get incomplete Fibonacci numbers [2]Some special cases of (8) are
1198650
119896119899= 119896119899minus1
(119899 ge 1)
1198651
119896119899= 119896119899minus1
+ (119899 minus 2) 119896119899minus3
(119899 ge 3)
1198652
119896119899= 119896119899minus1
+ (119899 minus 2) 119896119899minus3
+(119899 minus 4) (119899 minus 3)
2119896119899minus5
(119899 ge 5)
119865lfloor(119899minus1)2rfloor
119896119899= 119865119896119899 (119899 ge 1)
119865lfloor(119899minus3)2rfloor
119896119899=
119865119896119899minus119899119896
2(119899 even)
119865119896119899minus 1 (119899 odd)
(119899 ge 3)
(10)
21 Some Recurrence Properties of the Numbers 119865119897119896119899
Proposition 2 The recurrence relation of the incomplete 119896-Fibonacci numbers 119865119897
119896119899is
119865119897+1
119896119899+2= 119896119865119897+1
119896119899+1+ 119865119897
119896119899 0 le 119897 le
119899 minus 2
2 (11)
The relation (11) can be transformed into the nonhomoge-neous recurrence relation
119865119897
119896119899+2= 119896119865119897
119896119899+1+ 119865119897
119896119899minus (119899 minus 1 minus 119897
119897) 119896119899minus1minus2119897
(12)
Proof Use Definition 1 to rewrite the right-hand side of (11)as
119896
119897+1
sum
119894=0
(119899 minus 119894
119894) 119896119899minus2119894
+
119897
sum
119894=0
(119899 minus 119894 minus 1
119894) 119896119899minus2119894minus1
=
119897+1
sum
119894=0
(119899 minus 119894
119894) 119896119899minus2119894+1
+
119897+1
sum
119894=1
(119899 minus 119894
119894 minus 1) 119896119899minus2119894+1
= 119896119899minus2119894+1
(
119897+1
sum
119894=0
[(119899 minus 119894
119894) + (
119899 minus 119894
119894 minus 1)]) minus 119896
119899+1
(119899
minus1)
=
119897+1
sum
119894=0
(119899 minus 119894 + 1
119894) 119896119899minus2119894+1
minus 0
= 119865119897
119896119899+2
(13)
Proposition 3 One has119904
sum
119894=0
(119904
119894) 119865119897+119894
119896119899+119894119896119894
= 119865119897+119904
119896119899+2119904 0 le 119897 le
119899 minus 119904 minus 1
2 (14)
Proof (by induction on 119904) Sum (14) clearly holds for 119904 = 0 and119904 = 1 (see (11)) Now suppose that the result is true for all119895 lt 119904 + 1 we prove it for 119904 + 1
119904+1
sum
119894=0
(119904 + 1
119894)119865119897+119894
119896119899+119894119896119894
=
119904+1
sum
119894=0
[(119904
119894) + (
119904
119894 minus 1)]119865119897+119894
119896119899+119894119896119894
=
119904+1
sum
119894=0
(119904
119894) 119865119897+119894
119896119899+119894119896119894
+
119904+1
sum
119894=0
(119904
119894 minus 1)119865119897+119894
119896119899+119894119896119894
= 119865119897+119904
119896119899+2119904+ (
119904
119904 + 1)119865119897+119904+1
119896119899+119904+1119896119904+1
+
119904
sum
119894=minus1
(119904
119894) 119865119897+119894+1
119896119899+119894+1119896119894+1
= 119865119897+119904
119896119899+2119904+ 0 +
119904
sum
119894=0
(119904
119894) 119865119897+119894+1
119896119899+119894+1119896119894+1
+ (119904
minus1)119865119897
119896119899
= 119865119897+119904
119896119899+2119904+ 119896
119904
sum
119894=0
(119904
119894) 119865119897+119894+1
119896119899+119894+1119896119894
+ 0
= 119865119897+119904
119896119899+2119904+ 119896119865119897+119904+1
119896119899+2119904+1
= 119865119897+119904+1
119896119899+2119904+2
(15)
Proposition 4 For 119899 ge 2119897 + 2119904minus1
sum
119894=0
119865119897
119896119899+119894119896119904minus1minus119894
= 119865119897+1
119896119899+119904+1minus 119896119904
119865119897+1
119896119899+1 (16)
Chinese Journal of Mathematics 3
Table 1 The numbers 119865119897119896119899 for 1 ⩽ 119899 ⩽ 10
nl 0 1 2 3 41 12 k3 k2 119896
2
+ 1
4 k3 1198963
+ 2119896
5 k4 1198964
+ 31198962
1198964
+ 31198962
+ 1
6 k5 1198965
+ 41198963
1198965
+ 41198963
+ 3119896
7 k6 1198966
+ 51198964
1198966
+ 51198964
+ 61198962
1198966
+ 51198964
+ 61198962
+ 1
8 k7 1198967
+ 61198965
1198967
+ 61198965
+ 101198963
1198967
+ 61198965
+ 101198963
+ 4119896
9 k8 1198968
+ 71198966
1198968
+ 71198966
+ 151198964
1198968
+ 71198966
+ 151198964
+ 101198962
1198968
+ 71198966
+ 151198964
+ 101198962
+ 1
10 k9 1198969
+ 81198967
1198969
+ 81198967
+ 211198965
1198969
+ 81198967
+ 211198965
+ 201198963
1198969
+ 81198967
+ 211198965
+ 201198963
+ 5119896
Proof (by induction on 119904) Sum (16) clearly holds for 119904 = 1(see (11)) Now suppose that the result is true for all 119895 lt 119904We prove it for 119904
119904
sum
119894=0
119865119897
119896119899+119894119896119904minus119894
= 119896
119904minus1
sum
119894=0
119865119897
119896119899+119894119896119904minus119894minus1
+ 119865119897
119896119899+119904
= 119896 (119865119897+1
119896119899+119904+1minus 119896119904
119865119897+1
119896119899+1) + 119865119897
119896119899+119904
= (119896119865119897+1
119896119899+119904+1+ 119865119897
119896119899+119904) minus 119896119904+1
119865119897+1
119896119899+1
= 119865119897+1
119896119899+119904+2minus 119896119904+1
119865119897+1
119896119899+1
(17)
Note that if 119896 in (4) is a real variable then 119865119896119899= 119865119909119899
andthey correspond to the Fibonacci polynomials defined by
119865119899+1(119909) =
1 if 119899 = 0119909 if 119899 = 1119909119865119899(119909) + 119865
119899minus1(119909) if 119899 gt 1
(18)
Lemma 5 One has
1198651015840
119899(119909) =
119899119865119899+1(119909) minus 119909119865
119899(119909) + 119899119865
119899minus1(119909)
1199092 + 4
=119899119871119899(119909) minus 119909119865
119899(119909)
1199092 + 4
(19)
See Proposition 13 of [12]
Lemma 6 One haslfloor(119899minus1)2rfloor
sum
119894=0
119894 (119899 minus 1 minus 119894
119894) 119896119899minus1minus2119894
=
((1198962
+ 4) 119899 minus 4) 119865119896119899minus 119899119896119871
119896119899
2 (1198962 + 4)
(20)
Proof From (6) we have that
119896119865119896119899=
lfloor(119899minus1)2rfloor
sum
119894=0
(119899 minus 1 minus 119894
119894) 119896119899minus2119894
(21)
By deriving into the previous equation (respect to 119896) it isobtained
119865119896119899+ 1198961198651015840
119896119899=
lfloor(119899minus1)2rfloor
sum
119894=0
(119899 minus 2119894) (119899 minus 1 minus 119894
119894) 119896119899minus2119894minus1
= 119899119865119896119899minus 2
lfloor(119899minus1)2rfloor
sum
119894=0
119894 (119899 minus 1 minus 119894
119894) 119896119899minus2119894minus1
(22)
From Lemma 5
119865119896119899+ 119896(
119899119871119896119899minus 119896119865119896119899
1198962 + 4)
= 119899119865119896119899minus 2
lfloor(119899minus1)2rfloor
sum
119894=0
119894 (119899 minus 1 minus 119894
119894) 119896119899minus2119894minus1
(23)
From where after some algebra (20) is obtained
Proposition 7 One has
lfloor(119899minus1)2rfloor
sum
119897=0
119865119897
119896119899=
4119865119896119899+ 119899119896119871
119896119899
2 (1198962 + 4)(119899 even)
(1198962
+ 8) 119865119896119899+ 119899119896119871
119896119899
2 (1198962 + 4)(119899 odd)
(24)
4 Chinese Journal of Mathematics
Table 2 The numbers 119871119897119896119899 for 1 ⩽ 119899 ⩽ 9
nl 0 1 2 3 41 k2 119896
2
1198962
+ 2
3 1198963
1198963
+ 3119896
4 1198964
1198964
+ 41198962
1198964
+ 41198962
+ 2
5 1198965
1198965
+ 51198963
1198965
+ 51198963
+ 5119896
6 1198966
1198966
+ 61198964
1198966
+ 61198964
+ 91198962
1198966
+ 61198964
+ 91198962
+ 2
7 1198967
1198967
+ 71198965
1198967
+ 71198965
+ 141198963
1198967
+ 71198965
+ 141198963
+ 7119896
8 1198968
1198968
+ 81198966
1198968
+ 81198966
+ 201198964
1198968
+ 81198966
+ 201198964
+ 161198962
1198968
+ 81198966
+ 201198964
+ 161198962
+ 2
9 1198969
1198969
+ 91198967
1198969
+ 91198967
+ 271198965
1198969
+ 91198967
+ 271198965
+ 301198963
1198969
+ 91198967
+ 271198965
+ 301198963
+ 9119896
Prooflfloor(119899minus1)2rfloor
sum
119897=0
119865119897
119896119899
= 1198650
119896119899+ 1198651
119896119899+ sdot sdot sdot + 119865
lfloor(119899minus1)2rfloor
119896119899
= (119899 minus 1 minus 0
0) 119896119899minus1
+ [(119899 minus 1 minus 0
0) 119896119899minus1
+ (119899 minus 1 minus 1
1) 119896119899minus3
]
+ sdot sdot sdot +
[[[[
[
(119899 minus 1 minus 0
0) 119896119899minus1
+ (119899 minus 1 minus 1
1) 119896119899minus3
+ sdot sdot sdot +(
119899 minus 1 minus lfloor119899 minus 1
2rfloor
lfloor119899 minus 1
2rfloor
)119896119899minus1minus2lfloor(119899minus1)2rfloor
]]]]
]
= (lfloor119899 minus 1
2rfloor + 1)(
119899 minus 1 minus 0
0) 119896119899minus1
+ lfloor119899 minus 1
2rfloor(119899 minus 1 minus 1
1) 119896119899minus3
+ sdot sdot sdot +(
119899 minus 1 minus lfloor119899 minus 1
2rfloor
lfloor119899 minus 1
2rfloor
)119896119899minus1minus2lfloor(119899minus1)2rfloor
=
lfloor(119899minus1)2rfloor
sum
119894=0
(lfloor119899 minus 1
2rfloor + 1 minus 119894) (
119899 minus 1 minus 119894
119894) 119896119899minus1minus2119894
=
lfloor(119899minus1)2rfloor
sum
119894=0
(lfloor119899 minus 1
2rfloor + 1)(
119899 minus 1 minus 119894
119894) 119896119899minus1minus2119894
minus
lfloor(119899minus1)2rfloor
sum
119894=0
119894 (119899 minus 1 minus 119894
119894) 119896119899minus1minus2119894
= (lfloor119899 minus 1
2rfloor + 1)119865
119896119899minus
lfloor(119899minus1)2rfloor
sum
119894=0
119894 (119899 minus 1 minus 119894
119894) 119896119899minus1minus2119894
(25)
From Lemma 6 (24) is obtained
3 The Incomplete 119896-Lucas Numbers
Definition 8 The incomplete 119896-Lucas numbers are defined by
119871119897
119896119899=
119897
sum
119894=0
119899
119899 minus 119894(119899 minus 119894
119894) 119896119899minus2119894
0 le 119897 le lfloor119899
2rfloor (26)
InTable 2 somenumbers of incomplete 119896-Lucas numbersare provided
We note that
119871lfloor1198992rfloor
1119899= 119871119899 (27)
Some special cases of (26) are
1198710
119896119899= 119896119899
(119899 ge 1)
1198711
119896119899= 119896119899
+ 119899119896119899minus2
(119899 ge 2)
1198712
119896119899= 119896119899
+ 119899119896119899minus2
+119899 (119899 minus 3)
2119896119899minus4
(119899 ge 4)
119871lfloor1198992rfloor
119896119899= 119871119896119899 (119899 ge 1)
119871lfloor(119899minus2)2rfloor
119896119899=
119871119896119899minus 2 (119899 even)
119871119896119899minus 119899119896 (119899 odd)
(119899 ge 2)
(28)
Chinese Journal of Mathematics 5
31 Some Recurrence Properties of the Numbers 119871119897119896119899
Proposition 9 One has
119871119897
119896119899= 119865119897minus1
119896119899minus1+ 119865119897
119896119899+1 0 le 119897 le lfloor
119899
2rfloor (29)
Proof By (8) rewrite the right-hand side of (29) as
119897minus1
sum
119894=0
(119899 minus 2 minus 119894
119894) 119896119899minus2minus2119894
+
119897
sum
119894=0
(119899 minus 119894
119894) 119896119899minus2119894
=
119897
sum
119894=1
(119899 minus 1 minus 119894
119894 minus 1) 119896119899minus2119894
+
119897
sum
119894=0
(119899 minus 119894
119894) 119896119899minus2119894
=
119897
sum
119894=0
[(119899 minus 1 minus 119894
119894 minus 1) + (
119899 minus 119894
119894)] 119896119899minus2119894
minus (119899 minus 1
minus1)
=
119897
sum
119894=0
119899
119899 minus 119894(119899 minus 119894
119894) 119896119899minus2119894
+ 0
= 119871119897
119896119899
(30)
Proposition 10 The recurrence relation of the incomplete 119896-Lucas numbers 119871119897
119896119899is
119871119897+1
119896119899+2= 119896119871119897+1
119896119899+1+ 119871119897
119896119899 0 le 119897 le lfloor
119899
2rfloor (31)
The relation (31) can be transformed into the nonhomoge-neous recurrence relation
119871119897
119896119899+2= 119896119871119897
119896119899+1+ 119871119897
119896119899minus119899
119899 minus 119897(119899 minus 119897
119897) 119896119899minus2119897
(32)
Proof Using (29) and (11) we write
119871119897+1
119896119899+2= 119865119897
119896119899+1+ 119865119897+1
119896119899+3
= 119896119865119897
119896119899+ 119865119897minus1
119896119899minus1+ 119896119865119897+1
119896119899+2+ 119865119897
119896119899+1
= 119896 (119865119897
119896119899+ 119865119897+1
119896119899+2) + 119865119897minus1
119896119899minus1+ 119865119897
119896119899+1
= 119896119871119897+1
119896119899+1+ 119871119897
119896119899
(33)
Proposition 11 One has
119896119871119897
119896119899= 119865119897
119896119899+2minus 119865119897minus2
119896119899minus2 0 le 119897 le lfloor
119899 minus 1
2rfloor (34)
Proof By (29)
119865119897
119896119899+2= 119871119897
119896119899+1minus 119865119897minus1
119896119899 119865
119897minus2
119896119899minus2= 119871119897minus1
119896119899minus1minus 119865119897minus1
119896119899 (35)
whence from (31)
119865119897
119896119899+2minus 119865119897minus2
119896119899minus2= 119871119897
119896119899+1minus 119871119897minus1
119896119899minus1= 119896119871119897
119896119899 (36)
Proposition 12 One has
119904
sum
119894=0
(119904
119894) 119871119897+119894
119896119899+119894119896119894
= 119871119897+119904
119896119899+2119904 0 le 119897 le
119899 minus 119904
2 (37)
Proof Using (29) and (14) we write
119904
sum
119894=0
(119904
119894) 119871119897+119894
119896119899+119894119896119894
=
119904
sum
119894=0
(119904
119894) [119865119897+119894minus1
119896119899+119894minus1+ 119865119897+119894
119896119899+119894+1] 119896119894
=
119904
sum
119894=0
(119904
119894) 119865119897+119894minus1
119896119899+119894minus1119896119894
+
119904
sum
119894=0
(119904
119894) 119865119897+119894
119896119899+119894+1119896119894
= 119865119897minus1+119904
119896119899minus1+2119904+ 119865119897+119904
119896119899+1+2119904= 119871119897+119904
119896119899+2119904
(38)
Proposition 13 For 119899 ge 2119897 + 1
119904minus1
sum
119894=0
119871119897
119896119899+119894119896119904minus1minus119894
= 119871119897+1
119896119899+119904+1minus 119896119904
119871119897+1
119896119899+1 (39)
The proof can be done by using (31) and induction on 119904
Lemma 14 One has
lfloor1198992rfloor
sum
119894=0
119894119899
119899 minus 119894(119899 minus 119894
119894) 119896119899minus2119894
=119899
2[119871119896119899minus 119896119865119896119899] (40)
The proof is similar to Lemma 6
Proposition 15 One has
lfloor1198992rfloor
sum
119897=0
119871119897
119896119899=
119871119896119899+119899119896
2119865119896119899
(119899 even)1
2(119871119896119899+ 119899119896119865
119896119899) (119899 odd)
(41)
Proof An argument analogous to that of the proof ofProposition 7 yields
lfloor1198992rfloor
sum
119897=0
119871119897
119896119899= (lfloor
119899
2rfloor + 1) 119871
119896119899minus
lfloor1198992rfloor
sum
119894=0
119894119899
119899 minus 119894(119899 minus 119894
119894) 119896119899minus2119894
(42)
From Lemma 14 (41) is obtained
4 Generating Functions of the Incomplete119896-Fibonacci and 119896-Lucas Number
In this section we give the generating functions of incomplete119896-Fibonacci and 119896-Lucas numbers
Lemma 16 (see [3 page 592]) Let 119904119899infin
119899=0be a complex
sequence satisfying the following nonhomogeneous recurrencerelation
119904119899= 119886119904119899minus1+ 119887119904119899minus2+ 119903119899(119899 gt 1) (43)
6 Chinese Journal of Mathematics
where 119886 and 119887 are complex numbers and 119903119899 is a given complex
sequence Then the generating function 119880(119905) of the sequence119904119899 is
119880 (119905) =119866 (119905) + 119904
0minus 1199030+ (1199041minus 1199040119886 minus 1199031) 119905
1 minus 119886119905 minus 1198871199052 (44)
where 119866(119905) denotes the generating function of 119903119899
Theorem 17 The generating function of the incomplete 119896-Fibonacci numbers 119865119897
119896119899is given by
119877119896119897(119909) =
infin
sum
119894=0
119865119897
119896119894119905119894
= 1199052119897+1
[1198651198962119897+1
+ (1198651198962119897+2
minus 1198961198651198962119897+1
) 119905 minus1199052
(1 minus 119896119905)119897+1
]
times [1 minus 119896119905 minus 1199052
]minus1
(45)
Proof Let 119897 be a fixed positive integer From (8) and (12)119865119897
119896119899= 0 for 0 le 119899 lt 2119897 + 1 119865119897
1198962119897+1= 1198651198962119897+1
and 1198651198971198962119897+2
=
1198651198962119897+2
and
119865119897
119896119899= 119896119865119897
119896119899minus1+ 119865119897
119896119899minus2minus (119899 minus 3 minus 119897
119897) 119896119899minus3minus2119897
(46)
Now let
1199040= 119865119897
1198962119897+1 119904
1= 119865119897
1198962119897+2 119904
119899= 119865119897
119896119899+2119897+1 (47)
Also let
1199030= 1199031= 0 119903
119899= (119899 + 119897 minus 2
119899 minus 2) 119896119899minus2
(48)
The generating function of the sequence 119903119899 is119866(119905) = 1199052(1minus
119896119905)119897+1 (see [16 page 355]) Thus from Lemma 16 we get the
generating function 119877119896119897(119909) of sequence 119904
119899
Theorem 18 The generating function of the incomplete 119896-Lucas numbers 119871119897
119896119899is given by
119878119896119897(119909) =
infin
sum
119894=0
119871119897
119896119894119905119894
= 1199052119897
[1198711198962119897+ (1198711198962119897+1
minus 1198961198711198962119897) 119905 minus
1199052
(2 minus 119905)
(1 minus 119896119905)119897+1
]
times [1 minus 119896119905 minus 1199052
]minus1
(49)
Proof The proof of this theorem is similar to the proof ofTheorem 17 Let 119897 be a fixed positive integer From (26) and(32)119871119897
119896119899= 0 for 0 le 119899 lt 2119897119871119897
1198962119897= 1198711198962119897
and1198711198971198962119897+1
= 1198711198962119897+1
and
119871119897
119896119899= 119896119871119897
119896119899minus1+ 119871119897
119896119899minus2minus119899 minus 2
119899 minus 2 minus 119897(119899 minus 2 minus 119897
119899 minus 2 minus 2119897) 119896119899minus2minus2119897
(50)
Now let
1199040= 119871119897
1198962119897 119904
1= 119871119897
1198962119897+1 119904
119899= 119871119897
119896119899+2119897 (51)
Also let
1199030= 1199031= 0 119903
119899= (119899 + 2119897 minus 2
119899 + 119897 minus 2) 119896119899+2119897minus2
(52)
The generating function of the sequence 119903119899 is 119866(119905) = 1199052(2 minus
119905)(1 minus 119896119905)119897+1 (see [16 page 355]) Thus from Lemma 16 we
get the generating function 119878119896119897(119909) of sequence 119904
119899
5 Conclusion
In this paper we introduce incomplete 119896-Fibonacci and119896-Lucas numbers and we obtain new identities In [17]the authors introduced the ℎ(119909)-Fibonacci polynomialsThat generalizes Catalanrsquos Fibonacci polynomials and the119896-Fibonacci numbers Let ℎ(119909) be a polynomial with realcoefficientsThe ℎ(119909)-Fibonacci polynomials 119865
ℎ119899(119909)119899isinN are
defined by the recurrence relation
119865ℎ0(119909) = 0 119865
ℎ1(119909) = 1
119865ℎ119899+1
(119909) = ℎ (119909) 119865ℎ119899(119909) + 119865
ℎ119899minus1(119909) 119899 ⩾ 1
(53)
It would be interesting to study a definition of incompleteℎ(119909)-Fibonacci polynomials and research their properties
Acknowledgments
The author would like to thank the anonymous referees fortheir helpful comments The author was partially supportedby Universidad Sergio Arboleda under Grant no USA-II-2011-0059
References
[1] T Koshy Fibonacci and Lucas Numbers with ApplicationsWiley-Interscience 2001
[2] P Filipponi ldquoIncomplete Fibonacci and Lucas numbersrdquoRendi-conti del CircoloMatematico di Palermo vol 45 no 1 pp 37ndash561996
[3] A Pinter and H M Srivastava ldquoGenerating functions ofthe incomplete Fibonacci and Lucas numbersrdquo Rendiconti delCircolo Matematico di Palermo vol 48 no 3 pp 591ndash596 1999
[4] G B Djordjevic ldquoGenerating functions of the incomplete gen-eralized Fibonacci and generalized Lucas numbersrdquo FibonacciQuarterly vol 42 no 2 pp 106ndash113 2004
[5] G B Djordjevic and H M Srivastava ldquoIncomplete generalizedJacobsthal and Jacobsthal-Lucas numbersrdquo Mathematical andComputer Modelling vol 42 no 9-10 pp 1049ndash1056 2005
[6] D Tasci and M Cetin Firengiz ldquoIncomplete Fibonacci andLucas p-numbersrdquoMathematical and Computer Modelling vol52 no 9-10 pp 1763ndash1770 2010
[7] D Tasci M Cetin Firengiz andN Tuglu ldquoIncomplete bivariateFibonaccu and Lucas p-polynomialsrdquo Discrete Dynamics inNature and Society vol 2012 Article ID 840345 11 pages 2012
[8] S Falcon and A Plaza ldquoOn the Fibonacci k-numbersrdquo ChaosSolitons and Fractals vol 32 no 5 pp 1615ndash1624 2007
Chinese Journal of Mathematics 7
[9] C Bolat and H Kose ldquoOn the properties of k-Fibonaccinumbersrdquo International Journal of Contemporary MathematicalSciences vol 22 no 5 pp 1097ndash1105 2010
[10] S Falcon ldquoThe k-Fibonacci matrix and the Pascal matrixrdquoCentral European Journal ofMathematics vol 9 no 6 pp 1403ndash1410 2011
[11] S Falcon and A Plaza ldquoThe k-Fibonacci sequence and thePascal 2-trianglerdquo Chaos Solitons and Fractals vol 33 no 1 pp38ndash49 2007
[12] S Falcon and A Plaza ldquoOn k-Fibonacci sequences and polyno-mials and their derivativesrdquoChaos Solitons and Fractals vol 39no 3 pp 1005ndash1019 2009
[13] J Ramırez ldquoSome properties of convolved k-Fibonacci num-bersrdquo ISRN Combinatorics vol 2013 Article ID 759641 5 pages2013
[14] A Salas ldquoAbout k-Fibonacci numbers and their associatednumbersrdquo International Mathematical Forum vol 50 no 6 pp2473ndash2479 2011
[15] S Falcon ldquoOn the k-Lucas numbersrdquo International Journal ofContemporary Mathematical Sciences vol 21 no 6 pp 1039ndash1050 2011
[16] H M Srivastava and H L Manocha A Treatise on GeneratingFunctions Halsted Press Ellis Horwood Limited ChichesterUK John Wiley amp Sons New York NY USA 1984
[17] A Nalli and P Haukkanen ldquoOn generalized Fibonacci andLucas polynomialsrdquo Chaos Solitons and Fractals vol 42 no 5pp 3179ndash3186 2009
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
2 Chinese Journal of Mathematics
From (6) and (7) we introduce the incomplete 119896-Fibonacci and 119896-Lucas numbers and we obtain new recurrentrelations new identities and their generating functions
2 The Incomplete 119896-Fibonacci Numbers
Definition 1 The incomplete 119896-Fibonacci numbers aredefined by
119865119897
119896119899=
119897
sum
119894=0
(119899 minus 1 minus 119894
119894) 119896119899minus2119894minus1
0 le 119897 le lfloor119899 minus 1
2rfloor (8)
In Table 1 some values of incomplete 119896-Fibonacci num-bers are provided
We note that
119865lfloor(119899minus1)2rfloor
1119899= 119865119899 (9)
For 119896 = 1 we get incomplete Fibonacci numbers [2]Some special cases of (8) are
1198650
119896119899= 119896119899minus1
(119899 ge 1)
1198651
119896119899= 119896119899minus1
+ (119899 minus 2) 119896119899minus3
(119899 ge 3)
1198652
119896119899= 119896119899minus1
+ (119899 minus 2) 119896119899minus3
+(119899 minus 4) (119899 minus 3)
2119896119899minus5
(119899 ge 5)
119865lfloor(119899minus1)2rfloor
119896119899= 119865119896119899 (119899 ge 1)
119865lfloor(119899minus3)2rfloor
119896119899=
119865119896119899minus119899119896
2(119899 even)
119865119896119899minus 1 (119899 odd)
(119899 ge 3)
(10)
21 Some Recurrence Properties of the Numbers 119865119897119896119899
Proposition 2 The recurrence relation of the incomplete 119896-Fibonacci numbers 119865119897
119896119899is
119865119897+1
119896119899+2= 119896119865119897+1
119896119899+1+ 119865119897
119896119899 0 le 119897 le
119899 minus 2
2 (11)
The relation (11) can be transformed into the nonhomoge-neous recurrence relation
119865119897
119896119899+2= 119896119865119897
119896119899+1+ 119865119897
119896119899minus (119899 minus 1 minus 119897
119897) 119896119899minus1minus2119897
(12)
Proof Use Definition 1 to rewrite the right-hand side of (11)as
119896
119897+1
sum
119894=0
(119899 minus 119894
119894) 119896119899minus2119894
+
119897
sum
119894=0
(119899 minus 119894 minus 1
119894) 119896119899minus2119894minus1
=
119897+1
sum
119894=0
(119899 minus 119894
119894) 119896119899minus2119894+1
+
119897+1
sum
119894=1
(119899 minus 119894
119894 minus 1) 119896119899minus2119894+1
= 119896119899minus2119894+1
(
119897+1
sum
119894=0
[(119899 minus 119894
119894) + (
119899 minus 119894
119894 minus 1)]) minus 119896
119899+1
(119899
minus1)
=
119897+1
sum
119894=0
(119899 minus 119894 + 1
119894) 119896119899minus2119894+1
minus 0
= 119865119897
119896119899+2
(13)
Proposition 3 One has119904
sum
119894=0
(119904
119894) 119865119897+119894
119896119899+119894119896119894
= 119865119897+119904
119896119899+2119904 0 le 119897 le
119899 minus 119904 minus 1
2 (14)
Proof (by induction on 119904) Sum (14) clearly holds for 119904 = 0 and119904 = 1 (see (11)) Now suppose that the result is true for all119895 lt 119904 + 1 we prove it for 119904 + 1
119904+1
sum
119894=0
(119904 + 1
119894)119865119897+119894
119896119899+119894119896119894
=
119904+1
sum
119894=0
[(119904
119894) + (
119904
119894 minus 1)]119865119897+119894
119896119899+119894119896119894
=
119904+1
sum
119894=0
(119904
119894) 119865119897+119894
119896119899+119894119896119894
+
119904+1
sum
119894=0
(119904
119894 minus 1)119865119897+119894
119896119899+119894119896119894
= 119865119897+119904
119896119899+2119904+ (
119904
119904 + 1)119865119897+119904+1
119896119899+119904+1119896119904+1
+
119904
sum
119894=minus1
(119904
119894) 119865119897+119894+1
119896119899+119894+1119896119894+1
= 119865119897+119904
119896119899+2119904+ 0 +
119904
sum
119894=0
(119904
119894) 119865119897+119894+1
119896119899+119894+1119896119894+1
+ (119904
minus1)119865119897
119896119899
= 119865119897+119904
119896119899+2119904+ 119896
119904
sum
119894=0
(119904
119894) 119865119897+119894+1
119896119899+119894+1119896119894
+ 0
= 119865119897+119904
119896119899+2119904+ 119896119865119897+119904+1
119896119899+2119904+1
= 119865119897+119904+1
119896119899+2119904+2
(15)
Proposition 4 For 119899 ge 2119897 + 2119904minus1
sum
119894=0
119865119897
119896119899+119894119896119904minus1minus119894
= 119865119897+1
119896119899+119904+1minus 119896119904
119865119897+1
119896119899+1 (16)
Chinese Journal of Mathematics 3
Table 1 The numbers 119865119897119896119899 for 1 ⩽ 119899 ⩽ 10
nl 0 1 2 3 41 12 k3 k2 119896
2
+ 1
4 k3 1198963
+ 2119896
5 k4 1198964
+ 31198962
1198964
+ 31198962
+ 1
6 k5 1198965
+ 41198963
1198965
+ 41198963
+ 3119896
7 k6 1198966
+ 51198964
1198966
+ 51198964
+ 61198962
1198966
+ 51198964
+ 61198962
+ 1
8 k7 1198967
+ 61198965
1198967
+ 61198965
+ 101198963
1198967
+ 61198965
+ 101198963
+ 4119896
9 k8 1198968
+ 71198966
1198968
+ 71198966
+ 151198964
1198968
+ 71198966
+ 151198964
+ 101198962
1198968
+ 71198966
+ 151198964
+ 101198962
+ 1
10 k9 1198969
+ 81198967
1198969
+ 81198967
+ 211198965
1198969
+ 81198967
+ 211198965
+ 201198963
1198969
+ 81198967
+ 211198965
+ 201198963
+ 5119896
Proof (by induction on 119904) Sum (16) clearly holds for 119904 = 1(see (11)) Now suppose that the result is true for all 119895 lt 119904We prove it for 119904
119904
sum
119894=0
119865119897
119896119899+119894119896119904minus119894
= 119896
119904minus1
sum
119894=0
119865119897
119896119899+119894119896119904minus119894minus1
+ 119865119897
119896119899+119904
= 119896 (119865119897+1
119896119899+119904+1minus 119896119904
119865119897+1
119896119899+1) + 119865119897
119896119899+119904
= (119896119865119897+1
119896119899+119904+1+ 119865119897
119896119899+119904) minus 119896119904+1
119865119897+1
119896119899+1
= 119865119897+1
119896119899+119904+2minus 119896119904+1
119865119897+1
119896119899+1
(17)
Note that if 119896 in (4) is a real variable then 119865119896119899= 119865119909119899
andthey correspond to the Fibonacci polynomials defined by
119865119899+1(119909) =
1 if 119899 = 0119909 if 119899 = 1119909119865119899(119909) + 119865
119899minus1(119909) if 119899 gt 1
(18)
Lemma 5 One has
1198651015840
119899(119909) =
119899119865119899+1(119909) minus 119909119865
119899(119909) + 119899119865
119899minus1(119909)
1199092 + 4
=119899119871119899(119909) minus 119909119865
119899(119909)
1199092 + 4
(19)
See Proposition 13 of [12]
Lemma 6 One haslfloor(119899minus1)2rfloor
sum
119894=0
119894 (119899 minus 1 minus 119894
119894) 119896119899minus1minus2119894
=
((1198962
+ 4) 119899 minus 4) 119865119896119899minus 119899119896119871
119896119899
2 (1198962 + 4)
(20)
Proof From (6) we have that
119896119865119896119899=
lfloor(119899minus1)2rfloor
sum
119894=0
(119899 minus 1 minus 119894
119894) 119896119899minus2119894
(21)
By deriving into the previous equation (respect to 119896) it isobtained
119865119896119899+ 1198961198651015840
119896119899=
lfloor(119899minus1)2rfloor
sum
119894=0
(119899 minus 2119894) (119899 minus 1 minus 119894
119894) 119896119899minus2119894minus1
= 119899119865119896119899minus 2
lfloor(119899minus1)2rfloor
sum
119894=0
119894 (119899 minus 1 minus 119894
119894) 119896119899minus2119894minus1
(22)
From Lemma 5
119865119896119899+ 119896(
119899119871119896119899minus 119896119865119896119899
1198962 + 4)
= 119899119865119896119899minus 2
lfloor(119899minus1)2rfloor
sum
119894=0
119894 (119899 minus 1 minus 119894
119894) 119896119899minus2119894minus1
(23)
From where after some algebra (20) is obtained
Proposition 7 One has
lfloor(119899minus1)2rfloor
sum
119897=0
119865119897
119896119899=
4119865119896119899+ 119899119896119871
119896119899
2 (1198962 + 4)(119899 even)
(1198962
+ 8) 119865119896119899+ 119899119896119871
119896119899
2 (1198962 + 4)(119899 odd)
(24)
4 Chinese Journal of Mathematics
Table 2 The numbers 119871119897119896119899 for 1 ⩽ 119899 ⩽ 9
nl 0 1 2 3 41 k2 119896
2
1198962
+ 2
3 1198963
1198963
+ 3119896
4 1198964
1198964
+ 41198962
1198964
+ 41198962
+ 2
5 1198965
1198965
+ 51198963
1198965
+ 51198963
+ 5119896
6 1198966
1198966
+ 61198964
1198966
+ 61198964
+ 91198962
1198966
+ 61198964
+ 91198962
+ 2
7 1198967
1198967
+ 71198965
1198967
+ 71198965
+ 141198963
1198967
+ 71198965
+ 141198963
+ 7119896
8 1198968
1198968
+ 81198966
1198968
+ 81198966
+ 201198964
1198968
+ 81198966
+ 201198964
+ 161198962
1198968
+ 81198966
+ 201198964
+ 161198962
+ 2
9 1198969
1198969
+ 91198967
1198969
+ 91198967
+ 271198965
1198969
+ 91198967
+ 271198965
+ 301198963
1198969
+ 91198967
+ 271198965
+ 301198963
+ 9119896
Prooflfloor(119899minus1)2rfloor
sum
119897=0
119865119897
119896119899
= 1198650
119896119899+ 1198651
119896119899+ sdot sdot sdot + 119865
lfloor(119899minus1)2rfloor
119896119899
= (119899 minus 1 minus 0
0) 119896119899minus1
+ [(119899 minus 1 minus 0
0) 119896119899minus1
+ (119899 minus 1 minus 1
1) 119896119899minus3
]
+ sdot sdot sdot +
[[[[
[
(119899 minus 1 minus 0
0) 119896119899minus1
+ (119899 minus 1 minus 1
1) 119896119899minus3
+ sdot sdot sdot +(
119899 minus 1 minus lfloor119899 minus 1
2rfloor
lfloor119899 minus 1
2rfloor
)119896119899minus1minus2lfloor(119899minus1)2rfloor
]]]]
]
= (lfloor119899 minus 1
2rfloor + 1)(
119899 minus 1 minus 0
0) 119896119899minus1
+ lfloor119899 minus 1
2rfloor(119899 minus 1 minus 1
1) 119896119899minus3
+ sdot sdot sdot +(
119899 minus 1 minus lfloor119899 minus 1
2rfloor
lfloor119899 minus 1
2rfloor
)119896119899minus1minus2lfloor(119899minus1)2rfloor
=
lfloor(119899minus1)2rfloor
sum
119894=0
(lfloor119899 minus 1
2rfloor + 1 minus 119894) (
119899 minus 1 minus 119894
119894) 119896119899minus1minus2119894
=
lfloor(119899minus1)2rfloor
sum
119894=0
(lfloor119899 minus 1
2rfloor + 1)(
119899 minus 1 minus 119894
119894) 119896119899minus1minus2119894
minus
lfloor(119899minus1)2rfloor
sum
119894=0
119894 (119899 minus 1 minus 119894
119894) 119896119899minus1minus2119894
= (lfloor119899 minus 1
2rfloor + 1)119865
119896119899minus
lfloor(119899minus1)2rfloor
sum
119894=0
119894 (119899 minus 1 minus 119894
119894) 119896119899minus1minus2119894
(25)
From Lemma 6 (24) is obtained
3 The Incomplete 119896-Lucas Numbers
Definition 8 The incomplete 119896-Lucas numbers are defined by
119871119897
119896119899=
119897
sum
119894=0
119899
119899 minus 119894(119899 minus 119894
119894) 119896119899minus2119894
0 le 119897 le lfloor119899
2rfloor (26)
InTable 2 somenumbers of incomplete 119896-Lucas numbersare provided
We note that
119871lfloor1198992rfloor
1119899= 119871119899 (27)
Some special cases of (26) are
1198710
119896119899= 119896119899
(119899 ge 1)
1198711
119896119899= 119896119899
+ 119899119896119899minus2
(119899 ge 2)
1198712
119896119899= 119896119899
+ 119899119896119899minus2
+119899 (119899 minus 3)
2119896119899minus4
(119899 ge 4)
119871lfloor1198992rfloor
119896119899= 119871119896119899 (119899 ge 1)
119871lfloor(119899minus2)2rfloor
119896119899=
119871119896119899minus 2 (119899 even)
119871119896119899minus 119899119896 (119899 odd)
(119899 ge 2)
(28)
Chinese Journal of Mathematics 5
31 Some Recurrence Properties of the Numbers 119871119897119896119899
Proposition 9 One has
119871119897
119896119899= 119865119897minus1
119896119899minus1+ 119865119897
119896119899+1 0 le 119897 le lfloor
119899
2rfloor (29)
Proof By (8) rewrite the right-hand side of (29) as
119897minus1
sum
119894=0
(119899 minus 2 minus 119894
119894) 119896119899minus2minus2119894
+
119897
sum
119894=0
(119899 minus 119894
119894) 119896119899minus2119894
=
119897
sum
119894=1
(119899 minus 1 minus 119894
119894 minus 1) 119896119899minus2119894
+
119897
sum
119894=0
(119899 minus 119894
119894) 119896119899minus2119894
=
119897
sum
119894=0
[(119899 minus 1 minus 119894
119894 minus 1) + (
119899 minus 119894
119894)] 119896119899minus2119894
minus (119899 minus 1
minus1)
=
119897
sum
119894=0
119899
119899 minus 119894(119899 minus 119894
119894) 119896119899minus2119894
+ 0
= 119871119897
119896119899
(30)
Proposition 10 The recurrence relation of the incomplete 119896-Lucas numbers 119871119897
119896119899is
119871119897+1
119896119899+2= 119896119871119897+1
119896119899+1+ 119871119897
119896119899 0 le 119897 le lfloor
119899
2rfloor (31)
The relation (31) can be transformed into the nonhomoge-neous recurrence relation
119871119897
119896119899+2= 119896119871119897
119896119899+1+ 119871119897
119896119899minus119899
119899 minus 119897(119899 minus 119897
119897) 119896119899minus2119897
(32)
Proof Using (29) and (11) we write
119871119897+1
119896119899+2= 119865119897
119896119899+1+ 119865119897+1
119896119899+3
= 119896119865119897
119896119899+ 119865119897minus1
119896119899minus1+ 119896119865119897+1
119896119899+2+ 119865119897
119896119899+1
= 119896 (119865119897
119896119899+ 119865119897+1
119896119899+2) + 119865119897minus1
119896119899minus1+ 119865119897
119896119899+1
= 119896119871119897+1
119896119899+1+ 119871119897
119896119899
(33)
Proposition 11 One has
119896119871119897
119896119899= 119865119897
119896119899+2minus 119865119897minus2
119896119899minus2 0 le 119897 le lfloor
119899 minus 1
2rfloor (34)
Proof By (29)
119865119897
119896119899+2= 119871119897
119896119899+1minus 119865119897minus1
119896119899 119865
119897minus2
119896119899minus2= 119871119897minus1
119896119899minus1minus 119865119897minus1
119896119899 (35)
whence from (31)
119865119897
119896119899+2minus 119865119897minus2
119896119899minus2= 119871119897
119896119899+1minus 119871119897minus1
119896119899minus1= 119896119871119897
119896119899 (36)
Proposition 12 One has
119904
sum
119894=0
(119904
119894) 119871119897+119894
119896119899+119894119896119894
= 119871119897+119904
119896119899+2119904 0 le 119897 le
119899 minus 119904
2 (37)
Proof Using (29) and (14) we write
119904
sum
119894=0
(119904
119894) 119871119897+119894
119896119899+119894119896119894
=
119904
sum
119894=0
(119904
119894) [119865119897+119894minus1
119896119899+119894minus1+ 119865119897+119894
119896119899+119894+1] 119896119894
=
119904
sum
119894=0
(119904
119894) 119865119897+119894minus1
119896119899+119894minus1119896119894
+
119904
sum
119894=0
(119904
119894) 119865119897+119894
119896119899+119894+1119896119894
= 119865119897minus1+119904
119896119899minus1+2119904+ 119865119897+119904
119896119899+1+2119904= 119871119897+119904
119896119899+2119904
(38)
Proposition 13 For 119899 ge 2119897 + 1
119904minus1
sum
119894=0
119871119897
119896119899+119894119896119904minus1minus119894
= 119871119897+1
119896119899+119904+1minus 119896119904
119871119897+1
119896119899+1 (39)
The proof can be done by using (31) and induction on 119904
Lemma 14 One has
lfloor1198992rfloor
sum
119894=0
119894119899
119899 minus 119894(119899 minus 119894
119894) 119896119899minus2119894
=119899
2[119871119896119899minus 119896119865119896119899] (40)
The proof is similar to Lemma 6
Proposition 15 One has
lfloor1198992rfloor
sum
119897=0
119871119897
119896119899=
119871119896119899+119899119896
2119865119896119899
(119899 even)1
2(119871119896119899+ 119899119896119865
119896119899) (119899 odd)
(41)
Proof An argument analogous to that of the proof ofProposition 7 yields
lfloor1198992rfloor
sum
119897=0
119871119897
119896119899= (lfloor
119899
2rfloor + 1) 119871
119896119899minus
lfloor1198992rfloor
sum
119894=0
119894119899
119899 minus 119894(119899 minus 119894
119894) 119896119899minus2119894
(42)
From Lemma 14 (41) is obtained
4 Generating Functions of the Incomplete119896-Fibonacci and 119896-Lucas Number
In this section we give the generating functions of incomplete119896-Fibonacci and 119896-Lucas numbers
Lemma 16 (see [3 page 592]) Let 119904119899infin
119899=0be a complex
sequence satisfying the following nonhomogeneous recurrencerelation
119904119899= 119886119904119899minus1+ 119887119904119899minus2+ 119903119899(119899 gt 1) (43)
6 Chinese Journal of Mathematics
where 119886 and 119887 are complex numbers and 119903119899 is a given complex
sequence Then the generating function 119880(119905) of the sequence119904119899 is
119880 (119905) =119866 (119905) + 119904
0minus 1199030+ (1199041minus 1199040119886 minus 1199031) 119905
1 minus 119886119905 minus 1198871199052 (44)
where 119866(119905) denotes the generating function of 119903119899
Theorem 17 The generating function of the incomplete 119896-Fibonacci numbers 119865119897
119896119899is given by
119877119896119897(119909) =
infin
sum
119894=0
119865119897
119896119894119905119894
= 1199052119897+1
[1198651198962119897+1
+ (1198651198962119897+2
minus 1198961198651198962119897+1
) 119905 minus1199052
(1 minus 119896119905)119897+1
]
times [1 minus 119896119905 minus 1199052
]minus1
(45)
Proof Let 119897 be a fixed positive integer From (8) and (12)119865119897
119896119899= 0 for 0 le 119899 lt 2119897 + 1 119865119897
1198962119897+1= 1198651198962119897+1
and 1198651198971198962119897+2
=
1198651198962119897+2
and
119865119897
119896119899= 119896119865119897
119896119899minus1+ 119865119897
119896119899minus2minus (119899 minus 3 minus 119897
119897) 119896119899minus3minus2119897
(46)
Now let
1199040= 119865119897
1198962119897+1 119904
1= 119865119897
1198962119897+2 119904
119899= 119865119897
119896119899+2119897+1 (47)
Also let
1199030= 1199031= 0 119903
119899= (119899 + 119897 minus 2
119899 minus 2) 119896119899minus2
(48)
The generating function of the sequence 119903119899 is119866(119905) = 1199052(1minus
119896119905)119897+1 (see [16 page 355]) Thus from Lemma 16 we get the
generating function 119877119896119897(119909) of sequence 119904
119899
Theorem 18 The generating function of the incomplete 119896-Lucas numbers 119871119897
119896119899is given by
119878119896119897(119909) =
infin
sum
119894=0
119871119897
119896119894119905119894
= 1199052119897
[1198711198962119897+ (1198711198962119897+1
minus 1198961198711198962119897) 119905 minus
1199052
(2 minus 119905)
(1 minus 119896119905)119897+1
]
times [1 minus 119896119905 minus 1199052
]minus1
(49)
Proof The proof of this theorem is similar to the proof ofTheorem 17 Let 119897 be a fixed positive integer From (26) and(32)119871119897
119896119899= 0 for 0 le 119899 lt 2119897119871119897
1198962119897= 1198711198962119897
and1198711198971198962119897+1
= 1198711198962119897+1
and
119871119897
119896119899= 119896119871119897
119896119899minus1+ 119871119897
119896119899minus2minus119899 minus 2
119899 minus 2 minus 119897(119899 minus 2 minus 119897
119899 minus 2 minus 2119897) 119896119899minus2minus2119897
(50)
Now let
1199040= 119871119897
1198962119897 119904
1= 119871119897
1198962119897+1 119904
119899= 119871119897
119896119899+2119897 (51)
Also let
1199030= 1199031= 0 119903
119899= (119899 + 2119897 minus 2
119899 + 119897 minus 2) 119896119899+2119897minus2
(52)
The generating function of the sequence 119903119899 is 119866(119905) = 1199052(2 minus
119905)(1 minus 119896119905)119897+1 (see [16 page 355]) Thus from Lemma 16 we
get the generating function 119878119896119897(119909) of sequence 119904
119899
5 Conclusion
In this paper we introduce incomplete 119896-Fibonacci and119896-Lucas numbers and we obtain new identities In [17]the authors introduced the ℎ(119909)-Fibonacci polynomialsThat generalizes Catalanrsquos Fibonacci polynomials and the119896-Fibonacci numbers Let ℎ(119909) be a polynomial with realcoefficientsThe ℎ(119909)-Fibonacci polynomials 119865
ℎ119899(119909)119899isinN are
defined by the recurrence relation
119865ℎ0(119909) = 0 119865
ℎ1(119909) = 1
119865ℎ119899+1
(119909) = ℎ (119909) 119865ℎ119899(119909) + 119865
ℎ119899minus1(119909) 119899 ⩾ 1
(53)
It would be interesting to study a definition of incompleteℎ(119909)-Fibonacci polynomials and research their properties
Acknowledgments
The author would like to thank the anonymous referees fortheir helpful comments The author was partially supportedby Universidad Sergio Arboleda under Grant no USA-II-2011-0059
References
[1] T Koshy Fibonacci and Lucas Numbers with ApplicationsWiley-Interscience 2001
[2] P Filipponi ldquoIncomplete Fibonacci and Lucas numbersrdquoRendi-conti del CircoloMatematico di Palermo vol 45 no 1 pp 37ndash561996
[3] A Pinter and H M Srivastava ldquoGenerating functions ofthe incomplete Fibonacci and Lucas numbersrdquo Rendiconti delCircolo Matematico di Palermo vol 48 no 3 pp 591ndash596 1999
[4] G B Djordjevic ldquoGenerating functions of the incomplete gen-eralized Fibonacci and generalized Lucas numbersrdquo FibonacciQuarterly vol 42 no 2 pp 106ndash113 2004
[5] G B Djordjevic and H M Srivastava ldquoIncomplete generalizedJacobsthal and Jacobsthal-Lucas numbersrdquo Mathematical andComputer Modelling vol 42 no 9-10 pp 1049ndash1056 2005
[6] D Tasci and M Cetin Firengiz ldquoIncomplete Fibonacci andLucas p-numbersrdquoMathematical and Computer Modelling vol52 no 9-10 pp 1763ndash1770 2010
[7] D Tasci M Cetin Firengiz andN Tuglu ldquoIncomplete bivariateFibonaccu and Lucas p-polynomialsrdquo Discrete Dynamics inNature and Society vol 2012 Article ID 840345 11 pages 2012
[8] S Falcon and A Plaza ldquoOn the Fibonacci k-numbersrdquo ChaosSolitons and Fractals vol 32 no 5 pp 1615ndash1624 2007
Chinese Journal of Mathematics 7
[9] C Bolat and H Kose ldquoOn the properties of k-Fibonaccinumbersrdquo International Journal of Contemporary MathematicalSciences vol 22 no 5 pp 1097ndash1105 2010
[10] S Falcon ldquoThe k-Fibonacci matrix and the Pascal matrixrdquoCentral European Journal ofMathematics vol 9 no 6 pp 1403ndash1410 2011
[11] S Falcon and A Plaza ldquoThe k-Fibonacci sequence and thePascal 2-trianglerdquo Chaos Solitons and Fractals vol 33 no 1 pp38ndash49 2007
[12] S Falcon and A Plaza ldquoOn k-Fibonacci sequences and polyno-mials and their derivativesrdquoChaos Solitons and Fractals vol 39no 3 pp 1005ndash1019 2009
[13] J Ramırez ldquoSome properties of convolved k-Fibonacci num-bersrdquo ISRN Combinatorics vol 2013 Article ID 759641 5 pages2013
[14] A Salas ldquoAbout k-Fibonacci numbers and their associatednumbersrdquo International Mathematical Forum vol 50 no 6 pp2473ndash2479 2011
[15] S Falcon ldquoOn the k-Lucas numbersrdquo International Journal ofContemporary Mathematical Sciences vol 21 no 6 pp 1039ndash1050 2011
[16] H M Srivastava and H L Manocha A Treatise on GeneratingFunctions Halsted Press Ellis Horwood Limited ChichesterUK John Wiley amp Sons New York NY USA 1984
[17] A Nalli and P Haukkanen ldquoOn generalized Fibonacci andLucas polynomialsrdquo Chaos Solitons and Fractals vol 42 no 5pp 3179ndash3186 2009
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Chinese Journal of Mathematics 3
Table 1 The numbers 119865119897119896119899 for 1 ⩽ 119899 ⩽ 10
nl 0 1 2 3 41 12 k3 k2 119896
2
+ 1
4 k3 1198963
+ 2119896
5 k4 1198964
+ 31198962
1198964
+ 31198962
+ 1
6 k5 1198965
+ 41198963
1198965
+ 41198963
+ 3119896
7 k6 1198966
+ 51198964
1198966
+ 51198964
+ 61198962
1198966
+ 51198964
+ 61198962
+ 1
8 k7 1198967
+ 61198965
1198967
+ 61198965
+ 101198963
1198967
+ 61198965
+ 101198963
+ 4119896
9 k8 1198968
+ 71198966
1198968
+ 71198966
+ 151198964
1198968
+ 71198966
+ 151198964
+ 101198962
1198968
+ 71198966
+ 151198964
+ 101198962
+ 1
10 k9 1198969
+ 81198967
1198969
+ 81198967
+ 211198965
1198969
+ 81198967
+ 211198965
+ 201198963
1198969
+ 81198967
+ 211198965
+ 201198963
+ 5119896
Proof (by induction on 119904) Sum (16) clearly holds for 119904 = 1(see (11)) Now suppose that the result is true for all 119895 lt 119904We prove it for 119904
119904
sum
119894=0
119865119897
119896119899+119894119896119904minus119894
= 119896
119904minus1
sum
119894=0
119865119897
119896119899+119894119896119904minus119894minus1
+ 119865119897
119896119899+119904
= 119896 (119865119897+1
119896119899+119904+1minus 119896119904
119865119897+1
119896119899+1) + 119865119897
119896119899+119904
= (119896119865119897+1
119896119899+119904+1+ 119865119897
119896119899+119904) minus 119896119904+1
119865119897+1
119896119899+1
= 119865119897+1
119896119899+119904+2minus 119896119904+1
119865119897+1
119896119899+1
(17)
Note that if 119896 in (4) is a real variable then 119865119896119899= 119865119909119899
andthey correspond to the Fibonacci polynomials defined by
119865119899+1(119909) =
1 if 119899 = 0119909 if 119899 = 1119909119865119899(119909) + 119865
119899minus1(119909) if 119899 gt 1
(18)
Lemma 5 One has
1198651015840
119899(119909) =
119899119865119899+1(119909) minus 119909119865
119899(119909) + 119899119865
119899minus1(119909)
1199092 + 4
=119899119871119899(119909) minus 119909119865
119899(119909)
1199092 + 4
(19)
See Proposition 13 of [12]
Lemma 6 One haslfloor(119899minus1)2rfloor
sum
119894=0
119894 (119899 minus 1 minus 119894
119894) 119896119899minus1minus2119894
=
((1198962
+ 4) 119899 minus 4) 119865119896119899minus 119899119896119871
119896119899
2 (1198962 + 4)
(20)
Proof From (6) we have that
119896119865119896119899=
lfloor(119899minus1)2rfloor
sum
119894=0
(119899 minus 1 minus 119894
119894) 119896119899minus2119894
(21)
By deriving into the previous equation (respect to 119896) it isobtained
119865119896119899+ 1198961198651015840
119896119899=
lfloor(119899minus1)2rfloor
sum
119894=0
(119899 minus 2119894) (119899 minus 1 minus 119894
119894) 119896119899minus2119894minus1
= 119899119865119896119899minus 2
lfloor(119899minus1)2rfloor
sum
119894=0
119894 (119899 minus 1 minus 119894
119894) 119896119899minus2119894minus1
(22)
From Lemma 5
119865119896119899+ 119896(
119899119871119896119899minus 119896119865119896119899
1198962 + 4)
= 119899119865119896119899minus 2
lfloor(119899minus1)2rfloor
sum
119894=0
119894 (119899 minus 1 minus 119894
119894) 119896119899minus2119894minus1
(23)
From where after some algebra (20) is obtained
Proposition 7 One has
lfloor(119899minus1)2rfloor
sum
119897=0
119865119897
119896119899=
4119865119896119899+ 119899119896119871
119896119899
2 (1198962 + 4)(119899 even)
(1198962
+ 8) 119865119896119899+ 119899119896119871
119896119899
2 (1198962 + 4)(119899 odd)
(24)
4 Chinese Journal of Mathematics
Table 2 The numbers 119871119897119896119899 for 1 ⩽ 119899 ⩽ 9
nl 0 1 2 3 41 k2 119896
2
1198962
+ 2
3 1198963
1198963
+ 3119896
4 1198964
1198964
+ 41198962
1198964
+ 41198962
+ 2
5 1198965
1198965
+ 51198963
1198965
+ 51198963
+ 5119896
6 1198966
1198966
+ 61198964
1198966
+ 61198964
+ 91198962
1198966
+ 61198964
+ 91198962
+ 2
7 1198967
1198967
+ 71198965
1198967
+ 71198965
+ 141198963
1198967
+ 71198965
+ 141198963
+ 7119896
8 1198968
1198968
+ 81198966
1198968
+ 81198966
+ 201198964
1198968
+ 81198966
+ 201198964
+ 161198962
1198968
+ 81198966
+ 201198964
+ 161198962
+ 2
9 1198969
1198969
+ 91198967
1198969
+ 91198967
+ 271198965
1198969
+ 91198967
+ 271198965
+ 301198963
1198969
+ 91198967
+ 271198965
+ 301198963
+ 9119896
Prooflfloor(119899minus1)2rfloor
sum
119897=0
119865119897
119896119899
= 1198650
119896119899+ 1198651
119896119899+ sdot sdot sdot + 119865
lfloor(119899minus1)2rfloor
119896119899
= (119899 minus 1 minus 0
0) 119896119899minus1
+ [(119899 minus 1 minus 0
0) 119896119899minus1
+ (119899 minus 1 minus 1
1) 119896119899minus3
]
+ sdot sdot sdot +
[[[[
[
(119899 minus 1 minus 0
0) 119896119899minus1
+ (119899 minus 1 minus 1
1) 119896119899minus3
+ sdot sdot sdot +(
119899 minus 1 minus lfloor119899 minus 1
2rfloor
lfloor119899 minus 1
2rfloor
)119896119899minus1minus2lfloor(119899minus1)2rfloor
]]]]
]
= (lfloor119899 minus 1
2rfloor + 1)(
119899 minus 1 minus 0
0) 119896119899minus1
+ lfloor119899 minus 1
2rfloor(119899 minus 1 minus 1
1) 119896119899minus3
+ sdot sdot sdot +(
119899 minus 1 minus lfloor119899 minus 1
2rfloor
lfloor119899 minus 1
2rfloor
)119896119899minus1minus2lfloor(119899minus1)2rfloor
=
lfloor(119899minus1)2rfloor
sum
119894=0
(lfloor119899 minus 1
2rfloor + 1 minus 119894) (
119899 minus 1 minus 119894
119894) 119896119899minus1minus2119894
=
lfloor(119899minus1)2rfloor
sum
119894=0
(lfloor119899 minus 1
2rfloor + 1)(
119899 minus 1 minus 119894
119894) 119896119899minus1minus2119894
minus
lfloor(119899minus1)2rfloor
sum
119894=0
119894 (119899 minus 1 minus 119894
119894) 119896119899minus1minus2119894
= (lfloor119899 minus 1
2rfloor + 1)119865
119896119899minus
lfloor(119899minus1)2rfloor
sum
119894=0
119894 (119899 minus 1 minus 119894
119894) 119896119899minus1minus2119894
(25)
From Lemma 6 (24) is obtained
3 The Incomplete 119896-Lucas Numbers
Definition 8 The incomplete 119896-Lucas numbers are defined by
119871119897
119896119899=
119897
sum
119894=0
119899
119899 minus 119894(119899 minus 119894
119894) 119896119899minus2119894
0 le 119897 le lfloor119899
2rfloor (26)
InTable 2 somenumbers of incomplete 119896-Lucas numbersare provided
We note that
119871lfloor1198992rfloor
1119899= 119871119899 (27)
Some special cases of (26) are
1198710
119896119899= 119896119899
(119899 ge 1)
1198711
119896119899= 119896119899
+ 119899119896119899minus2
(119899 ge 2)
1198712
119896119899= 119896119899
+ 119899119896119899minus2
+119899 (119899 minus 3)
2119896119899minus4
(119899 ge 4)
119871lfloor1198992rfloor
119896119899= 119871119896119899 (119899 ge 1)
119871lfloor(119899minus2)2rfloor
119896119899=
119871119896119899minus 2 (119899 even)
119871119896119899minus 119899119896 (119899 odd)
(119899 ge 2)
(28)
Chinese Journal of Mathematics 5
31 Some Recurrence Properties of the Numbers 119871119897119896119899
Proposition 9 One has
119871119897
119896119899= 119865119897minus1
119896119899minus1+ 119865119897
119896119899+1 0 le 119897 le lfloor
119899
2rfloor (29)
Proof By (8) rewrite the right-hand side of (29) as
119897minus1
sum
119894=0
(119899 minus 2 minus 119894
119894) 119896119899minus2minus2119894
+
119897
sum
119894=0
(119899 minus 119894
119894) 119896119899minus2119894
=
119897
sum
119894=1
(119899 minus 1 minus 119894
119894 minus 1) 119896119899minus2119894
+
119897
sum
119894=0
(119899 minus 119894
119894) 119896119899minus2119894
=
119897
sum
119894=0
[(119899 minus 1 minus 119894
119894 minus 1) + (
119899 minus 119894
119894)] 119896119899minus2119894
minus (119899 minus 1
minus1)
=
119897
sum
119894=0
119899
119899 minus 119894(119899 minus 119894
119894) 119896119899minus2119894
+ 0
= 119871119897
119896119899
(30)
Proposition 10 The recurrence relation of the incomplete 119896-Lucas numbers 119871119897
119896119899is
119871119897+1
119896119899+2= 119896119871119897+1
119896119899+1+ 119871119897
119896119899 0 le 119897 le lfloor
119899
2rfloor (31)
The relation (31) can be transformed into the nonhomoge-neous recurrence relation
119871119897
119896119899+2= 119896119871119897
119896119899+1+ 119871119897
119896119899minus119899
119899 minus 119897(119899 minus 119897
119897) 119896119899minus2119897
(32)
Proof Using (29) and (11) we write
119871119897+1
119896119899+2= 119865119897
119896119899+1+ 119865119897+1
119896119899+3
= 119896119865119897
119896119899+ 119865119897minus1
119896119899minus1+ 119896119865119897+1
119896119899+2+ 119865119897
119896119899+1
= 119896 (119865119897
119896119899+ 119865119897+1
119896119899+2) + 119865119897minus1
119896119899minus1+ 119865119897
119896119899+1
= 119896119871119897+1
119896119899+1+ 119871119897
119896119899
(33)
Proposition 11 One has
119896119871119897
119896119899= 119865119897
119896119899+2minus 119865119897minus2
119896119899minus2 0 le 119897 le lfloor
119899 minus 1
2rfloor (34)
Proof By (29)
119865119897
119896119899+2= 119871119897
119896119899+1minus 119865119897minus1
119896119899 119865
119897minus2
119896119899minus2= 119871119897minus1
119896119899minus1minus 119865119897minus1
119896119899 (35)
whence from (31)
119865119897
119896119899+2minus 119865119897minus2
119896119899minus2= 119871119897
119896119899+1minus 119871119897minus1
119896119899minus1= 119896119871119897
119896119899 (36)
Proposition 12 One has
119904
sum
119894=0
(119904
119894) 119871119897+119894
119896119899+119894119896119894
= 119871119897+119904
119896119899+2119904 0 le 119897 le
119899 minus 119904
2 (37)
Proof Using (29) and (14) we write
119904
sum
119894=0
(119904
119894) 119871119897+119894
119896119899+119894119896119894
=
119904
sum
119894=0
(119904
119894) [119865119897+119894minus1
119896119899+119894minus1+ 119865119897+119894
119896119899+119894+1] 119896119894
=
119904
sum
119894=0
(119904
119894) 119865119897+119894minus1
119896119899+119894minus1119896119894
+
119904
sum
119894=0
(119904
119894) 119865119897+119894
119896119899+119894+1119896119894
= 119865119897minus1+119904
119896119899minus1+2119904+ 119865119897+119904
119896119899+1+2119904= 119871119897+119904
119896119899+2119904
(38)
Proposition 13 For 119899 ge 2119897 + 1
119904minus1
sum
119894=0
119871119897
119896119899+119894119896119904minus1minus119894
= 119871119897+1
119896119899+119904+1minus 119896119904
119871119897+1
119896119899+1 (39)
The proof can be done by using (31) and induction on 119904
Lemma 14 One has
lfloor1198992rfloor
sum
119894=0
119894119899
119899 minus 119894(119899 minus 119894
119894) 119896119899minus2119894
=119899
2[119871119896119899minus 119896119865119896119899] (40)
The proof is similar to Lemma 6
Proposition 15 One has
lfloor1198992rfloor
sum
119897=0
119871119897
119896119899=
119871119896119899+119899119896
2119865119896119899
(119899 even)1
2(119871119896119899+ 119899119896119865
119896119899) (119899 odd)
(41)
Proof An argument analogous to that of the proof ofProposition 7 yields
lfloor1198992rfloor
sum
119897=0
119871119897
119896119899= (lfloor
119899
2rfloor + 1) 119871
119896119899minus
lfloor1198992rfloor
sum
119894=0
119894119899
119899 minus 119894(119899 minus 119894
119894) 119896119899minus2119894
(42)
From Lemma 14 (41) is obtained
4 Generating Functions of the Incomplete119896-Fibonacci and 119896-Lucas Number
In this section we give the generating functions of incomplete119896-Fibonacci and 119896-Lucas numbers
Lemma 16 (see [3 page 592]) Let 119904119899infin
119899=0be a complex
sequence satisfying the following nonhomogeneous recurrencerelation
119904119899= 119886119904119899minus1+ 119887119904119899minus2+ 119903119899(119899 gt 1) (43)
6 Chinese Journal of Mathematics
where 119886 and 119887 are complex numbers and 119903119899 is a given complex
sequence Then the generating function 119880(119905) of the sequence119904119899 is
119880 (119905) =119866 (119905) + 119904
0minus 1199030+ (1199041minus 1199040119886 minus 1199031) 119905
1 minus 119886119905 minus 1198871199052 (44)
where 119866(119905) denotes the generating function of 119903119899
Theorem 17 The generating function of the incomplete 119896-Fibonacci numbers 119865119897
119896119899is given by
119877119896119897(119909) =
infin
sum
119894=0
119865119897
119896119894119905119894
= 1199052119897+1
[1198651198962119897+1
+ (1198651198962119897+2
minus 1198961198651198962119897+1
) 119905 minus1199052
(1 minus 119896119905)119897+1
]
times [1 minus 119896119905 minus 1199052
]minus1
(45)
Proof Let 119897 be a fixed positive integer From (8) and (12)119865119897
119896119899= 0 for 0 le 119899 lt 2119897 + 1 119865119897
1198962119897+1= 1198651198962119897+1
and 1198651198971198962119897+2
=
1198651198962119897+2
and
119865119897
119896119899= 119896119865119897
119896119899minus1+ 119865119897
119896119899minus2minus (119899 minus 3 minus 119897
119897) 119896119899minus3minus2119897
(46)
Now let
1199040= 119865119897
1198962119897+1 119904
1= 119865119897
1198962119897+2 119904
119899= 119865119897
119896119899+2119897+1 (47)
Also let
1199030= 1199031= 0 119903
119899= (119899 + 119897 minus 2
119899 minus 2) 119896119899minus2
(48)
The generating function of the sequence 119903119899 is119866(119905) = 1199052(1minus
119896119905)119897+1 (see [16 page 355]) Thus from Lemma 16 we get the
generating function 119877119896119897(119909) of sequence 119904
119899
Theorem 18 The generating function of the incomplete 119896-Lucas numbers 119871119897
119896119899is given by
119878119896119897(119909) =
infin
sum
119894=0
119871119897
119896119894119905119894
= 1199052119897
[1198711198962119897+ (1198711198962119897+1
minus 1198961198711198962119897) 119905 minus
1199052
(2 minus 119905)
(1 minus 119896119905)119897+1
]
times [1 minus 119896119905 minus 1199052
]minus1
(49)
Proof The proof of this theorem is similar to the proof ofTheorem 17 Let 119897 be a fixed positive integer From (26) and(32)119871119897
119896119899= 0 for 0 le 119899 lt 2119897119871119897
1198962119897= 1198711198962119897
and1198711198971198962119897+1
= 1198711198962119897+1
and
119871119897
119896119899= 119896119871119897
119896119899minus1+ 119871119897
119896119899minus2minus119899 minus 2
119899 minus 2 minus 119897(119899 minus 2 minus 119897
119899 minus 2 minus 2119897) 119896119899minus2minus2119897
(50)
Now let
1199040= 119871119897
1198962119897 119904
1= 119871119897
1198962119897+1 119904
119899= 119871119897
119896119899+2119897 (51)
Also let
1199030= 1199031= 0 119903
119899= (119899 + 2119897 minus 2
119899 + 119897 minus 2) 119896119899+2119897minus2
(52)
The generating function of the sequence 119903119899 is 119866(119905) = 1199052(2 minus
119905)(1 minus 119896119905)119897+1 (see [16 page 355]) Thus from Lemma 16 we
get the generating function 119878119896119897(119909) of sequence 119904
119899
5 Conclusion
In this paper we introduce incomplete 119896-Fibonacci and119896-Lucas numbers and we obtain new identities In [17]the authors introduced the ℎ(119909)-Fibonacci polynomialsThat generalizes Catalanrsquos Fibonacci polynomials and the119896-Fibonacci numbers Let ℎ(119909) be a polynomial with realcoefficientsThe ℎ(119909)-Fibonacci polynomials 119865
ℎ119899(119909)119899isinN are
defined by the recurrence relation
119865ℎ0(119909) = 0 119865
ℎ1(119909) = 1
119865ℎ119899+1
(119909) = ℎ (119909) 119865ℎ119899(119909) + 119865
ℎ119899minus1(119909) 119899 ⩾ 1
(53)
It would be interesting to study a definition of incompleteℎ(119909)-Fibonacci polynomials and research their properties
Acknowledgments
The author would like to thank the anonymous referees fortheir helpful comments The author was partially supportedby Universidad Sergio Arboleda under Grant no USA-II-2011-0059
References
[1] T Koshy Fibonacci and Lucas Numbers with ApplicationsWiley-Interscience 2001
[2] P Filipponi ldquoIncomplete Fibonacci and Lucas numbersrdquoRendi-conti del CircoloMatematico di Palermo vol 45 no 1 pp 37ndash561996
[3] A Pinter and H M Srivastava ldquoGenerating functions ofthe incomplete Fibonacci and Lucas numbersrdquo Rendiconti delCircolo Matematico di Palermo vol 48 no 3 pp 591ndash596 1999
[4] G B Djordjevic ldquoGenerating functions of the incomplete gen-eralized Fibonacci and generalized Lucas numbersrdquo FibonacciQuarterly vol 42 no 2 pp 106ndash113 2004
[5] G B Djordjevic and H M Srivastava ldquoIncomplete generalizedJacobsthal and Jacobsthal-Lucas numbersrdquo Mathematical andComputer Modelling vol 42 no 9-10 pp 1049ndash1056 2005
[6] D Tasci and M Cetin Firengiz ldquoIncomplete Fibonacci andLucas p-numbersrdquoMathematical and Computer Modelling vol52 no 9-10 pp 1763ndash1770 2010
[7] D Tasci M Cetin Firengiz andN Tuglu ldquoIncomplete bivariateFibonaccu and Lucas p-polynomialsrdquo Discrete Dynamics inNature and Society vol 2012 Article ID 840345 11 pages 2012
[8] S Falcon and A Plaza ldquoOn the Fibonacci k-numbersrdquo ChaosSolitons and Fractals vol 32 no 5 pp 1615ndash1624 2007
Chinese Journal of Mathematics 7
[9] C Bolat and H Kose ldquoOn the properties of k-Fibonaccinumbersrdquo International Journal of Contemporary MathematicalSciences vol 22 no 5 pp 1097ndash1105 2010
[10] S Falcon ldquoThe k-Fibonacci matrix and the Pascal matrixrdquoCentral European Journal ofMathematics vol 9 no 6 pp 1403ndash1410 2011
[11] S Falcon and A Plaza ldquoThe k-Fibonacci sequence and thePascal 2-trianglerdquo Chaos Solitons and Fractals vol 33 no 1 pp38ndash49 2007
[12] S Falcon and A Plaza ldquoOn k-Fibonacci sequences and polyno-mials and their derivativesrdquoChaos Solitons and Fractals vol 39no 3 pp 1005ndash1019 2009
[13] J Ramırez ldquoSome properties of convolved k-Fibonacci num-bersrdquo ISRN Combinatorics vol 2013 Article ID 759641 5 pages2013
[14] A Salas ldquoAbout k-Fibonacci numbers and their associatednumbersrdquo International Mathematical Forum vol 50 no 6 pp2473ndash2479 2011
[15] S Falcon ldquoOn the k-Lucas numbersrdquo International Journal ofContemporary Mathematical Sciences vol 21 no 6 pp 1039ndash1050 2011
[16] H M Srivastava and H L Manocha A Treatise on GeneratingFunctions Halsted Press Ellis Horwood Limited ChichesterUK John Wiley amp Sons New York NY USA 1984
[17] A Nalli and P Haukkanen ldquoOn generalized Fibonacci andLucas polynomialsrdquo Chaos Solitons and Fractals vol 42 no 5pp 3179ndash3186 2009
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
4 Chinese Journal of Mathematics
Table 2 The numbers 119871119897119896119899 for 1 ⩽ 119899 ⩽ 9
nl 0 1 2 3 41 k2 119896
2
1198962
+ 2
3 1198963
1198963
+ 3119896
4 1198964
1198964
+ 41198962
1198964
+ 41198962
+ 2
5 1198965
1198965
+ 51198963
1198965
+ 51198963
+ 5119896
6 1198966
1198966
+ 61198964
1198966
+ 61198964
+ 91198962
1198966
+ 61198964
+ 91198962
+ 2
7 1198967
1198967
+ 71198965
1198967
+ 71198965
+ 141198963
1198967
+ 71198965
+ 141198963
+ 7119896
8 1198968
1198968
+ 81198966
1198968
+ 81198966
+ 201198964
1198968
+ 81198966
+ 201198964
+ 161198962
1198968
+ 81198966
+ 201198964
+ 161198962
+ 2
9 1198969
1198969
+ 91198967
1198969
+ 91198967
+ 271198965
1198969
+ 91198967
+ 271198965
+ 301198963
1198969
+ 91198967
+ 271198965
+ 301198963
+ 9119896
Prooflfloor(119899minus1)2rfloor
sum
119897=0
119865119897
119896119899
= 1198650
119896119899+ 1198651
119896119899+ sdot sdot sdot + 119865
lfloor(119899minus1)2rfloor
119896119899
= (119899 minus 1 minus 0
0) 119896119899minus1
+ [(119899 minus 1 minus 0
0) 119896119899minus1
+ (119899 minus 1 minus 1
1) 119896119899minus3
]
+ sdot sdot sdot +
[[[[
[
(119899 minus 1 minus 0
0) 119896119899minus1
+ (119899 minus 1 minus 1
1) 119896119899minus3
+ sdot sdot sdot +(
119899 minus 1 minus lfloor119899 minus 1
2rfloor
lfloor119899 minus 1
2rfloor
)119896119899minus1minus2lfloor(119899minus1)2rfloor
]]]]
]
= (lfloor119899 minus 1
2rfloor + 1)(
119899 minus 1 minus 0
0) 119896119899minus1
+ lfloor119899 minus 1
2rfloor(119899 minus 1 minus 1
1) 119896119899minus3
+ sdot sdot sdot +(
119899 minus 1 minus lfloor119899 minus 1
2rfloor
lfloor119899 minus 1
2rfloor
)119896119899minus1minus2lfloor(119899minus1)2rfloor
=
lfloor(119899minus1)2rfloor
sum
119894=0
(lfloor119899 minus 1
2rfloor + 1 minus 119894) (
119899 minus 1 minus 119894
119894) 119896119899minus1minus2119894
=
lfloor(119899minus1)2rfloor
sum
119894=0
(lfloor119899 minus 1
2rfloor + 1)(
119899 minus 1 minus 119894
119894) 119896119899minus1minus2119894
minus
lfloor(119899minus1)2rfloor
sum
119894=0
119894 (119899 minus 1 minus 119894
119894) 119896119899minus1minus2119894
= (lfloor119899 minus 1
2rfloor + 1)119865
119896119899minus
lfloor(119899minus1)2rfloor
sum
119894=0
119894 (119899 minus 1 minus 119894
119894) 119896119899minus1minus2119894
(25)
From Lemma 6 (24) is obtained
3 The Incomplete 119896-Lucas Numbers
Definition 8 The incomplete 119896-Lucas numbers are defined by
119871119897
119896119899=
119897
sum
119894=0
119899
119899 minus 119894(119899 minus 119894
119894) 119896119899minus2119894
0 le 119897 le lfloor119899
2rfloor (26)
InTable 2 somenumbers of incomplete 119896-Lucas numbersare provided
We note that
119871lfloor1198992rfloor
1119899= 119871119899 (27)
Some special cases of (26) are
1198710
119896119899= 119896119899
(119899 ge 1)
1198711
119896119899= 119896119899
+ 119899119896119899minus2
(119899 ge 2)
1198712
119896119899= 119896119899
+ 119899119896119899minus2
+119899 (119899 minus 3)
2119896119899minus4
(119899 ge 4)
119871lfloor1198992rfloor
119896119899= 119871119896119899 (119899 ge 1)
119871lfloor(119899minus2)2rfloor
119896119899=
119871119896119899minus 2 (119899 even)
119871119896119899minus 119899119896 (119899 odd)
(119899 ge 2)
(28)
Chinese Journal of Mathematics 5
31 Some Recurrence Properties of the Numbers 119871119897119896119899
Proposition 9 One has
119871119897
119896119899= 119865119897minus1
119896119899minus1+ 119865119897
119896119899+1 0 le 119897 le lfloor
119899
2rfloor (29)
Proof By (8) rewrite the right-hand side of (29) as
119897minus1
sum
119894=0
(119899 minus 2 minus 119894
119894) 119896119899minus2minus2119894
+
119897
sum
119894=0
(119899 minus 119894
119894) 119896119899minus2119894
=
119897
sum
119894=1
(119899 minus 1 minus 119894
119894 minus 1) 119896119899minus2119894
+
119897
sum
119894=0
(119899 minus 119894
119894) 119896119899minus2119894
=
119897
sum
119894=0
[(119899 minus 1 minus 119894
119894 minus 1) + (
119899 minus 119894
119894)] 119896119899minus2119894
minus (119899 minus 1
minus1)
=
119897
sum
119894=0
119899
119899 minus 119894(119899 minus 119894
119894) 119896119899minus2119894
+ 0
= 119871119897
119896119899
(30)
Proposition 10 The recurrence relation of the incomplete 119896-Lucas numbers 119871119897
119896119899is
119871119897+1
119896119899+2= 119896119871119897+1
119896119899+1+ 119871119897
119896119899 0 le 119897 le lfloor
119899
2rfloor (31)
The relation (31) can be transformed into the nonhomoge-neous recurrence relation
119871119897
119896119899+2= 119896119871119897
119896119899+1+ 119871119897
119896119899minus119899
119899 minus 119897(119899 minus 119897
119897) 119896119899minus2119897
(32)
Proof Using (29) and (11) we write
119871119897+1
119896119899+2= 119865119897
119896119899+1+ 119865119897+1
119896119899+3
= 119896119865119897
119896119899+ 119865119897minus1
119896119899minus1+ 119896119865119897+1
119896119899+2+ 119865119897
119896119899+1
= 119896 (119865119897
119896119899+ 119865119897+1
119896119899+2) + 119865119897minus1
119896119899minus1+ 119865119897
119896119899+1
= 119896119871119897+1
119896119899+1+ 119871119897
119896119899
(33)
Proposition 11 One has
119896119871119897
119896119899= 119865119897
119896119899+2minus 119865119897minus2
119896119899minus2 0 le 119897 le lfloor
119899 minus 1
2rfloor (34)
Proof By (29)
119865119897
119896119899+2= 119871119897
119896119899+1minus 119865119897minus1
119896119899 119865
119897minus2
119896119899minus2= 119871119897minus1
119896119899minus1minus 119865119897minus1
119896119899 (35)
whence from (31)
119865119897
119896119899+2minus 119865119897minus2
119896119899minus2= 119871119897
119896119899+1minus 119871119897minus1
119896119899minus1= 119896119871119897
119896119899 (36)
Proposition 12 One has
119904
sum
119894=0
(119904
119894) 119871119897+119894
119896119899+119894119896119894
= 119871119897+119904
119896119899+2119904 0 le 119897 le
119899 minus 119904
2 (37)
Proof Using (29) and (14) we write
119904
sum
119894=0
(119904
119894) 119871119897+119894
119896119899+119894119896119894
=
119904
sum
119894=0
(119904
119894) [119865119897+119894minus1
119896119899+119894minus1+ 119865119897+119894
119896119899+119894+1] 119896119894
=
119904
sum
119894=0
(119904
119894) 119865119897+119894minus1
119896119899+119894minus1119896119894
+
119904
sum
119894=0
(119904
119894) 119865119897+119894
119896119899+119894+1119896119894
= 119865119897minus1+119904
119896119899minus1+2119904+ 119865119897+119904
119896119899+1+2119904= 119871119897+119904
119896119899+2119904
(38)
Proposition 13 For 119899 ge 2119897 + 1
119904minus1
sum
119894=0
119871119897
119896119899+119894119896119904minus1minus119894
= 119871119897+1
119896119899+119904+1minus 119896119904
119871119897+1
119896119899+1 (39)
The proof can be done by using (31) and induction on 119904
Lemma 14 One has
lfloor1198992rfloor
sum
119894=0
119894119899
119899 minus 119894(119899 minus 119894
119894) 119896119899minus2119894
=119899
2[119871119896119899minus 119896119865119896119899] (40)
The proof is similar to Lemma 6
Proposition 15 One has
lfloor1198992rfloor
sum
119897=0
119871119897
119896119899=
119871119896119899+119899119896
2119865119896119899
(119899 even)1
2(119871119896119899+ 119899119896119865
119896119899) (119899 odd)
(41)
Proof An argument analogous to that of the proof ofProposition 7 yields
lfloor1198992rfloor
sum
119897=0
119871119897
119896119899= (lfloor
119899
2rfloor + 1) 119871
119896119899minus
lfloor1198992rfloor
sum
119894=0
119894119899
119899 minus 119894(119899 minus 119894
119894) 119896119899minus2119894
(42)
From Lemma 14 (41) is obtained
4 Generating Functions of the Incomplete119896-Fibonacci and 119896-Lucas Number
In this section we give the generating functions of incomplete119896-Fibonacci and 119896-Lucas numbers
Lemma 16 (see [3 page 592]) Let 119904119899infin
119899=0be a complex
sequence satisfying the following nonhomogeneous recurrencerelation
119904119899= 119886119904119899minus1+ 119887119904119899minus2+ 119903119899(119899 gt 1) (43)
6 Chinese Journal of Mathematics
where 119886 and 119887 are complex numbers and 119903119899 is a given complex
sequence Then the generating function 119880(119905) of the sequence119904119899 is
119880 (119905) =119866 (119905) + 119904
0minus 1199030+ (1199041minus 1199040119886 minus 1199031) 119905
1 minus 119886119905 minus 1198871199052 (44)
where 119866(119905) denotes the generating function of 119903119899
Theorem 17 The generating function of the incomplete 119896-Fibonacci numbers 119865119897
119896119899is given by
119877119896119897(119909) =
infin
sum
119894=0
119865119897
119896119894119905119894
= 1199052119897+1
[1198651198962119897+1
+ (1198651198962119897+2
minus 1198961198651198962119897+1
) 119905 minus1199052
(1 minus 119896119905)119897+1
]
times [1 minus 119896119905 minus 1199052
]minus1
(45)
Proof Let 119897 be a fixed positive integer From (8) and (12)119865119897
119896119899= 0 for 0 le 119899 lt 2119897 + 1 119865119897
1198962119897+1= 1198651198962119897+1
and 1198651198971198962119897+2
=
1198651198962119897+2
and
119865119897
119896119899= 119896119865119897
119896119899minus1+ 119865119897
119896119899minus2minus (119899 minus 3 minus 119897
119897) 119896119899minus3minus2119897
(46)
Now let
1199040= 119865119897
1198962119897+1 119904
1= 119865119897
1198962119897+2 119904
119899= 119865119897
119896119899+2119897+1 (47)
Also let
1199030= 1199031= 0 119903
119899= (119899 + 119897 minus 2
119899 minus 2) 119896119899minus2
(48)
The generating function of the sequence 119903119899 is119866(119905) = 1199052(1minus
119896119905)119897+1 (see [16 page 355]) Thus from Lemma 16 we get the
generating function 119877119896119897(119909) of sequence 119904
119899
Theorem 18 The generating function of the incomplete 119896-Lucas numbers 119871119897
119896119899is given by
119878119896119897(119909) =
infin
sum
119894=0
119871119897
119896119894119905119894
= 1199052119897
[1198711198962119897+ (1198711198962119897+1
minus 1198961198711198962119897) 119905 minus
1199052
(2 minus 119905)
(1 minus 119896119905)119897+1
]
times [1 minus 119896119905 minus 1199052
]minus1
(49)
Proof The proof of this theorem is similar to the proof ofTheorem 17 Let 119897 be a fixed positive integer From (26) and(32)119871119897
119896119899= 0 for 0 le 119899 lt 2119897119871119897
1198962119897= 1198711198962119897
and1198711198971198962119897+1
= 1198711198962119897+1
and
119871119897
119896119899= 119896119871119897
119896119899minus1+ 119871119897
119896119899minus2minus119899 minus 2
119899 minus 2 minus 119897(119899 minus 2 minus 119897
119899 minus 2 minus 2119897) 119896119899minus2minus2119897
(50)
Now let
1199040= 119871119897
1198962119897 119904
1= 119871119897
1198962119897+1 119904
119899= 119871119897
119896119899+2119897 (51)
Also let
1199030= 1199031= 0 119903
119899= (119899 + 2119897 minus 2
119899 + 119897 minus 2) 119896119899+2119897minus2
(52)
The generating function of the sequence 119903119899 is 119866(119905) = 1199052(2 minus
119905)(1 minus 119896119905)119897+1 (see [16 page 355]) Thus from Lemma 16 we
get the generating function 119878119896119897(119909) of sequence 119904
119899
5 Conclusion
In this paper we introduce incomplete 119896-Fibonacci and119896-Lucas numbers and we obtain new identities In [17]the authors introduced the ℎ(119909)-Fibonacci polynomialsThat generalizes Catalanrsquos Fibonacci polynomials and the119896-Fibonacci numbers Let ℎ(119909) be a polynomial with realcoefficientsThe ℎ(119909)-Fibonacci polynomials 119865
ℎ119899(119909)119899isinN are
defined by the recurrence relation
119865ℎ0(119909) = 0 119865
ℎ1(119909) = 1
119865ℎ119899+1
(119909) = ℎ (119909) 119865ℎ119899(119909) + 119865
ℎ119899minus1(119909) 119899 ⩾ 1
(53)
It would be interesting to study a definition of incompleteℎ(119909)-Fibonacci polynomials and research their properties
Acknowledgments
The author would like to thank the anonymous referees fortheir helpful comments The author was partially supportedby Universidad Sergio Arboleda under Grant no USA-II-2011-0059
References
[1] T Koshy Fibonacci and Lucas Numbers with ApplicationsWiley-Interscience 2001
[2] P Filipponi ldquoIncomplete Fibonacci and Lucas numbersrdquoRendi-conti del CircoloMatematico di Palermo vol 45 no 1 pp 37ndash561996
[3] A Pinter and H M Srivastava ldquoGenerating functions ofthe incomplete Fibonacci and Lucas numbersrdquo Rendiconti delCircolo Matematico di Palermo vol 48 no 3 pp 591ndash596 1999
[4] G B Djordjevic ldquoGenerating functions of the incomplete gen-eralized Fibonacci and generalized Lucas numbersrdquo FibonacciQuarterly vol 42 no 2 pp 106ndash113 2004
[5] G B Djordjevic and H M Srivastava ldquoIncomplete generalizedJacobsthal and Jacobsthal-Lucas numbersrdquo Mathematical andComputer Modelling vol 42 no 9-10 pp 1049ndash1056 2005
[6] D Tasci and M Cetin Firengiz ldquoIncomplete Fibonacci andLucas p-numbersrdquoMathematical and Computer Modelling vol52 no 9-10 pp 1763ndash1770 2010
[7] D Tasci M Cetin Firengiz andN Tuglu ldquoIncomplete bivariateFibonaccu and Lucas p-polynomialsrdquo Discrete Dynamics inNature and Society vol 2012 Article ID 840345 11 pages 2012
[8] S Falcon and A Plaza ldquoOn the Fibonacci k-numbersrdquo ChaosSolitons and Fractals vol 32 no 5 pp 1615ndash1624 2007
Chinese Journal of Mathematics 7
[9] C Bolat and H Kose ldquoOn the properties of k-Fibonaccinumbersrdquo International Journal of Contemporary MathematicalSciences vol 22 no 5 pp 1097ndash1105 2010
[10] S Falcon ldquoThe k-Fibonacci matrix and the Pascal matrixrdquoCentral European Journal ofMathematics vol 9 no 6 pp 1403ndash1410 2011
[11] S Falcon and A Plaza ldquoThe k-Fibonacci sequence and thePascal 2-trianglerdquo Chaos Solitons and Fractals vol 33 no 1 pp38ndash49 2007
[12] S Falcon and A Plaza ldquoOn k-Fibonacci sequences and polyno-mials and their derivativesrdquoChaos Solitons and Fractals vol 39no 3 pp 1005ndash1019 2009
[13] J Ramırez ldquoSome properties of convolved k-Fibonacci num-bersrdquo ISRN Combinatorics vol 2013 Article ID 759641 5 pages2013
[14] A Salas ldquoAbout k-Fibonacci numbers and their associatednumbersrdquo International Mathematical Forum vol 50 no 6 pp2473ndash2479 2011
[15] S Falcon ldquoOn the k-Lucas numbersrdquo International Journal ofContemporary Mathematical Sciences vol 21 no 6 pp 1039ndash1050 2011
[16] H M Srivastava and H L Manocha A Treatise on GeneratingFunctions Halsted Press Ellis Horwood Limited ChichesterUK John Wiley amp Sons New York NY USA 1984
[17] A Nalli and P Haukkanen ldquoOn generalized Fibonacci andLucas polynomialsrdquo Chaos Solitons and Fractals vol 42 no 5pp 3179ndash3186 2009
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Chinese Journal of Mathematics 5
31 Some Recurrence Properties of the Numbers 119871119897119896119899
Proposition 9 One has
119871119897
119896119899= 119865119897minus1
119896119899minus1+ 119865119897
119896119899+1 0 le 119897 le lfloor
119899
2rfloor (29)
Proof By (8) rewrite the right-hand side of (29) as
119897minus1
sum
119894=0
(119899 minus 2 minus 119894
119894) 119896119899minus2minus2119894
+
119897
sum
119894=0
(119899 minus 119894
119894) 119896119899minus2119894
=
119897
sum
119894=1
(119899 minus 1 minus 119894
119894 minus 1) 119896119899minus2119894
+
119897
sum
119894=0
(119899 minus 119894
119894) 119896119899minus2119894
=
119897
sum
119894=0
[(119899 minus 1 minus 119894
119894 minus 1) + (
119899 minus 119894
119894)] 119896119899minus2119894
minus (119899 minus 1
minus1)
=
119897
sum
119894=0
119899
119899 minus 119894(119899 minus 119894
119894) 119896119899minus2119894
+ 0
= 119871119897
119896119899
(30)
Proposition 10 The recurrence relation of the incomplete 119896-Lucas numbers 119871119897
119896119899is
119871119897+1
119896119899+2= 119896119871119897+1
119896119899+1+ 119871119897
119896119899 0 le 119897 le lfloor
119899
2rfloor (31)
The relation (31) can be transformed into the nonhomoge-neous recurrence relation
119871119897
119896119899+2= 119896119871119897
119896119899+1+ 119871119897
119896119899minus119899
119899 minus 119897(119899 minus 119897
119897) 119896119899minus2119897
(32)
Proof Using (29) and (11) we write
119871119897+1
119896119899+2= 119865119897
119896119899+1+ 119865119897+1
119896119899+3
= 119896119865119897
119896119899+ 119865119897minus1
119896119899minus1+ 119896119865119897+1
119896119899+2+ 119865119897
119896119899+1
= 119896 (119865119897
119896119899+ 119865119897+1
119896119899+2) + 119865119897minus1
119896119899minus1+ 119865119897
119896119899+1
= 119896119871119897+1
119896119899+1+ 119871119897
119896119899
(33)
Proposition 11 One has
119896119871119897
119896119899= 119865119897
119896119899+2minus 119865119897minus2
119896119899minus2 0 le 119897 le lfloor
119899 minus 1
2rfloor (34)
Proof By (29)
119865119897
119896119899+2= 119871119897
119896119899+1minus 119865119897minus1
119896119899 119865
119897minus2
119896119899minus2= 119871119897minus1
119896119899minus1minus 119865119897minus1
119896119899 (35)
whence from (31)
119865119897
119896119899+2minus 119865119897minus2
119896119899minus2= 119871119897
119896119899+1minus 119871119897minus1
119896119899minus1= 119896119871119897
119896119899 (36)
Proposition 12 One has
119904
sum
119894=0
(119904
119894) 119871119897+119894
119896119899+119894119896119894
= 119871119897+119904
119896119899+2119904 0 le 119897 le
119899 minus 119904
2 (37)
Proof Using (29) and (14) we write
119904
sum
119894=0
(119904
119894) 119871119897+119894
119896119899+119894119896119894
=
119904
sum
119894=0
(119904
119894) [119865119897+119894minus1
119896119899+119894minus1+ 119865119897+119894
119896119899+119894+1] 119896119894
=
119904
sum
119894=0
(119904
119894) 119865119897+119894minus1
119896119899+119894minus1119896119894
+
119904
sum
119894=0
(119904
119894) 119865119897+119894
119896119899+119894+1119896119894
= 119865119897minus1+119904
119896119899minus1+2119904+ 119865119897+119904
119896119899+1+2119904= 119871119897+119904
119896119899+2119904
(38)
Proposition 13 For 119899 ge 2119897 + 1
119904minus1
sum
119894=0
119871119897
119896119899+119894119896119904minus1minus119894
= 119871119897+1
119896119899+119904+1minus 119896119904
119871119897+1
119896119899+1 (39)
The proof can be done by using (31) and induction on 119904
Lemma 14 One has
lfloor1198992rfloor
sum
119894=0
119894119899
119899 minus 119894(119899 minus 119894
119894) 119896119899minus2119894
=119899
2[119871119896119899minus 119896119865119896119899] (40)
The proof is similar to Lemma 6
Proposition 15 One has
lfloor1198992rfloor
sum
119897=0
119871119897
119896119899=
119871119896119899+119899119896
2119865119896119899
(119899 even)1
2(119871119896119899+ 119899119896119865
119896119899) (119899 odd)
(41)
Proof An argument analogous to that of the proof ofProposition 7 yields
lfloor1198992rfloor
sum
119897=0
119871119897
119896119899= (lfloor
119899
2rfloor + 1) 119871
119896119899minus
lfloor1198992rfloor
sum
119894=0
119894119899
119899 minus 119894(119899 minus 119894
119894) 119896119899minus2119894
(42)
From Lemma 14 (41) is obtained
4 Generating Functions of the Incomplete119896-Fibonacci and 119896-Lucas Number
In this section we give the generating functions of incomplete119896-Fibonacci and 119896-Lucas numbers
Lemma 16 (see [3 page 592]) Let 119904119899infin
119899=0be a complex
sequence satisfying the following nonhomogeneous recurrencerelation
119904119899= 119886119904119899minus1+ 119887119904119899minus2+ 119903119899(119899 gt 1) (43)
6 Chinese Journal of Mathematics
where 119886 and 119887 are complex numbers and 119903119899 is a given complex
sequence Then the generating function 119880(119905) of the sequence119904119899 is
119880 (119905) =119866 (119905) + 119904
0minus 1199030+ (1199041minus 1199040119886 minus 1199031) 119905
1 minus 119886119905 minus 1198871199052 (44)
where 119866(119905) denotes the generating function of 119903119899
Theorem 17 The generating function of the incomplete 119896-Fibonacci numbers 119865119897
119896119899is given by
119877119896119897(119909) =
infin
sum
119894=0
119865119897
119896119894119905119894
= 1199052119897+1
[1198651198962119897+1
+ (1198651198962119897+2
minus 1198961198651198962119897+1
) 119905 minus1199052
(1 minus 119896119905)119897+1
]
times [1 minus 119896119905 minus 1199052
]minus1
(45)
Proof Let 119897 be a fixed positive integer From (8) and (12)119865119897
119896119899= 0 for 0 le 119899 lt 2119897 + 1 119865119897
1198962119897+1= 1198651198962119897+1
and 1198651198971198962119897+2
=
1198651198962119897+2
and
119865119897
119896119899= 119896119865119897
119896119899minus1+ 119865119897
119896119899minus2minus (119899 minus 3 minus 119897
119897) 119896119899minus3minus2119897
(46)
Now let
1199040= 119865119897
1198962119897+1 119904
1= 119865119897
1198962119897+2 119904
119899= 119865119897
119896119899+2119897+1 (47)
Also let
1199030= 1199031= 0 119903
119899= (119899 + 119897 minus 2
119899 minus 2) 119896119899minus2
(48)
The generating function of the sequence 119903119899 is119866(119905) = 1199052(1minus
119896119905)119897+1 (see [16 page 355]) Thus from Lemma 16 we get the
generating function 119877119896119897(119909) of sequence 119904
119899
Theorem 18 The generating function of the incomplete 119896-Lucas numbers 119871119897
119896119899is given by
119878119896119897(119909) =
infin
sum
119894=0
119871119897
119896119894119905119894
= 1199052119897
[1198711198962119897+ (1198711198962119897+1
minus 1198961198711198962119897) 119905 minus
1199052
(2 minus 119905)
(1 minus 119896119905)119897+1
]
times [1 minus 119896119905 minus 1199052
]minus1
(49)
Proof The proof of this theorem is similar to the proof ofTheorem 17 Let 119897 be a fixed positive integer From (26) and(32)119871119897
119896119899= 0 for 0 le 119899 lt 2119897119871119897
1198962119897= 1198711198962119897
and1198711198971198962119897+1
= 1198711198962119897+1
and
119871119897
119896119899= 119896119871119897
119896119899minus1+ 119871119897
119896119899minus2minus119899 minus 2
119899 minus 2 minus 119897(119899 minus 2 minus 119897
119899 minus 2 minus 2119897) 119896119899minus2minus2119897
(50)
Now let
1199040= 119871119897
1198962119897 119904
1= 119871119897
1198962119897+1 119904
119899= 119871119897
119896119899+2119897 (51)
Also let
1199030= 1199031= 0 119903
119899= (119899 + 2119897 minus 2
119899 + 119897 minus 2) 119896119899+2119897minus2
(52)
The generating function of the sequence 119903119899 is 119866(119905) = 1199052(2 minus
119905)(1 minus 119896119905)119897+1 (see [16 page 355]) Thus from Lemma 16 we
get the generating function 119878119896119897(119909) of sequence 119904
119899
5 Conclusion
In this paper we introduce incomplete 119896-Fibonacci and119896-Lucas numbers and we obtain new identities In [17]the authors introduced the ℎ(119909)-Fibonacci polynomialsThat generalizes Catalanrsquos Fibonacci polynomials and the119896-Fibonacci numbers Let ℎ(119909) be a polynomial with realcoefficientsThe ℎ(119909)-Fibonacci polynomials 119865
ℎ119899(119909)119899isinN are
defined by the recurrence relation
119865ℎ0(119909) = 0 119865
ℎ1(119909) = 1
119865ℎ119899+1
(119909) = ℎ (119909) 119865ℎ119899(119909) + 119865
ℎ119899minus1(119909) 119899 ⩾ 1
(53)
It would be interesting to study a definition of incompleteℎ(119909)-Fibonacci polynomials and research their properties
Acknowledgments
The author would like to thank the anonymous referees fortheir helpful comments The author was partially supportedby Universidad Sergio Arboleda under Grant no USA-II-2011-0059
References
[1] T Koshy Fibonacci and Lucas Numbers with ApplicationsWiley-Interscience 2001
[2] P Filipponi ldquoIncomplete Fibonacci and Lucas numbersrdquoRendi-conti del CircoloMatematico di Palermo vol 45 no 1 pp 37ndash561996
[3] A Pinter and H M Srivastava ldquoGenerating functions ofthe incomplete Fibonacci and Lucas numbersrdquo Rendiconti delCircolo Matematico di Palermo vol 48 no 3 pp 591ndash596 1999
[4] G B Djordjevic ldquoGenerating functions of the incomplete gen-eralized Fibonacci and generalized Lucas numbersrdquo FibonacciQuarterly vol 42 no 2 pp 106ndash113 2004
[5] G B Djordjevic and H M Srivastava ldquoIncomplete generalizedJacobsthal and Jacobsthal-Lucas numbersrdquo Mathematical andComputer Modelling vol 42 no 9-10 pp 1049ndash1056 2005
[6] D Tasci and M Cetin Firengiz ldquoIncomplete Fibonacci andLucas p-numbersrdquoMathematical and Computer Modelling vol52 no 9-10 pp 1763ndash1770 2010
[7] D Tasci M Cetin Firengiz andN Tuglu ldquoIncomplete bivariateFibonaccu and Lucas p-polynomialsrdquo Discrete Dynamics inNature and Society vol 2012 Article ID 840345 11 pages 2012
[8] S Falcon and A Plaza ldquoOn the Fibonacci k-numbersrdquo ChaosSolitons and Fractals vol 32 no 5 pp 1615ndash1624 2007
Chinese Journal of Mathematics 7
[9] C Bolat and H Kose ldquoOn the properties of k-Fibonaccinumbersrdquo International Journal of Contemporary MathematicalSciences vol 22 no 5 pp 1097ndash1105 2010
[10] S Falcon ldquoThe k-Fibonacci matrix and the Pascal matrixrdquoCentral European Journal ofMathematics vol 9 no 6 pp 1403ndash1410 2011
[11] S Falcon and A Plaza ldquoThe k-Fibonacci sequence and thePascal 2-trianglerdquo Chaos Solitons and Fractals vol 33 no 1 pp38ndash49 2007
[12] S Falcon and A Plaza ldquoOn k-Fibonacci sequences and polyno-mials and their derivativesrdquoChaos Solitons and Fractals vol 39no 3 pp 1005ndash1019 2009
[13] J Ramırez ldquoSome properties of convolved k-Fibonacci num-bersrdquo ISRN Combinatorics vol 2013 Article ID 759641 5 pages2013
[14] A Salas ldquoAbout k-Fibonacci numbers and their associatednumbersrdquo International Mathematical Forum vol 50 no 6 pp2473ndash2479 2011
[15] S Falcon ldquoOn the k-Lucas numbersrdquo International Journal ofContemporary Mathematical Sciences vol 21 no 6 pp 1039ndash1050 2011
[16] H M Srivastava and H L Manocha A Treatise on GeneratingFunctions Halsted Press Ellis Horwood Limited ChichesterUK John Wiley amp Sons New York NY USA 1984
[17] A Nalli and P Haukkanen ldquoOn generalized Fibonacci andLucas polynomialsrdquo Chaos Solitons and Fractals vol 42 no 5pp 3179ndash3186 2009
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
6 Chinese Journal of Mathematics
where 119886 and 119887 are complex numbers and 119903119899 is a given complex
sequence Then the generating function 119880(119905) of the sequence119904119899 is
119880 (119905) =119866 (119905) + 119904
0minus 1199030+ (1199041minus 1199040119886 minus 1199031) 119905
1 minus 119886119905 minus 1198871199052 (44)
where 119866(119905) denotes the generating function of 119903119899
Theorem 17 The generating function of the incomplete 119896-Fibonacci numbers 119865119897
119896119899is given by
119877119896119897(119909) =
infin
sum
119894=0
119865119897
119896119894119905119894
= 1199052119897+1
[1198651198962119897+1
+ (1198651198962119897+2
minus 1198961198651198962119897+1
) 119905 minus1199052
(1 minus 119896119905)119897+1
]
times [1 minus 119896119905 minus 1199052
]minus1
(45)
Proof Let 119897 be a fixed positive integer From (8) and (12)119865119897
119896119899= 0 for 0 le 119899 lt 2119897 + 1 119865119897
1198962119897+1= 1198651198962119897+1
and 1198651198971198962119897+2
=
1198651198962119897+2
and
119865119897
119896119899= 119896119865119897
119896119899minus1+ 119865119897
119896119899minus2minus (119899 minus 3 minus 119897
119897) 119896119899minus3minus2119897
(46)
Now let
1199040= 119865119897
1198962119897+1 119904
1= 119865119897
1198962119897+2 119904
119899= 119865119897
119896119899+2119897+1 (47)
Also let
1199030= 1199031= 0 119903
119899= (119899 + 119897 minus 2
119899 minus 2) 119896119899minus2
(48)
The generating function of the sequence 119903119899 is119866(119905) = 1199052(1minus
119896119905)119897+1 (see [16 page 355]) Thus from Lemma 16 we get the
generating function 119877119896119897(119909) of sequence 119904
119899
Theorem 18 The generating function of the incomplete 119896-Lucas numbers 119871119897
119896119899is given by
119878119896119897(119909) =
infin
sum
119894=0
119871119897
119896119894119905119894
= 1199052119897
[1198711198962119897+ (1198711198962119897+1
minus 1198961198711198962119897) 119905 minus
1199052
(2 minus 119905)
(1 minus 119896119905)119897+1
]
times [1 minus 119896119905 minus 1199052
]minus1
(49)
Proof The proof of this theorem is similar to the proof ofTheorem 17 Let 119897 be a fixed positive integer From (26) and(32)119871119897
119896119899= 0 for 0 le 119899 lt 2119897119871119897
1198962119897= 1198711198962119897
and1198711198971198962119897+1
= 1198711198962119897+1
and
119871119897
119896119899= 119896119871119897
119896119899minus1+ 119871119897
119896119899minus2minus119899 minus 2
119899 minus 2 minus 119897(119899 minus 2 minus 119897
119899 minus 2 minus 2119897) 119896119899minus2minus2119897
(50)
Now let
1199040= 119871119897
1198962119897 119904
1= 119871119897
1198962119897+1 119904
119899= 119871119897
119896119899+2119897 (51)
Also let
1199030= 1199031= 0 119903
119899= (119899 + 2119897 minus 2
119899 + 119897 minus 2) 119896119899+2119897minus2
(52)
The generating function of the sequence 119903119899 is 119866(119905) = 1199052(2 minus
119905)(1 minus 119896119905)119897+1 (see [16 page 355]) Thus from Lemma 16 we
get the generating function 119878119896119897(119909) of sequence 119904
119899
5 Conclusion
In this paper we introduce incomplete 119896-Fibonacci and119896-Lucas numbers and we obtain new identities In [17]the authors introduced the ℎ(119909)-Fibonacci polynomialsThat generalizes Catalanrsquos Fibonacci polynomials and the119896-Fibonacci numbers Let ℎ(119909) be a polynomial with realcoefficientsThe ℎ(119909)-Fibonacci polynomials 119865
ℎ119899(119909)119899isinN are
defined by the recurrence relation
119865ℎ0(119909) = 0 119865
ℎ1(119909) = 1
119865ℎ119899+1
(119909) = ℎ (119909) 119865ℎ119899(119909) + 119865
ℎ119899minus1(119909) 119899 ⩾ 1
(53)
It would be interesting to study a definition of incompleteℎ(119909)-Fibonacci polynomials and research their properties
Acknowledgments
The author would like to thank the anonymous referees fortheir helpful comments The author was partially supportedby Universidad Sergio Arboleda under Grant no USA-II-2011-0059
References
[1] T Koshy Fibonacci and Lucas Numbers with ApplicationsWiley-Interscience 2001
[2] P Filipponi ldquoIncomplete Fibonacci and Lucas numbersrdquoRendi-conti del CircoloMatematico di Palermo vol 45 no 1 pp 37ndash561996
[3] A Pinter and H M Srivastava ldquoGenerating functions ofthe incomplete Fibonacci and Lucas numbersrdquo Rendiconti delCircolo Matematico di Palermo vol 48 no 3 pp 591ndash596 1999
[4] G B Djordjevic ldquoGenerating functions of the incomplete gen-eralized Fibonacci and generalized Lucas numbersrdquo FibonacciQuarterly vol 42 no 2 pp 106ndash113 2004
[5] G B Djordjevic and H M Srivastava ldquoIncomplete generalizedJacobsthal and Jacobsthal-Lucas numbersrdquo Mathematical andComputer Modelling vol 42 no 9-10 pp 1049ndash1056 2005
[6] D Tasci and M Cetin Firengiz ldquoIncomplete Fibonacci andLucas p-numbersrdquoMathematical and Computer Modelling vol52 no 9-10 pp 1763ndash1770 2010
[7] D Tasci M Cetin Firengiz andN Tuglu ldquoIncomplete bivariateFibonaccu and Lucas p-polynomialsrdquo Discrete Dynamics inNature and Society vol 2012 Article ID 840345 11 pages 2012
[8] S Falcon and A Plaza ldquoOn the Fibonacci k-numbersrdquo ChaosSolitons and Fractals vol 32 no 5 pp 1615ndash1624 2007
Chinese Journal of Mathematics 7
[9] C Bolat and H Kose ldquoOn the properties of k-Fibonaccinumbersrdquo International Journal of Contemporary MathematicalSciences vol 22 no 5 pp 1097ndash1105 2010
[10] S Falcon ldquoThe k-Fibonacci matrix and the Pascal matrixrdquoCentral European Journal ofMathematics vol 9 no 6 pp 1403ndash1410 2011
[11] S Falcon and A Plaza ldquoThe k-Fibonacci sequence and thePascal 2-trianglerdquo Chaos Solitons and Fractals vol 33 no 1 pp38ndash49 2007
[12] S Falcon and A Plaza ldquoOn k-Fibonacci sequences and polyno-mials and their derivativesrdquoChaos Solitons and Fractals vol 39no 3 pp 1005ndash1019 2009
[13] J Ramırez ldquoSome properties of convolved k-Fibonacci num-bersrdquo ISRN Combinatorics vol 2013 Article ID 759641 5 pages2013
[14] A Salas ldquoAbout k-Fibonacci numbers and their associatednumbersrdquo International Mathematical Forum vol 50 no 6 pp2473ndash2479 2011
[15] S Falcon ldquoOn the k-Lucas numbersrdquo International Journal ofContemporary Mathematical Sciences vol 21 no 6 pp 1039ndash1050 2011
[16] H M Srivastava and H L Manocha A Treatise on GeneratingFunctions Halsted Press Ellis Horwood Limited ChichesterUK John Wiley amp Sons New York NY USA 1984
[17] A Nalli and P Haukkanen ldquoOn generalized Fibonacci andLucas polynomialsrdquo Chaos Solitons and Fractals vol 42 no 5pp 3179ndash3186 2009
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Chinese Journal of Mathematics 7
[9] C Bolat and H Kose ldquoOn the properties of k-Fibonaccinumbersrdquo International Journal of Contemporary MathematicalSciences vol 22 no 5 pp 1097ndash1105 2010
[10] S Falcon ldquoThe k-Fibonacci matrix and the Pascal matrixrdquoCentral European Journal ofMathematics vol 9 no 6 pp 1403ndash1410 2011
[11] S Falcon and A Plaza ldquoThe k-Fibonacci sequence and thePascal 2-trianglerdquo Chaos Solitons and Fractals vol 33 no 1 pp38ndash49 2007
[12] S Falcon and A Plaza ldquoOn k-Fibonacci sequences and polyno-mials and their derivativesrdquoChaos Solitons and Fractals vol 39no 3 pp 1005ndash1019 2009
[13] J Ramırez ldquoSome properties of convolved k-Fibonacci num-bersrdquo ISRN Combinatorics vol 2013 Article ID 759641 5 pages2013
[14] A Salas ldquoAbout k-Fibonacci numbers and their associatednumbersrdquo International Mathematical Forum vol 50 no 6 pp2473ndash2479 2011
[15] S Falcon ldquoOn the k-Lucas numbersrdquo International Journal ofContemporary Mathematical Sciences vol 21 no 6 pp 1039ndash1050 2011
[16] H M Srivastava and H L Manocha A Treatise on GeneratingFunctions Halsted Press Ellis Horwood Limited ChichesterUK John Wiley amp Sons New York NY USA 1984
[17] A Nalli and P Haukkanen ldquoOn generalized Fibonacci andLucas polynomialsrdquo Chaos Solitons and Fractals vol 42 no 5pp 3179ndash3186 2009
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of