a survey on bessel type functions · interesting relations between these integrals and derivatives...

International Journal of Applied Mathematics————————————————————–Volume 32 No. 3 2019, 357-380ISSN: 1311-1728 (printed version); ISSN: 1314-8060 (on-line version)doi: http://dx.doi.org/10.12732/ijam.v32i3.1

A SURVEY ON BESSEL TYPE FUNCTIONS

AS MULTI-INDEX MITTAG-LEFFLER FUNCTIONS:

DIFFERENTIAL AND INTEGRAL RELATIONS

Jordanka Paneva-Konovska1,2

1 Faculty of Applied Mathematics and InformaticsTechnical University of Sofia

8 Kliment Ohridski, Sofia – 1000, BULGARIA2 Institute of Mathematics and Informatics

Bulgarian Academy of Sciences‘Acad. G. Bontchev’ Street, Block 8

Sofia – 1113, BULGARIA

Abstract: As recently observed by Bazhlekova and Dimovski [1], the n-th derivative of the 2-parametric Mittag-Leffler function gives a 3-parametricMittag-Leffler function, known as the Prabhakar function. Following the anal-ogy, the n-th derivatives of the Bessel type functions are obtained, and it turnsout that usually they are expressed in terms of the Bessel type functions withthe same or more number of indices, up to a matching power function. Further,some special cases of the fractional order Riemann–Liouville derivatives andintegrals of the Bessel type functions are calculated and interesting relationsare proved.

AMS Subject Classification: 33E12, 33C10, 26A33Key Words: Bessel functions and generalizations, Mittag-Leffler functions,multi-index Mittag-Leffler functions, fractional calculus, Riemann–Liouville frac-tional integral and derivative

Received: March 3, 2019 c© 2019 Academic Publications

358 J. Paneva-Konovska

1. Introduction

This is a survey paper on a part of author’s publications, referring to the dif-ferentiation and integration of an arbitrary order of the Bessel type functions.In the beginning we consider both the classical Bessel functions of the firstkind and the Bessel–Clifford functions, which are closely related. The first aredefined by the series

Jν(z) =

∞∑

k=0

(−1)k(z/2)2k+ν

k! Γ(k + ν + 1), z ∈ C \ (−∞, 0]; ν ∈ C, (1)

and the second, which are entire functions, are defined by the power series

Cν(z) =

∞∑

k=0

(−1)k(z)k

k! Γ(k + ν + 1), z ∈ C; ν ∈ C. (2)

Originating from concrete problems in mechanics and astronomy, these func-tions have plenty of applications. In that reason they both have various gener-alizations with more indices (parameters).

In contrast, the Mittag-Leffler functions, which are entire functions, definedby the power series

Eα, β(z) =

∞∑

k=0

zk

Γ(αk + β)(z ∈ C; α, β ∈ C, Re(α) > 0), (3)

originated from a purely theoretical problem connected with an analytical con-tinuation of a function. Arising in the beginning of 20th century (initially asEα(z), for β = 1) they remained unremarked and unused for a long time (almostfor a half of century). Nevertheless, nowadays they also have numerous gen-eralizations, related to applications in Fractional Calculus (FC) and fractionalorder differential equations and systems.

2. Generalizations of the Bessel functions

Here we consider the generalizations of the Bessel and Bessel–Clifford functionswith two, three and four indices, or briefly said Bessel type functions. We beginwith the entire functions

Jµν (z) =

∞∑

k=0

(−z)k

k! Γ(µk + ν + 1), z ∈ C; ν ∈ C and µ > −1, (4)

introduced by the British mathematician Sir Edward MaitlandWright [26] and known in the literature as Bessel–Maitland functions (namedafter his second name). Wright firstly introduced them for µ > 0 and on alater stage, studying the asymptotic behavior of the Bessel–Maitland functions

A SURVEY ON BESSEL TYPE FUNCTIONS AS... 359

for large values of |z|, he extended their definition for µ > −1 ([27], see also[13] and [14]). Let us mention, that for µ = 1 the Bessel–Maitland functionbecomes Bessel–Clifford function, i.e.

Cν(z) = J1ν (z).

We also consider the generalized Bessel–Maitland (or Bessel–Wright) functionswith 3 parameters (introduced by R.S. Pathak [22]):

Jµν,σ(z) = (z/2)ν+2σ

∞∑

k=0

(−1)k(z/2)2k

Γ(k + σ + 1)Γ(µk + σ + ν + 1),

z ∈ C \ (−∞, 0] ; ν, σ ∈ C, µ > 0,

(5)

and along with it

Jµν,σ(2

√z) = zν/2+σ

∞∑

k=0

(−z)k

Γ(k + σ + 1)Γ(µk + σ + ν + 1),

z ∈ C \ (−∞, 0] ; ν, σ ∈ C, µ > 0.

(6)

At last, the 4-index generalizations (Prieto and all [24]) are given by the formula

Jµ,mν,σ (z) = (z/2)ν+2σ

∞∑

k=0

(−1)k(z/2)2k

(Γ(k + σ + 1))mΓ(µk + σ + ν + 1),

z ∈ C \ (−∞, 0] ; ν, σ ∈ C, µ > 0, m ∈ N,

(7)

and along with it

Jµ,mν,σ (2

√z) = zν/2+σ

∞∑

k=0

(−z)k

(Γ(k + σ + 1))mΓ(µk + σ + ν + 1),

z ∈ C \ (−∞, 0] ; ν, σ ∈ C, µ > 0, m ∈ N.

(8)

It is easily seen there is a simple relation between the two-index Bessel–Maitland functions Jµ

ν (z) and the generalized Bessel–Maitland functions with3 indices, namely

Jµν (z) = z−ν/2 Jµ

ν,0(2√z), z 6= 0; ν ∈ C, µ > 0, (9)

and in particular, for µ = 1 the relation is between Cν and J1ν,0 = Jν , i.e.

Cν(z) = z−ν/2Jν(2√z), z 6= 0, ν ∈ C. (10)

Studying the properties of the functions, mentioned above, integrals andderivatives of an arbitrary order of them are found. As a result, differentinteresting relations between these integrals and derivatives and functions ofthe considered kind are often obtained. The corresponding special cases forthe Bessel functions of the first kind (1), Bessel–Clifford functions (2), andMittag-Leffler functions (3), merely follow as corollaries. Such type of resultsare also obtained for other Special Functions of Fractional Calculus, for more


general functions see e.g. [7]. Various useful results for the generalized fractionalcalculus operators of special functions are given in the interesting paper [11]. Apart of the results connected with the 2-parametric Bessel–Maitland functionsare directly obtained by the author in [21]. Other type of properties for thefunctions families of the kinds (3) and (11) can be seen in the survey papers[17] and [18] by the author.

3. Generalizations of the Mittag-Leffler functions

Regardless of their purely theoretical origin, nowadays the Mittag-Leffler func-tions enjoy many generalizations and wide applications. Among them is the3-parametric Mittag-Leffler function

Eγα, β(z) =

∞∑

k=0

(γ)kΓ(αk + β)

zk

k!(z ∈ C; α, β, γ ∈ C, Re(α) > 0), (11)

introduced by Prabhakar [23] and known in the literature also as the Prabhakarfunction. It is clear that

E1α, β(z) = Eα, β(z), E1

α, 1(z) = Eα, 1(z) = Eα(z).

Other generalizations are the multi-index Mittag-Leffler functions. Firstof them is the multi-index Mittag-Leffler function with 2m parameters (m =1, 2, 3, ...):

E(αi), (βi)(z) = Em(αi), (βi)

(z) =∞∑

k=0

zk

Γ(α1k + β1) . . .Γ(αmk + βm). (12)

It was introduced by Luchko and Yakubovich [28] and Kiryakova [8] and studiedin details by Kiryakova [8, 10]. Originally defined for αi > 0 and βi arbitraryreal (complex) numbers, later its definition is extended for complex parameterswith Re(αi) > 0 (see e.g. [8] for the first case and the works [3, 4] for the secondone). For the applications of this class of special functions in the solutions offractional order differential equations and models, see e.g. in Kiryakova andLuchko [12]. The survey by Kilbas et al. [5] describes the historical developmentof the theory of these multi-index (2m-parametric) Mittag-Leffler functions asa subclass of the Wright generalized hypergeometric functions pΨq(z). Themethod of Mellin–Barnes type integral representations allows these authors toextend the considered functions and to study them in the case of arbitraryvalues of parameters.

The next is the multi-index Mittag-Leffler function defined by means of 3mparameters:


E(γi)(αi), (βi)

(z) = E(γi), m(αi), (βi)

(z) =

∞∑

k=0

(γ1)k . . . (γm)kΓ(α1k + β1) . . .Γ(αmk + βm)

zk

(k!)m(13)

(z ∈ C; αi, βi, γi ∈ C, Re(αi) > 0),

and was introduced by the author in [16, 19]. It generalizes both the 3-parametric Prabhakar function (11) and the 2m-parametric Mittag-Leffler func-tion (12).

More precisely, taking m = 1 in (13), the Prabhakar function (11) is pro-duced. If γ1 = · · · = γm = 1, then (13) is reduced to the 2m-parametricfunction (12).

All of these listed functions can be expressed by the more general Wrightgeneralized hypergeometric functions pΨq:

pΨq

[(a1, A1) . . . (ap, Ap)

(b1, B1) . . . (bq, Bq)

∣∣∣∣σ]=

∞∑

k=0

Γ(a1 + kA1) . . .Γ(ap + kAp)

Γ(b1 + kB1) . . .Γ(bq + kBq)

σk

k!, (14)

with Ai > 0, Bi > 0, ai, bi ∈ C, i = 1, . . . ,m.

4. The Bessel type functions as multi-index Mittag-Leffler functions

The Bessel functions and their generalizations are also connected with the multi-index Mittag-Leffler functions. The relations between them are given in thissection.

Again, let γ1 = · · · = γm = 1. A special case of (13) (for m ≥ 2) is thegeneralized Lommel–Wright function with 4 indices (µ > 0, q ∈ N, ν, λ ∈ C),introduced by de Oteiza, Kalla and Conde (for details see [9]):

This is an interesting example of a multi-index Mittag-Leffler function witharbitrary m = q + 1, q ∈ N. It is clear that for α1 = µ, α2 = 1, . . . , αq+1 = 1,β1 = σ+ ν +1, β2 = σ+1 , . . . , βq+1 = σ+1, the generalized Lommel–Wrightfunction can be expressed by the multi-index Mittag-Leffler functions (12) and(13) as follows:

Jµ,qν,σ (z) = (z/2)ν+2σE

(1,...,1), q+1(µ,1,...,1), (σ+ν+1,σ+1,...,σ+1)(−

z2

4)

= (z/2)ν+2σE q+1(µ,1,...,1), (σ+ν+1,σ+1,...,σ+1)(−

z2

4),

z ∈ C \ (−∞, 0] ; ν, σ ∈ C, µ > 0,

(15)


respectively

Jµ,qν,σ (2

√z) = zν/2+σE

(1,...,1), q+1(µ,1,...,1), (σ+ν+1,σ+1,...,σ+1)(−z)

= zν/2+σE q+1(µ,1,...,1), (σ+ν+1,σ+1,...,σ+1)(−z),

z ∈ C \ (−∞, 0] ; ν, σ ∈ C, µ > 0.

(16)

Further, for q = 1, α1 = µ, α2 = 1, β1 = σ+ν+1, β2 = σ+1, the equalities(15) and (16) give:

Jµν,σ(z) = (z/2)ν+2σE

(1,1)(µ,1), (σ+ν+1,σ+1)(−

z2

4)

= (z/2)ν+2σE(µ,1), (σ+ν+1,σ+1)(−z2

4),

z ∈ C \ (−∞, 0] ; ν, σ ∈ C, µ > 0,

(17)

respectively

Jµν,σ(2

√z) = zν/2+σE

(1,1)(µ,1), (σ+ν+1,σ+1)(−z)

= zν/2+σE(µ,1), (σ+ν+1,σ+1)(−z),

z ∈ C \ (−∞, 0] ; ν, σ ∈ C, µ > 0.

(18)

Additionally, if σ = 0, i. e. for q = 1, α1 = µ, α2 = 1, β1 = ν + 1, β2 = 1,the relations (15) and (17) produce:

Jµν (z) = E

(1,1)(µ,1), (ν+1,1)(−z) = E(µ,1), (ν+1,1)(−z),

z ∈ C; ν ∈ C and µ > 0,(19)

and finally, if α1 = µ = 1, the formula (19) is reduced to

Cν(z) = E(1,1)(1,1), (ν+1,1)

(−z) = E(1,1), (ν+1,1)(−z), z ∈ C; ν ∈ C. (20)

Thus, all the considered functions in the preceding sections, both the gen-eralizations of the Bessel and Mittag-Leffler functions, are expressed by themulti-index Mittag-Leffler functions (13) (and the most of them by (12)), notonly by the Wright generalized hypergeometric functions (14). In what followswe essentially use these relations.


5. Integer order derivatives

In order to continue our study, we firstly need some results for the integer orderderivatives.

Recently, it has been obtained by Bazhlekova and Dimovski [1] that then-th derivative of the 2-parametric Mittag-Leffler function gives a 3-parametricMittag-Leffler function, introduced by Prabhakar (up to a constant), namely

E(n)α,β(z) = n!En+1

α,β+nα(z). (21)

Following the analogy, the n-th derivative of the generalized Bessel functionsin Section 2 are calculated. For this purpose we are based on the followingresult giving the n-th derivative of the multi-index Mittag-Leffler function (13),multiplied by a suitable power function (for details and proofs see [16] and [19]).

Theorem 1. Let z ∈ C\(−∞, 0] be a complex variable and let αi, βi, γi, ω ∈C, Re(αi) > 0, Re(βi0) > 0, i = 1, . . . ,m, 1 ≤ i0 ≤ m, i0 ∈ N, then for anyn ∈ N the following identity holds:

Dn[zβi0−1E

(γi), m(αi), (βi)

(ωzαi0 )] =

(d

dz

)n

[zβi0−1E

(γi),m(αi), (βi)

(ωzαi0 )]

= zβi0−n−1E

(γi), m

(αi), (β̃i)(ωzαi0 ), | arg z| < π, (22)

with β̃i0 = βi0 − n and β̃i = βi, if i 6= i0.

Remark 2. In particular, as a result of (22), we obtain the followingidentities:

Dn[zβ−1Eγα, β(ωz

α)] = zβ−n−1Eγα, β−n(ωz

α),

Dn[zβi0−1Em

(αi), (βi)(ωzαi0 )] = zβi0

−n−1Em(αi), (β̃i)

(ωzαi0 ), (23)

with β̃i0 = βi0 − n and β̃i = βi, if i 6= i0. These results follow, takinginto account (11) and (12), if m = 1, respectively, γ1 = · · · = γm = 1. Thementioned relations are well known, for which see Refs. [6] and [8].

The results below refer to the n-th derivatives of the generalized Bessel–Lomel functions (7) and (8). It turns out that, in the most cases, these integerorder derivatives are expressed by functions of the same kind, up to a constant.Further, due to the relation Jµ,1

ν,σ = Jµν,σ, the results obtained can be reduced


to the 3-parametric generalized Bessel–Maitland functions (5) and (6), up to apower function.

Theorem 3. Let z ∈ C \ (−∞, 0] be a complex variable, and let ν, σ ∈ C,q ∈ N, be parameters. Then the following equalities hold true for all the valuesof n ∈ N0 :

Dn[z−σ J2,q

ν,σ(z)]=

dn

dzn[z−σ J2,q

ν,σ(z)]= 2−nz−σ J2,q

ν−n,σ(z), (24)

for Re(ν + σ + 1) > 0,

Dn[zν/2 J1,q

ν,σ(2√z)]= zν/2−n/2 J1,q

ν−n,σ(2√z), (25)

for Re(ν + σ + 1) > 0, and

Dn[z−ν/2 Jµ,q

ν,σ (2√z)]= zσ−nE q+1

(µ,1,...,1), (σ+ν+1, σ−n+1, σ+1, ..., σ+1)(−z), (26)

with Re(σ + 1) > 0.

Proof. In what follows we denote for convenience

J̃µ,qν,σ (z) = (z/2)−ν−2σJµ,q

ν,σ (z), z 6= 0,

J̃µ,qν,σ (2

√z) = z−ν/2−σJµ,q

ν,σ (2√z), z 6= 0.

(27)

Let us start with proving (24). Using the first relation (27), the functionz−σ J2,q

ν,σ(z) can be represented as follows

z−σ J2,qν,σ(z) = z−σ (z/2)ν+2σ J̃2,q

ν,σ(z) = 2−ν−2σ zν+σ J̃2,qν,σ(z). (28)

Then, in view of (15), applying the formula (22) with ω = −1/4, αi0 = 2, andthe corresponding βi0 = σ + ν + 1, we obtain

Dn[zσ+ν J̃ 2,q

ν,σ (z)]= zσ+ν−n J̃ 2,q

ν−n,σ(z) =z−σ

2−(ν+2σ)2nJ 2,qν−n,σ(z). (29)

Now, using (28), the equality (24) immediately follows.

Further, for proving (25), we express zν/2 J1,qν,σ(2

√z) in an analogical way,

and we obtain:zν/2 J1,q

ν,σ(2√z) = zν+σ J̃ 1,q

ν,σ (2√z). (30)

Then, in view of (16) with ω = −1, αi0 = 1, and βi0 = σ + ν + 1, we can writedown that

Dn[zσ+ν J̃ 1,q

ν,σ (2√z)]= zσ+ν−n J̃ 1,q

ν−n,σ(2√z) = zν/2−n/2 J 1,q

ν−n,σ(2√z), (31)


from where

Dn[zν/2 J1,q

ν,σ(2√z)]= zσ+ν−n J̃ 1,q

ν−n,σ(2√z) = zν/2−n/2 J 1,q

ν−n,σ(2√z). (32)

In order to prove (26), we apply the formula (23) for ω = −1, αi0 = 1, andthe corresponding βi0 = σ + 1. Then, following the same idea as above, weobtain consequently

z−ν/2 J µ,qν,σ (2

√z) = z−ν/2 zν/2+σ J̃ µ,q

ν,σ (2√z) = zσ J̃ µ,q

ν,σ (2√z),

= zσ∞∑

k=0

(−z)k

Γ(µk + σ + ν + 1)(Γ(k + σ + 1))q.

ThereforeDn

[z−ν/2 Jµ,q

ν,σ (2√z)]= Dn

[zσ J̃ µ,q

ν,σ (2√z)]

= zσ−n∞∑

k=0

(−z)k

Γ(µk + σ + ν + 1)Γ(k + σ − n+ 1)(Γ(k + σ + 1))q−1

= zσ−nE q+1(µ,1,...,1), (σ+ν+1, σ−n+1, σ+1, ..., σ+1)(−z),

which proves (26).

Remark 4. Putting q = 1 in the equalities (24), (25) and (26), they givethe corresponding results for the functions (5) and (6).

Corollary 5. Let z ∈ C \ (−∞, 0] be a complex variable and let ν ∈ C bea parameter with Re(ν +1) > 0. Then the following equalities hold true for allthe values of n ∈ N0 :

Dn[zν/2 Jν(2

√z)]= zν/2−n/2 J1,1

ν−n,0(2√z) = zν/2−n/2 Jν−n(2

√z). (33)

Proof. The truthfulness of (33) immediately follows from (25), taking q = 1and σ = 0.

Theorem 6. Let z ∈ C \ (−∞, 0] be a complex variable and let µ, ν, bethe parameters, satisfying the conditions µ > 0, ν ∈ C. Then the followingequalities hold true for all the values of n ∈ N0 :


Dn [Jµν (z)] = z−(ν+n)/2 Jµ

ν+n,−n (2√z), (34)

and

Dn [zνCν (z)] = zν−nCν−n (z) with Re(ν + 1) > 0. (35)

Proof. Bearing in mind that Jµν (z) can be obtained from (12) taking α1 = 1,

β1 = 1, α2 = µ and β2 = ν+1, we apply (23) with αi0 = 1, βi0 = 1 and ω = −1.Before proving (34), we firstly note that its left-hand-side can also be writtenin the next form:

Dn [Jµν (z)] = z−n

∞∑

k=0

(−z)k

Γ(k − n+ 1)Γ(µk + ν + 1)

= z−n J̃µν+n,−n (2

√z).

(36)

Then, in view of (27), it follows

Dn [Jµν (z)] = z−(ν+n)/2 Jµ

ν+n,−n (2√z),

which proves (34).

Further putting µ = 1 in Jµν leads to Cν . Applying after that the second

formula (23) with αi0 = 1 and βi0 = ν + 1 leads to the following result

Dn [zνCν (z)] = zν−n∞∑

k=0

(−z)k

Γ(k + 1)Γ(k + ν − n+ 1)

= zν−nCν−n (z).

(37)

Thus the correctness of (35) follows.

In particular, taking µ = 1 in (34), Theorem 6 gives result referring to theclassical Bessel–Clifford functions, i.e. we produce the following corollary.

Corollary 7. Let the variable z ∈ C \ (−∞, 0] and the parameter ν ∈ C.Then the following equalities hold true for all the values of n ∈ N0 and z 6= 0 :

DnCν(z) = [Cν(z)](n) = z−(ν+n)/2 J1

ν+n,−n (2√z). (38)


Proof. Putting µ = 1, the function Jµν (z) becomes Cν(z). Therefore the

following logical conclusions can be deduced from (4) and (34):

DnCν(z) = [Cν(z)](n) =

[J1ν (z)

](n)= z−(ν+n)/2 J1

ν+n,−n (2√z),

which complete the proof.

Another interesting result, connecting the multi-index function (13) and then-th derivative of (12) is formulated below.

Theorem 8. Let αi, βi, z ∈ C and let Re(αi) > 0 for i = 1, . . . ,m. Thenthe following equality holds true for all the values of n ∈ N0:

Dn[E(αi), (βi) (wz)

]= wn Γ(n+ 1)E

(γi),m(αi), (βi+nαi)

(wz), (39)

with γ1 = n+ 1, γ2 = · · · = γm = 1.

Proof. The result has been proved for w = 1 in [20]. The proof for (39) iscompletely similar (it can be also made as the n-th derivative of a compositefunction) and because of that details are omitted.

Remark 9. Taking m = 1 and w = 1, the equality (39) gives the relation(21), obtained by Bazhlekova and Dimovski in [1].

Further, the corresponding results for the 2-, 3-, and 4-index generalizationsof the Bessel functions are given.

Theorem 10. Let z be a complex variable and let µ, ν, σ be the param-eters, satisfying the conditions ν, σ ∈ C, µ > 0. Then the following equalitieshold true for all the values of n ∈ N0 :

Dn[z−ν/2−σ Jµ,q

ν,σ (2√z)]= (−1)n Γ(n+ 1)E

(γi),q+1

(αi),(β̃i)(−z), (40)

for z 6= 0 and with

α1 = µ, α2 = · · · = αq+1 = 1, γ1 = n+ 1, γ2 = · · · = γq+1 = 1,

β̃1 = σ + ν + 1 + nµ, β̃2 = · · · = β̃q+1 = σ + n+ 1,(41)

Dn [Jµν (z)] =

dn

dzn[Jµ

ν (z)] = (−1)n Jµν+nµ (z), z ∈ C, (42)

and

Dn [Cν (z)] =dn

dzn[Cν (z)] = (−1)n Cν+nµ (z), z ∈ C. (43)


Proof. Let us begin with the proof of (40). Using (16) and (39) with w =−1, along with it, we have

Dn[z−ν/2−σ Jµ,q

ν,σ (2√z)]= Dn

[E q+1

(µ,1,...,1), (σ+ν+1,σ+1,...,σ+1)(−z)]

= (−1)n Γ(n+ 1)E(γi),q+1

(αi),(β̃i)(−z),

(44)

with the same αi, β̃i, γi as in (41) (i = 1, . . . , q + 1).In order to prove (42), we use (19) and (39) with w = −1, and we obtain

Dn Jµν (z) = DnE(µ,1), (ν+1,1)(−z)

= (−1)n Γ(n+ 1)E(n+1,1),2(µ,1), (nµ+ν+1,n+1) (−z) = (−1)n Jµ

ν+nµ (z),(45)

which is exactly (42).Now, if we put µ = 1 in the formula (42), it is reduced to (43).

Corollary 11. Let z be a complex variable and let µ, ν, q be the pa-rameters, satisfying the conditions ν ∈ C, µ > 0, q ∈ N. Then the followingequalities hold true for all the values of n ∈ N0 :

Dn[z−ν/2 Jµ,q

ν,0 (2√z)]= (−1)n E q+1

(αi),(β̃i)(−z), z 6= 0, (46)

withα1 = µ, α2 = · · · = αq+1 = 1,

β̃1 = ν + 1 + nµ, β̃2 = 1 + n, · · · = β̃q+1 = 1 + n.(47)

Proof. It immediately follows taking σ = 0 in (40) and (41).

Remark 12. Putting q = 1 in the equalities (40), and (46), they givethe corresponding results for the three-parametric generalized Bessel–Maitlandfunctions (6).

6. Fractional Riemann–Liouville

integral and derivative

The most popular definition for integration of arbitrary (i.e. not obligatorilyinteger) order λ ∈ C (Re(λ) > 0) is the Riemann–Liouville (R-L) fractional

integral [25]


Rλf(z) =1

Γ(λ)

z∫

0

(z − t)λ−1f(t)dt =zλ

Γ(λ)

1∫

0

(1− τ)λ−1f(zτ)dτ . (48)

The corresponding Riemann–Liouville fractional derivative of order λ isdefined as a composition of a derivative of integer order and an integral offractional order of the form (48), namely:

Dλf(z) := DnRn−λf(z), (49)

where n := [Re(λ)] + 1 > Re(λ), [Re(λ)] = integer part of Re(λ).

For the theory and applications of the fractional Riemann–Liouville integraland derivative, as well as of other integral and differential operators of fractionalcalculus (FC), see the FC encyclopedia [25] and monograph [7], for the evolutionand development of FC, see [15]. For miscellaneous useful applications of anumber operators of FC, see also the recent survey paper [2].

Theorem 13. Let the variable z ∈ C \ (−∞, 0] be a complex variable andlet αi, βi, γi, ω, λ ∈ C, Re(αi) > 0, Re(βi0) > 0, Re(λ) > 0, i = 1, . . . ,m,1 ≤ i0 ≤ m, i0 ∈ N. Then the following identity holds:

Rλ[zβi0−1E

(γi),m(αi), (βi)

(ωzαi0 )] = zβi0+λ−1E

(γi), m

(αi), (β̃i)(ωzαi0 ), (50)

with β̃i0 = βi0 + λ and β̃i = βi, if i 6= i0.


Rλ[zβ−1Eγα, β(ωz

α)] = zβ+λ−1Eγα, β+λ(ωz

α),

Rλ[zβi0−1Em


+λ−1Em(αi), (β̃i)

(ωzαi0 ), (51)

with β̃i0 = βi0 + λ and β̃i = βi, if i 6= i0. These results follow, taking intoaccount (11) and (12), if m = 1 respectively γ1 = · · · = γm = 1. The mentionedrelations are well known, for which see Refs. [6] and [8].

Using the relations in Section 4 along with Theorem 13, allows us to obtainthe corresponding results connected with the functions of Bessel type.


Theorem 15. Let z ∈ C \ (−∞, 0] be a complex variable and let µ, ν bethe parameters, satisfying the conditions µ > 0, ν, λ ∈ C and Re(λ) > 0. Thenthe following equalities hold true:

Rλ[z−σ J2,q

ν,σ(z)]= 2λz−σ J2,q

ν+λ,σ(z), (52)

Rλ[zν/2 J1,q

ν,σ(2√z)]= zν/2+λ/2 J1,q

ν+λ,σ(2√z), (53)

if additionally Re(σ + ν + 1) > 0. If Re(σ + 1) > 0, then

Rλ[z−ν/2 Jµ,q

ν,σ (2√z)]= zσ+λE q+1

(µ,1,...,1), (σ+ν+1, σ+λ+1, σ+1, ..., σ+1)(−z). (54)

Proof. In view of (28) we can write

z−σ J2,qν,σ(z) = 2−ν−2σ zν+σ J̃2,q

ν,σ(z). (55)

Applying after that the formula (50) for ω = −1/4, αi0 = 2, and the corre-sponding βi0 = σ + ν + 1, we have

Rλ[zσ+ν J̃ 2,q

ν,σ (z)]= zσ+ν+λ J̃ 2,q

ν+λ,σ(z) =2λ z−σ

2−(ν+2σ)J 2,qν+λ,σ(z).

Now, using (55), the equality (52) immediately follows. The proofs of (53) and(54) follow the same idea and they are omitted.

Remark 16. Putting q = 1 in the equalities (52), (53) and (54), they givethe corresponding results for the 3-index functions (5) and (6).

Corollary 17. Let z ∈ C\(−∞, 0] be a complex variable and let λ, ν ∈ C

be parameters, with Re(λ) > 0 and Re(ν+1) > 0. Then the following equalitieshold true

Rλ[zν/2 Jν(2

√z)]= zν/2+λ/2 J1,1

ν+λ,0(2√z) = zν/2+λ/2 Jν+λ(2

√z). (56)

Proof. It follows by (53) taking q = 1 and σ = 0.


Theorem 18. Let z ∈ C \ (−∞, 0] be a complex variable and let λ, µ, νbe the parameters, satisfying the conditions µ > 0, λ, ν ∈ C, Re(λ) > 0. Thenthe following equalities hold true

Rλ [Jµν (z)] = z−(ν−λ)/2 Jµ

ν−λ,λ (2√z), (57)

and

Rλ [zνCν (z)] = zν+λCν+λ (z) for Re(ν + 1) > 0. (58)

Proof. Since Jµν (z) can be obtained from (12) taking α1 = 1, β1 = 1, α2 = µ

and β2 = ν + 1, we apply the second formula (51) with αi0 = 1, βi0 = 1 andω = −1. Before proving (57), we firstly note that its left-hand-side can also bepresented as follows

Rλ [Jµν (z)] = zλ

∞∑

k=0

(−z)k

Γ(k + λ+ 1)Γ(µk + ν + 1)= zλJ̃µ

ν−λ,λ (2√z). (59)


Rλ [Jµν (z)] = z−(ν−λ)/2 Jµ

ν−λ,λ (2√z),

which proves (57).Further putting µ = 1 in Jµ

ν leads to Cν . Applying again the second formula(51) with αi0 = 1 and βi0 = ν + 1 leads to the following result

Rλ [zνCν(z)] = zν+λ∞∑

k=0

(−z)k

Γ(k + 1)Γ(k + ν + λ+ 1)= zν+λCν+λ(z). (60)


In particular, taking µ = 1, Theorem 18 gives a result referring to theclassical Bessel–Clifford functions, i.e. we produce the following corollary.

Corollary 19. Let the variable z ∈ C \ (−∞, 0] and the parameter ν ∈ C.Then the following equalities hold true

RλCν(z) = z−(ν−λ)/2 J1ν−λ,λ (2

√z). (61)


Other result, which we consider, refers to the Riemann–Liouville derivativeof the multi-index Mittag-Leffler function (13) (for details and proof see [16]and [19]).

Theorem 20. Let z ∈ C\(−∞, 0] be a complex variable and let αi, βi, γi, ω,λ ∈ C, Re(αi) > 0, Re(βi0) > 0, Re(λ) > 0, i = 1, . . . ,m, 1 ≤ i0 ≤ m, i0 ∈ N,then the following identity holds

Dλ[zβi0−1E

(γi),m(αi), (βi)

(ωzαi0 )] = zβi0−λ−1E

(γi), m

(αi), (β̃i)(ωzαi0 ), (62)

with β̃i = βi, if i 6= i0, and β̃i0 = βi0 − λ.


Dλ[zβ−1Eγα, β(ωz

α)] = zβ−λ−1Eγα, β−λ(ωz

α),

Dλ[zβi0−1Em


−λ−1Em(αi), (β̃i)

(ωzαi0 ), (63)

with β̃i0 = βi0 − λ and β̃i = βi, if i 6= i0. These results follow, takinginto account (11), if m = 1, and respectively (12), if γ1 = · · · = γm = 1. Thementioned relations are well known, for which see Refs. [6] and [8].

Theorem 22. Let z ∈ C \ (−∞, 0] be a complex variable and let ν ∈ C

be a parameter. Then the following equalities hold true

Dλ[z−σ J2,q

ν,σ(z)]= 2−λz−σ J2,q

ν−λ,σ(z), (64)

Dλ[zν/2 J1,q

ν,σ(2√z)]= zν/2−λ/2 J1,q

ν−λ,σ(2√z), (65)

if additionally Re(σ + ν + 1) > 0. If Re(σ + 1) > 0, then

Dλ[z−ν/2 Jµ,q

ν,σ (2√z)]= zσ−λE q+1

(µ,1,...,1), (σ+ν+1, σ−λ+1, σ+1, ..., σ+1)(−z), (66)

Proof. Using (27), the function z−σ J2,qν,σ(z) can be expressed as follows

z−σ J2,qν,σ(z) = 2−ν−2σ zν+σ J̃2,q

ν,σ(z). (67)


Then, in view of (15), applying the formula (63) with ω = −1/4, αi0 = 2, andthe corresponding βi0 = σ + ν + 1, we obtain

Dλ[zσ+ν J̃ 2,q

ν,σ (z)]= zσ+ν−λ J̃ 2,q

ν−λ,σ(z) =z−σ

2−(ν+2σ)2λJ 2,qν−λ,σ(z). (68)

Now, using (67), the equality (64) immediately follows.

Further, proving (65), we express zν/2 J1,qν,σ(2

√z) in an analogical way, and

we obtain:zν/2 J1,q

ν,σ(2√z) = zν+σ J̃ 1,q

ν,σ (2√z). (69)

Then, in view of (16) with ω = −1, αi0 = 1, and βi0 = σ + ν + 1, we can writedown that

Dλ[zσ+ν J̃ 1,q

ν,σ (2√z)]= zσ+ν−λ J̃ 1,q

ν−λ,σ(2√z) = zν/2−λ/2 J 1,q

ν−λ,σ(2√z), (70)

from where

Dλ[zν/2 J1,q

ν,σ(2√z)]= zν/2−λ/2 J 1,q

ν−λ,σ(2√z). (71)

In order to prove (66), we apply the formula (63) for ω = −1, αi0 = 1, andthe corresponding βi0 = σ + 1. Then, following the same idea as above, weobtain consequently

z−ν/2 J µ,qν,σ (2

√z) = z−ν/2 zν/2+σ J̃ µ,q

ν,σ (2√z) = zσ J̃ µ,q

ν,σ (2√z),

= zσ∞∑

k=0

(−z)k

Γ(µk + σ + ν + 1)(Γ(k + σ + 1))q.

ThereforeDλ

[z−ν/2 Jµ,q

ν,σ (2√z)]= Dλ

[zσ J̃ µ,q

ν,σ (2√z)]

= zσ−λ∞∑

k=0

(−z)k

Γ(µk + σ + ν + 1)Γ(k + σ − λ+ 1)(Γ(k + σ + 1))q−1

= zσ−λE q+1(µ,1,...,1), (σ+ν+1, σ−λ+1, σ+1, ..., σ+1)

(−z),

which proves (66).

Remark 23. Putting q = 1 in the equalities (64), (65) and (66), they givethe corresponding results for the corresponding 3-parametric functions (5) and(6).


In particular, if q = 1 and σ = 0, the formula (65) produces a result for theclassical Bessel function. Namely, the following corollary can be formulated.

Corollary 24. Let z ∈ C\ (−∞, 0] be a complex variable and let ν be theparameter, satisfying the conditions ν ∈ C, Re(ν + 1) > 0. Then the followingequalities hold true

Dλ[zν/2 Jν(2

√z)]= zν/2−λ/2 J1,1

ν−λ,0(2√z) = zν/2−λ/2 Jν−λ(2

√z). (72)

Theorem 25. Let z ∈ C \ (−∞, 0] be a complex variable and let µ, νbe the parameters, satisfying the conditions ν ∈ C, µ > 0. Then the followingequalities hold true

Dλ [Jµν (z)] = z−(ν+λ)/2 Jµ

ν+λ,−λ (2√z), (73)

andDλ [zνCν (z)] = zν−λ Cν−λ (z), for Re(ν + 1) > 0. (74)

Proof. Expressing Jµν (z) from (12) with α1 = 1, β1 = 1, α2 = µ and

β2 = ν + 1, and applying after that the second formula (63) with αi0 = 1,βi0 = 1 and ω = −1, we obtain

Dλ [Jµν (z)] = z−λ

∞∑

k=0

(−z)k

Γ(k − λ+ 1)Γ(µk + ν + 1)

= z−λ J̃µν+λ,−λ (2

√z).

(75)


Dλ [Jµν (z)] = z−(ν+λ)/2 Jµ

ν+λ,−λ (2√z),

which proves (73).Further setting µ = 1 in Jµ

ν gives Cν . Applying again the second formula(63) with αi0 = 1 and βi0 = ν + 1 leads to the following result

Dλ [zνCν (z)] = zν−λ∞∑

k=0

(−z)k

Γ(k + 1)Γ(k + ν − λ+ 1)

= zν−λ Cν−λ (z).

(76)



In particular, taking µ = 1, Theorem 25 gives a result referring to theclassical Bessel–Clifford functions, i.e. we produce the following corollary.

Corollary 26. Let z ∈ C \ (−∞, 0] be a complex variable and the param-eter ν ∈ C. Then the following equality holds true

DλCν(z) = z−(ν+λ)/2 J1ν+λ,−λ (2

√z). (77)

Another interesting result, also connected with the differentiation of anarbitrary order, is given by the theorem below.

Theorem 27. Let αi, βi, λ̃ ∈ C and Re(αi) > 0 for i = 1, . . . ,m, and letalso 0 < Re(λ̃) < 1, n ∈ N, and λ = n− 1 + λ̃. Then it holds

Dλ[zλ̃E(αi), (βi)(wz)] = wn−1 Γ(λ+ 1)E(γi)

(αi), (β̃i)(wz) (for | arg z| < π), (78)

with parameters β̃i = βi + (n− 1)αi and γ1 = λ+ 1, γ2 = · · · = γm = 1.

Proof. The proof of this theorem is omitted, because it follows completelythe same idea as that for w = 1 (for details see [20]).

Further, if 0 < Re(λ) < 1, then n = 1, ωn−1 = 1, λ̃ = λ and β̃i = βi. Thenthe following corollary can be produced.

Corollary 28. Let z ∈ C \ (−∞, 0] be a complex variable and the pa-rameters ν ∈ C, αi, βi, λ̃ ∈ C and Re(αi) > 0 for i = 1, . . . ,m, and let also0 < Re(λ) < 1. Then the following equality hold true

Dλ[zλE(αi), (βi)(wz)] = Γ(λ+ 1)E(γi)(αi), (βi)

(wz) (for | arg z| < π), (79)

with parameters γ1 = λ+ 1, γ2 = · · · = γm = 1.

Remark 29. If m = 1 in the equalities (78) and (79), then they give thecorresponding relations for the functions (11) and the derivative of order λ ofthe function (3). More precisely, (78) and (79) are reduced to

Dλ[zλ̃Eα, β(wz)] = wn−1 Γ(λ+ 1)Eγ

α, β̃(wz) (for | arg z| < π), (80)


with parameters β̃ = β + (n− 1)α and γ = λ+ 1, respectively

Dλ[zλEα, β(wz)] = Γ(λ+ 1)Eγα, β(wz) (for | arg z| < π), (81)

with parameter γ = λ+ 1.

Theorem 30. Let αi, βi, λ ∈ C and Re(αi) > 0 for i = 1, . . . ,m.Moreover, let 0 < Re(λ) < 1. Then

Rλ[z−λE(αi), (βi)(ωz)] = Γ(1− λ)E(γi)(αi), (βi)

(ωz) (for | arg z| < π), (82)

with parameters γ1 = 1− λ, γ2 = · · · = γm = 1.

The theorem is proved in [20]. As a corollary, taking m = 1 in Theorem 30,the corresponding result for the functions (3) and (11) can be formulated.

Corollary 31. Let α, β, λ ∈ C, and let Re(α) > 0 and 0 < Re(λ) < 1.Then

Rλ[z−λEα, β(ωz)] = Γ(1− λ)E1−λα, β (ωz) (for | arg z| < π). (83)

Now, the corresponding results, referring to the Bessel type functions canbe obtained due to the relations in Section 4, and they are given below.

Theorem 32. Let αi, βi, λ ∈ C and Re(αi) > 0 for i = 1, . . . ,m.Moreover, let 0 < Re(λ) < 1. Then

Rλ[z−λJµν (z)] = Γ(1− λ)E

(1−λ,1)(µ,1) (ν+1,1)(−z)

= Γ(1− λ)∞∑

k=0

(1− λ)k(1)kk! Γ(µk + ν + 1)

(−z)k

(k!)2(for | arg z| < π).

(84)

Proof. Using the formulae (19) and (82), we have

Rλ[z−λJµν (z)] = Rλ[z−λE(µ,1) (ν+1,1)(−z)]

= Γ(1− λ)E(1−λ,1)(µ,1) (ν+1,1)(−z)

= Γ(1− λ)

∞∑

k=0

(1− λ)k(1)kk! Γ(µk + ν + 1)

(−z)k

(k!)2,

which proves the theorem.


In the particular case, if µ = 1 and ν = −λ, Theorem 32 produces thefollowing corollary for the Bessel–Clifford functions.

Corollary 33. Let αi, βi, λ ∈ C and Re(αi) > 0 for i = 1, . . . ,m.Moreover, let 0 < Re(λ) < 1. Then

Rλ[z−λC−λ(z)] = C0(z) (for | arg z| < π). (85)

Proof. Using the formulae (19), (20) and (84), we have

Rλ[z−λC−λ(z)] = Rλ[z−λJ1

−λ(z)]

= Γ(1− λ)

∞∑

k=0

(1− λ)k(1)kk! Γ(k + 1− λ)

(−z)k

(k!)2=

∞∑

k=0

(−z)k

(k!)2= C0(z),

which proves the corollary.

In conclusion, we can summarize that this survey paper refers to variousdifferential and integral relations in the class of the multi-index Mittag-Lefflerfunctions. More precisely, a part of them connects the derivatives and integralsof the 2m-parametric Mittag-Leffler functions with the 3m-parametric Mittag-Leffler functions, and the other part is between functions with the same numberof indices. As special cases the relations are for different representatives of thisclass, all of them of Bessel type. In general the result of differentiation andintegration of a multi-index Mittag-Leffler function is also a function of thesame class (up to a constant and a power function). However, for the Besseltype functions the facts are sometimes different. There are three groups ofresults given.

The first group consists of relations, where the results of differentiations andintegrations are expressed by multi-index Mittag-Leffler functions but they inparticular are not of Bessel type. In the second group the relations are betweenBessel type functions with different number of indices, and in the third groupthey are with the same number of indices.

There are obtained interesting relations between the integrals and deriva-tives of the 4-index Bessel type functions Jµ,q

ν,σ , given by (7) and (8), on the onehand, and the multi-index Mittag-Leffler functions (12) and (13) on the otherhand, and also between (7) and (8) and the corresponding high-order derivativesand fractional Riemann–Liouville integrals and derivatives of them. In particu-lar, for q = 1 the results refer to the 3-index functions (5) and (6). We emphasize


that there are given simple relations between the 3-index generalized Bessel–Maitland functions (6) and the corresponding fractional Riemann–Liouville in-tegrals and derivatives of the 2-index Bessel–Maitland functions Jµ

ν (z), definedby (4). More precisely these integrals and derivatives are expressed by the 3-index functions (6), up to matching power functions. Besides, the derivativesof integer order n of (4) can also be expressed by a function of the kind (4),multiplied by the constant (−1)n. Finally, the results for the classical Besseland Bessel–Clifford functions merely follow as corollaries.

Acknowledgments

This paper is performed in the frames of the Bilateral Res. Projects ‘Analyticaland numerical methods for differential and integral equations and mathematicalmodels of arbitrary (fractional or high integer) order’ between BAS and SANUand ‘Analysis, Geometry and Topology’ between BAS and MANU.

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a survey on bessel type functions · interesting relations between these integrals and derivatives...

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