krawtchouk polynomials and the symmetrization of …people.virginia.edu/~der/pdf/der33.pdfsection 5...

SIAM J. MATH. ANAL.Vol. 5, No. 3, May 1974

KRAWTCHOUK POLYNOMIALS AND THE SYMMETRIZATION OFHYPERGROUPS*

CHARLES F. DUNKL AND DONALD E. RAMIREZ"

Abstract. This paper introduces the method of symmetrization of the measure algebra of a compactP,-hypergroup. This method is used to form a measure algebra whose characters are Krawtchoukpolynomials (these are the finite sets ofpolynomials orthogonal with respect to the binomial distributionon {0, 1, ..., N}). As a further application, one derives a theorem about nonnegative expansions of onefamily of Krawtchouk polynomials in terms of another family.

Introduction. The role of orthogonal polynomials in harmonic analysis haslong been of interest. For example, certain sets of orthogonal polynomials appearas sets of spherical functions of compact homogeneous spaces. In this paper weintroduce the method of symmetrization of a measure algebra and use it to form ameasure algebra whose characters are Krawtchouk polynomials (these are thefinite sets of polynomials orthogonal with respect to the binomial distribution on{0, 1, ..., N)). This method also makes it possible to prove a theorem about theexpansion of Krawtchouk polynomials of one family in series with positive co-efficient of the polynomials from another family.

The underlying structure is that ofa compact P,-hypergroup. This is a compactspace on which the space of finite regular Borel measures has a convolution struc-ture preserving the probability measures. Suppose further that a compact groupof automorphisms acts on the hypergroup; then the set of measures invariantunder the action of the automorphism group forms a subalgebra, which itselfhas the structure of the measure algebra of a hypergroup.

In the first section of the paper we give the definitions and basic theoremsabout compact P,-hypergroups, taken from Dunkl [23. In 2 we discuss theaction of a compact group of automorphisms on a compact P,-hypergroup,and show the existence of a symmetrization operator on the continuous functionson the hypergroup. In 3 and 4 we show that the algebra of measures invariantunder the automorphism group is the measure algebra of a P,-hypergroup.Some key formulas are also determined.

Section 5 contains the harmonic analysis structure of the Krawtchoukpolynomials. The idea is to take the N-fold Cartesian product of a two-point P,-hypergroup and let the permutation group on N letters act on the product bypermutation of the coordinates. The characters of the symmetrized product arethe Krawtchouk polynomials. To illustrate the techniques of 3 and 4 we computethe product theorems for these polynomials.

Section 6 discusses the two kinds of homomorphisms of P,-hypergroupsand uses the symmetrization technique to give nonnegative expansions of onefamily of Krawtchouk polynomials in terms of another family.

* Received by the editors November 21, 1972, and in revised form March 28, 1973."f Department of Mathematics, University of Virginia, Charlottesville, Virginia 22903. This work

was supported in part by the National Science Foundation under Contract GP-31483X.

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352 CHARLES F. DUNKL AND DONALD E. RAMIREZ

1. Basic definitions and facts about hypergroups.DEFINITION 1.1. A locally compact space H is called a hypergroup if there

exists a map :H H -, Mp(H) (the space of probability measures), with thefollowing properties:

(i) 2(x, y) 2(y, x), x, y 6 H (so H is a commutative hypergroup).(ii) For eachf6 Co(H) (the space of continuous functions on H with compact

support), the map (x, y)-,fnfd2(x, y) is in CB(H H) (the space of continuousbounded functions on H H), and the map x-,f d2(x, y) is in Cc(H) for eachyH.

(iii) The convolution on M(H) defined implicitly by

fu f d# * v fu d(x) fu dr(y)fufd2(x,y),#, v M(H), f Co(H) (the space of continuous functions vanishing at infinity)is associative.

(iv) There exists a unique point e H such that 2(x, e) di,, x H.Remark 1.2. A compact space H is a hypergroup if and only if the space M(H)

of regular Borel measures on H is a commutative Banach algebra and the spaceM,(H) of probability measures on H is a compact commutative topological semi-group with unit in the weak* topology [2].

DEFINITION 1.3. For the hypergroup H with f C(H), x H, and/ 6 M(H),define R(x)f 6 Cc(H) by R(x)f(y)= ff d2(x, y), y 6 H;and define the functiong(/a)f Co(H) by g(/)f(y) g(z)f(y) d#(z), y 6 H. (That R(/)f Co(H) isshown in [2, Thm. 1.10].)

DEFINITION 1.4. An invariant measure rn on the hypergroup H is a positivenonzero regular Borel measure on H which is finite on compact sets and such thatfdm= R(x)f din, x H, f Co(H).DEFINITION 1.5. If a hypergroup H has an invariant measure m, and a con-

tinuous involution x -, x’, x 6 H, such that fn (g(x)f), dm f{g(x’)g}- dm,f, g C(H), x H, and such that e 6 spt 2(x, x’), x 6 H and spt denoting the supportof a measure, then H is called a *-hypergroup.

Remark 1.6. The invariant measure m of a *-hypergroup is unique up to aconstant [2, Prop. 3.2]. If H is compact, we denote the normalized invariantmeasure ofH by ran, ran(H) 1.

DEFINITION 1.7. A nonzero function 4)6 CB(H) is called a character of thehypergroup H if the following formula holds:

4(x)ck(y) f,, 4 d,(x, y), x,yeH.

The set of all characters is denoted by/. (For e/-), Ib(x)l b(e) 1, x e H,[2, Prop. 2.2] .)

Remark 1.8. For compact *-hypergroups,/-) is an orthogonal basis for L2(H),and/ is discrete in the weak* topology from L2(H), [2, Thm. 3.5].

DEFINITION 1.9. If a compact *-hypergroup H has the further property that/// c co () (the convex hull of ), then H is called a compact P,-hypergroup.

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KRAWTCHOUK POLYNOMIALS 353

Remark 1.10. If H is a compact P.-hypergroup, then the linear span of/ issup-norm dense in C(H) (see [2, Prop. 3.8]). For th, e/, we write

(x)@(x) ,^ n(cp, @; co)co(x), x e H.H

DHNTION 1.11. For H a compact P,-hypergroup, and e , let

c(@) 1[2 dmn 1.

Then B (with the measure c and conjugation as the involution) is a ,-hypergroup[2, Thm. 3.9].

THEOREM 1.12. Let H be a compact P,-hypergroup. For x H,

s Z c()l(x)l

Proof. First note that &()=, x e H. By the Plancherel theorem forhypergroups [2, Thm. 3.5], eL(H)= l(R) if and only if S < . Also

L(H)if and only if m({x}) > 0. Finally, in this case,

1c(@)l@(xjl 2

m.({x}) ,COROLLARY 1.13. If H is a compact P,-hypergroup, then mH({e}) <-- mH({X}),

xeH.Poof. s <_ Z c4) s.COrOLLarY 1.14. If H is a denumerable compact P,-hypergroup, then the

identity e is a cluster point.Remark 1.15. The authors know of an example of a denumerable compact

P,-hypergroup, and will discuss it in a future paper. The hypergroup comes fromhaving the group of units of Ap (the p-adic integers) act on Ap.

2. Automorphisms on hypergroups. In this section H Will be a compact P.-hypergroup.

DEFINITION 2.1. Let W be a compact group of homeomorphisms on thecompact space X. The topology on W is the pointwise topology from X, and themap (x, z)-, z(x) of X x W - X is separately continuous.

For z W and f e C(X), define z lf C(X) by z lf(x) f(zx), x e X. Letbe the (weak* continuous) adjoint of z, that is, x fdz, x fo z dl,fe C(X),la e M(X).

DEFINITION 2.2. An automorphism on the compact P,-hypergroup H is ahomeomorphism such that ]’2(x, y) 2(zx, y), x, y e H. Thus for 4) e

fno d2(x, y), x,yeH.

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Also z(x)’ z(x’), x H" for

o (x’)= o (x)= (x)= ((x)’).

THEOREM 2.3. Let W be a compact group of automorphisms on the compactP,-hypergroup H. Then the set O(dp)= {d z I’z W}, d) I, called the orbitof d?, is afinite subset of.

Proof. The set O(b) is compact in the pointwise topology from H, and hencein the weak topology (as a subset of C(H)). Recall from Remark 1.8, is discretein the weak* topology (as a subset of L2(H)), and thus in the weak topology (as asubset of C(H)) and so 0(4)) is a finite subset of/.

THEOREM 2.4. Let W be a compact group ofautomorphisms on the compact P,-hypergroup H. Then the space W is totally disconnected.

Proof. For /, let A,= {z e W’b z }. The set A, is an open neighbor-hood ofthe identity ewin W;and for Zl,Z2 A4,,’c1"2 A4,’( -cl"c2)(x) # -c2(x)

b(x), x e H. Also A, is inverse closed" for z e A, and x s H with :(x) y, thenb(y) 4)o z(x) b(x)= 4)o z-l(y). Thus A, is an open (and closed) subgroupof W, and CI {A," 4 e/} {ew}. This implies that W is totally disconnected.

3. Symmetrization of hypergroups. In this section H will be a compactP,-hypergroup and W will be a compact group of automorphisms on H.

DEFINITION 3.1. Define the symmetrization operator 1 on C(H) by

fw f(’cx) dmw(’C), f e C(H), x e H,f(x)o"

where row,denotes the Haar measure on W. The function all is in C(H) (by theGrothendieck theorem that the pointwise and weak topologies are equivalent oncompact subsets of C(H); see also Glicksberg [5]). We let a :M(H) M(H) be the(weak* continuous)adjoint of a l. Note that a, a are projections.

Example 3.2. Let T denote the unit circle, and let W {ew, z} where (x) ,x T. Then for f C(T), 1 f(x) 1/2(f(x) + f(X)), x T.

DEFINITION 3.3. Let H be a compact P,-hypergroup and W a compact groupof automorphisms on H. We define the compact space Hw by identifying thepoints of H which are in the same orbit; that is, Hw H/ where x y if andonly if there exists : W such that zx y.

Let Cw(H) denote the space

{f e C(H):fo r f, all : e W}.

THEOREM 3.4 OlC(H) Cw(H).Proof. Let f e O C(H), x e H; then f O’lg, some g C(H), and f(x)w g(zx) draw(r,). Thus by the translation invariance of the Haar measure on W,

To p(x) w g(zpx) draw(Z) w g(zx) draw(Z) f(x), p e W; and so alC(H)c Cw(H).

Conversely, if f eCw(H), lf f since rlf(x)=wf(x)dmw()= f(x)w draw f(x), x e H. Thus Cw(H) al C(H).

DEFINITION 3.5. Let Mw(H p e M(H) fH f dp nf "c dp, all e W,fe C(H))}.

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THEOREM 3.6. Let H be a compact P,-hypergroup and W a compact group ofautomorphisms on H. Then Cw(H - C(Hw) and

aM(H) Mw(H) M(Hw).

Proof. That Cw(H) - C(Hw) is immediate from the definition of Hw. ThatMw(H) - M(Hw) follows from Theorem 3.4; that is,

M(Hw) Cw(H)* -(aC(H))* Mw(X).

Remark 3.7. For convenience, we often identify C(Hw), M(Hw) with Cw(H),Mw(H) respectively.

THEOREM 3.8. For #, v M(H), H a compact hypergroup, nd W a compactgroup ofautomorphisms, r(p * av) a * av.

Proof. For 2 e M(H), a2 is defined by n f da2 a f d2, f C(H). Sinceis weak* continuous, it will suffice to let 6, x e H.

Now for f C(H),

ffd * av ; dv(v) fda6(u)fd2(u, v)

fn dav(v) fn fw fu fd2(zu’v) dmw(z,d(u,

fH day,u, fW fHf d2’x’-lu’dmw’’

ffd( * v).

Cooa 3.9. The space M(H) M(H) is a commutative Banach algebra;and the space Mp(H) ofprobability measures in M(H) is isomorphic toand it is a commutative topological semigroup with unit (and thusH is a hypergroup).

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COROLLARY 3.10. amn mn and under the isomorphism of Mw(H) and M(Hw),mH is identified with mHw.

Proof. Since Hw is a compact hypergroup there exists an invariant measureon Hw which is unique [2, Thm. 1.12].

Let/ aM(H)p then amn * al a(mH * a#) a(mH * #) amH, and so amHis an invariant measure on Hw.

Further, for z W, Z*mH mH, since Z*mH is an invariant probability measureonH.

COROLLARY 3.11. If H’is afinite P,-hypergroup, then

mm({a})= L m({y)),yO(x)

where ax denotes the element ofnw (= n/) which contains x, and O(x) {y s H:there exists z W with zy x} the orbit ofx.

Proof. Let ZA denote the characteristic function of the set A, A H. Then

m({ax}) fu Z,x dmn

fu 7.o(x)dmu Z mu({y}).yO(x)

4. Duals of symmetrized hypergroups. In this section H is a compact P,-hypergroup and W is a compact group of automorphisms on H. Let a l, a be as in3.

PROPOSITION 4.1. For dp I7I, aidp . Iw.Proof. A continuous function g on Hw is a character if g defines a nonzero

multiplicative functional on M(nw) (see [2, Prop. 2.3]).For/ aM(H) - M(Hw),

a dp dla dp da# b d# #(b);

and so a$ /w.DEFINITION 4.2. DefineTHEOREM 4.3. Let dp, tct; then (#$). (#$) al(b- #$).Proof. Let x H; then

a,(dp. Oqt)(x) fw dp(,x)#q(xx) dmw(x)

#(x) fw dp(vx)draw(,)

,(x)(x).

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THEOREM 4.4. Let W be a compact group of automorphisms on the P,-hyper-group H. Then Hw is a compact *-hypergroup.

Proof. That Hw is a compact hypergroup is given in Corollary 3.9. Theinvariant measure on Hrv (as a hypergroup) is mn by Corollary 3.10.

The involution in Hrv is defined by (ax)’ a(x’), x H. This is well-definedsince (zx)’ z(x’), z W. This involution is continuous since the map a:M(H)

aM(H) is weak* continuous. Let f, g Cry(H), x H; then

fn (R(ox)f), dmn fn (R(ax)alf)alg dm

fn(R(x)af)agdmn= fnafR(x’)agdmnfHafR(ax’)ag dmH= fn f(R(ax)’g)dmn,,.

w

Since e e spt 2(x, x’), x e H, then e spt 2(ax, ax’). We have thus shown thatHrv is a compact ,-hypergroup.

THEOREM 4.5. Let W be a compact group of automorphisms on the compactP,-hypergroup H. Then Hw is a compact P,-hypergroup, and ff-Irv #ffI.

Proof. We show for #qb - #, b, , that nw #c.- dmnw O"

#ok. # dmnw a1(" ’1//)dmnwW

ck "r, dmv(z) dmn

fw fn ck" @ z dmndmv(z) 0

since distinct characters are orthogonal on H.Now let , with #b #ft. Thus

fHw acka---dmH fw fw f., ck(zx)l/l(z’x)dmH(x)dmrv(z)dmrv(’C’)

fw fw fn @(z(z’)-X) dmn(x)draw(z)dmw(z’,

m({ e "4 4} 1 am.We next show that. c . Let , 0 e ; then

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(Note that for r W, n(qS, if;og) n(qSo r, fro r;o9o ), b, , o/ since n(b,c(o)n ck dmn.) Thus Sp (#/) is dense in Cw(H).We have thus shown that #/ is a complete orthogonal basis for LZ(Hw),

and so/w #/. It follows that Hw is a compact P,-hypergroup.COROLLARY 4.6. For b6/_it, c(#qS)= c(b)]O(b)[, where [O(q)[ denotes the

cardinality of the set 0().We now derive the functional equation for symmetrized characters.THEOREM 4.7. Let C(H). Then dp ffIw if and only if thefollowing condition

holdsfor all x, y H:

() (x)(y) fw f,, cb d2(rx, y)dr.

Proof. Assume b /w. Then

R(y)d? dab., fn d(a6) * by

fn adp dab,, * by

--fnqdr(rx,Sy)= fttdPdoSx*abycb(x)cb(y).

Conversely, assume condition (1). Let xl xz. Then

q(xl)(y) fw fn ck d2(zx y) dmw(’C)

R(y)dp dab,,, fn R(y)dp da6,,

fw fn dpd2(’cx2’y) dmn(r’)=dp(x2)dp(Y)’

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Therefore b(x ) b(x2), and b e Cw(H). Furthermore,

qb d2(ax, ay) fn 4) da6x * a6y

fH 4) da(a6x * 6y)

f 4) d(a6,, * 6y)

4) d(a6,, * by) fu R(y) dp da6

R(y)c/) db,,, draw(r,)

R(y)dp d6,, dmw(’C)

4) d2(’cx, y) dmw(r)

(x)(y).

And so b e/w.Remark 4.8. As an application of Theorem 4.7, one can derive the functional

equation for the characters of a compact group. That is, for f C(G), f(x) X/nif and only if

f(x)f(y) fa f(zxz- y) dma(z)

for all x, y G (see Weil [8, p. 87]). The result is obtained by symmetrizing thenoncommutative hypergroup G by the compact group of inner automorphismsof G.

5. Krawtchouk polynomials. The Krawtchouk polynomials k,, 0 <- n =< N,are an orthogonal set of polynomials on the discrete set {0, 1,..., N}, where

O < p < l, N is a positive integer, and the weight function is (Nx )P(1- p)v-x,x 0, 1,.-., N. The polynomials are given in terms of the hypergeometricfunctions by

k,(x;p,N)=(1 -p)" )F[-n,x-N;x-n+ 1;p/(p- 1)]

p"( N).F[ n, x N l/p]

n!

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(see [1, 6]). We normalize the Krawtchouk polynomials by

K(x; p, N)

Szeg6 [6, p. 36] gives the explicit formula

kn(x p, N)kn(0; p, N)"

K,,(x p, N) -1)1- p

=o p (n j)!(x j)!(N + j x n)!N!j!

We now show how the symmetrized characters of the product of a two-point hypergroup yields the Krawtchouk polynomials.

Let -1 =< a < 0 and define H to be the two-point hypergroup with points{1, a} such that 2(1, 1) 61,2(1, a) 2(a, 1) dia,and.(a, a) -a61 + (a +The identity e is 1. The hypergroup H has two characters ;to, ;t defined by ;to 1,and ;tl(l)= 1 and ;tl(a)= a. Furthermore, H is a compact P.-hypergroup.For notational convenience, we write

H.=1

where the points of H are the rows and the characters are the columns. Let mdenote the invariant measure on H., defined by m({l}) 1 p and m({}) p.Then a (p 1)/p and 1/2 _< p < 1. Also c(;to) 1 and c(;t) p/(1 p) 1/a.

Let N be a positive integer, and let H. denote the hypergroup H. x /-/.x x H., (N times). The elements of//. are N-tuples of l’s and ’s, and theelements of (H)are N-tuples of )0’s and

The permutation group P on N letters acts naturally on the hypergroupH. as a compact group W of automorphisms. We let a, e respectively denote thesymmetrization operators on C(H.), M(H.) respectively.

For x H., let Ix] e {0, 1, ..., N} be the number of times appears in theN a),N- l’sN-tuple x. Let trx (Ha)w be represented by (1, 1, ..., 1, a, a,... Ix]

and Ix] a’s. For 0 =< n =< N, let ;t (Ha)w be represented by (;to, ;to,’", ;to,So;tl Z1 ", ;tl), N n Zo’S and n

For e e I-Ik__ {0, 1} (H)’, let ;t denote the element of (H)" associatedwith e; that is, ;t ;tl;t2"’" ;t,. For 0 __< n __< V, choose e VIk_ {0,1} withN n O’s and n l’s. We compute the symmetrized character ;t e -H),. Recallthat for x H,

z.(ax) a’STo compute trl;t we note that ;t, has n ;t and only the ;t’s need be evaluated.

Indeed to get the jth power of a, j of the Ix] a’s must be paired with j of theand n j of the N Ix] l’s must be paired with the remaining n j of the

The number of times this occurs isN Ix]| as the permutation group Pn-j

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acts on x. Since characters assume the value 1 at the identity, it follows that

),(crx)= X a1N-

1=0 n

0 _<_ n _<_ N, x H,. We have thus shown the following result.THEOREM 5.1. Let H, be the product of N copies of the finite P,-hypergroup

H,=( 1 la) -l < a < O" and let W be the finite group of automorphisms of H,given by the permutation group on N letters. Then the symmetrized characters of(H, )rr are given by

z(x) K.([x]; p, N),

xn,0nN,p= 1/(1 -a).We now derive a product theorem for Krawtchouk polynomials.Askey and Gasper [1], using a method of Eagleson [3], have shown that the

Krawtchouk polynomials satisfy the following linearization theorem"N

K(x; p, S)K(y; p, S) t(z, y, x; p, S)K(z; p, S),z=0

0x,y,nN, 0<p< 1, where

NI(z, y, x; p, N)

(1 p(2p 1)

ao (J- z)(j- y)(j- x)(z + + x- j)(- j)"

Therefore I(z, y, x; p, N) 0 for N P < 1.Since the Krawtchouk polynomials are the symmetrized characters of the

hypergroup H, a (p 1)/p, N P < 1, we will be able to obta this result forN p < 1 from the fact that characters satisfy the functional equation

(x, y e H a hypergroup) once a formula for 6x* y is obtained.

(1 la),N=l,2,.., andtrl, rRecall -1 <= a < O, a (p- 1)/p, H, - 1

are the symmetrization projections on C(H,), M(H,) respectively. For x e n,Ix] denotes the number of times a appears in the N-tuple x. The convolution struc-ture of M(H,) is given by * di , 6 6, . and i i -a6 + (1 + a)i,.For u, v e M(H,), 3. di (3u, 3o, x x (6u,, * 6vN).

We wish to compute adix*adiy for x, ye H,. For z e H,, we definee Mre(H,)by (z)= or(3, x 6, x... x di, x di. x . x... x .)= ai,(N- [z]di’s and [z] ft.’s). Now a3 * crfir a((x)* (5), (Theorem 3.8). For z e W, letdenote the number of coordinates where both x a and (zy) a. Hence for

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r W, 6x * 6 is a Cartesian product in which. -a61 + (1 + a)6. appears jr times,6 appears ([x] jr) + ([y] jr) times, and 6 appears N+jr-Ix]-[y]times. By applying the appropriate r’e W to 6x* 6y we may assume 6x, 6y

()N+Jr-[xl-[yl x (6a)[xl+[yl-zj X (--a6 + (1 + a)6.)J (where the exponentdenotes the number of successive appearances in the product). The symmetrization

/--

of the last jr factors is the same as that of o {J;l(--a)-k(l+a)k-k6Since the functions in Cw(H) are constant on the equivalence classes in H/,we have that

a((x) * by) . (Ix] jr + [y] jr + k) (-a)J-’(1 + a)W k 0

sinceIx]J

l- x]- [y] + 2j

(.._a)t.,,:l+tyl-j-/(1 + a)-tx-ty+2j

(let/= [x] + [y] 2fl: + k)

N

N N

/=0 j=O

N

[y] j (i Ix] ix] +

(_a)tX+t,-j-(1 + a)-tx-t:v+2j,

N

[y] j[y]!(N- [y])! is the number of re W for which jr j.

We have thus shown the following result.THEOREM 5.2. Let <= a < O, 1/2 <= p < 1, and x, y Hu,. Then

where

N

ax y Z J(t, [y], [x];p, N)(i),/=0

J(1, [y], Ix]; p, N)N

[Y]

N

[y] j l- [x] [y] + 2j

(_a)tX]+ty-j-l(1 + a)l-[x]-[y]+

’s and (a’S"and where (1) denotes 0"(61 1 a X X 6.), N 61Thus for 0 <= n <- N,

N

K,(x p, N)K,(y p, N) J(1, y, x; p, N)K,(I p, N),/=0

O<=x,y<=N.

PROPOSITION5.3. J(I,y,x;p,N) I(l,y,x;p,N),O <= l,y,x <= N,O < p < 1.

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Proof.

J(l, y, x; p, N)y(N y) N x!(N x)!j!

N! j=o j!(x-j)!(y-j)!(N- x- y +j)!(l- x y + 2j)!

(1 p)X+r-j-tp-X-r+j+t-t+x+r-2j(2p 1)/-x-r+ 2j

(x +y-l-j)!(since + a (2p- 1)/p)

N! N (1 p)t’-l(2p- 1)t+x+r-2’p

Y

(letk=x+y-j)

pk(1 p)k-l(2p 1)l- 2k

=o (k- y)!(k x)!(N k)!(l- 2k + x + y)!(k- l)!

I(l,y,x;p,N).

THEOREM 5.4 Let 1/2 <= p < 1, 0 <__ x, n <= N. Then

=o pX(1 p)-l/(x; p, g)l

Proof The left-hand side is

(c(K,(. ;p, N))) -1 f IK,([x]; p, N)I 2 damnS(x),a= (p 1)/p.

By Corollary 4.6, c(#b) c(b)IO()1, a character on a compact P,-hypergroupand a a symmetrization projection. Let ()o,’", o,,"’, )), N n

Xo’s and n Xl’S, be a character on H. Then lO(x)l (Nn), and(c(;)) -x Iz(x)l 2 dmt_ly(X)

Finally, recall that #)(x) K,([x] p, N).

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364 CHARELS F. DUNKL AND DONALD E. RAMIREZ

6. Homomorphisms between hypergroups. Let -1 __< a __< b < 0 and let

H"=( 11 la) 1 lb) bethetw’pintP*’hypergrupsassciatedwitha’b() (0a) andrespectively (as in 5). Define pl"C(Hb) C(H.) by p

b-a 1 -b-1-a 1-a

The map p: is the induced adjoint map of the point map p:H, -o Hb defined bypl landpa=b.

We compute now the action of p relative to the symmetrized hypergroups(H,)w, (H)w, where W is the permutation group on N letters, N 1, 2, ....

PROr’OSITION 6.1. p’Cw(n) -0 Cw(n).Proof That pCw(H)c Cw(Ha) follows since there exists a one-to-one

map a of W() into W(b) such that the following diagram commutes:

N

r(a) 1 1To see this, let f e C(H) with f ztb) f for all ztb) Wtb); then for x s Ha andzt) W), (pf) zt")(x) f pN(z()x) f(az() pN)(x) f(pCx) (pf)(x), andso pf Cw(H), which completes the proof.

For 0 =< n _< N, define Z"), Zt,b) to be the symmetrized characters on H, Hv,respectively, associated with N- nZto’s, Ztob)’s, respectively, and nz")’s, zb)’s,respectively. Let x s Ha and so x has Ix] a’s and N Ix] l’s.

We write Zb tr(tob), Zt0b), b),..., b). Similarly for Z. ThusN

N (b),o,z. (:,,:) o-, ]-1j=N-n+l

N

(b_aT.to)(xj)+ l _b,ta)tX)a "1 J!l-I 1- a 1-j=-+l

For 1/2 __< pl __< P2 < 1, let b (P2 1)/p2 and a (px 1)/p. Then -1 __< a=< b < 0, and the above remarks yield an expansion in Krawtchouk polynomials.

TI-IEOREM 6.2. Let 1/2 <_ p <= P2 < 1, 0 n <_ N. Then

K.(x; p2,N)j=0

K(x p, N)(P2 pl)n-jp

P2all x.

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Proof. Theorem 5.1 implies that

=o 1 K(x; p, N),

and the right-hand side is equal to

=o 1 -(Pl 1)/pl 1 (p 1)/p]

2.-’ P2- P "-J P JK(xj=0 P2 2] ;Pl, N)

(P: P)"-PK(x; p, N).

=o P,COROLLARY 6.3. Let x, n O, 1,..., N, and p P2 < 1. Then

K(; p, N K(y; p,.=o P

Proo The Krawtchouk polynomials are symmetric in x and n; that is,

g(x p, N) gx(n; p, g).

Remark 6.4. Theorem 6.2 shows that the Krawtchouk polynomials K(x; P2, N)can be expanded in a series of {Ki(x;p,N)}}= o with positive coefficients,< < 1. Corollary 6.3 shows each xP P2 < 0, 1,..., N can be represented

as a positive measure on {0, 1, -.., N} such that

g(x; P2, N) f K.(y; p, N) d.x(y),

n=0,1,...,N;)<plNP2 <Remark 6.5. Gasper [4, (5.13)] determined the expansion of K(.; p, N) re-

M andstricted to {0, 1,..., M} (for M < N) in terms of the set {K(., p, )}=oshowed that the coefficients were nonnegative. We can obtain this result with ourtechniques (one step at a time, that is, for M N 1). The idea is to first sym-metrize Hy over Px_ , the group of permutations of the first N 1 coordinates,thus obtaining an expression involving K(.; p, N- 1) (summing over Ps canbe done by summing over Ps_ and then summing over the two two-sided cosets ofPs_ in Ps). One obtains

Kn(m p, N)NnN

nK,,(m;p,N- 1)+ K,,_I(m;p,N- 1)

form=0,1,...,N- 1, n-- 0, 1,..., N,

and

Kn(m p, N)N

anKn(m 1; p, N 1)+-Kn_(m- 1; p, N 1)

form= 1,...,N, a=(p- 1)/p, n=0,1,2,...,N.

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By combining the two formulas one can derive the difference formula"

n(m- 1,p,N 1)K,(m p, N) K,(m 1 p, N) -(a 1)K,_

forn=0,1,...,N, m= 1,...,,N.

Historical remarks 6.6. Vere-Jones [7, p. 268] identified the Krawtchoukpolynomials (in the symmetric case p q 1/2) with the spherical zonal functionsassociated with a certain finite group and subgroup. He asked for which values ofp, 0 < p < 1, is there such a group interpretation.

Let G be a compact group and K a closed subgroup such that there existsprecisely two two-sided cosets K and KxK, x G. Let H {K, KxK}. ThenH has the structure of a finite P.-hypergroup (see [2, Ex. 4.2]). The invariantmeasure mn of H is positive on each element of H [2, Prop. 3.2]; indeed, sinceKxK is closed K is open. Using our previous symbolism, write

H=1 la)

where -1 =< a < 0. Now mn(KxK kmn(K), where k is the number of distinctleft K-cosets in KxK; and so k is a positive integer.

By the orthogonality of Zo and Z1, we have the equations

mn(K + amn(KxK 0, mn(K + akmH(K O,

and so

a 1/k.It follows that the only values of p, 0 < p < 1, for which there exists such a groupinterpretation is for p k/(k + 1), k 1, 2,..., for recall a (p- 1)/p; andindeed, let G Pk+l and K Pk, k 1, 2,....

REFERENCES

[1] R. AsIEY and G. GASPER,. Convolution structuresfor Laguerre polynomials, to appear.[2] C. DtNKL, The measure algebra of a locally compact hypergroup, Trans. Amer. Math. Soc., to

appear.[3] G. EAGLESON, A characterization theorem for positive definite sequences on the Krawtchouk poly-

nomials, Austral. J. Statist., 11 (1969), pp. 29-38.[4] G. GASPER, Projection formulasfor orthogonalpolynomials ofa discrete variable, to appear.[5] I. GLICKSBER6, Weak compactness and separate continuity, Pacific J. Math., 11 (1961), pp. 205-214.[6] G. SZEOS, Orthogonal Polynomials, Colloquium Publications, vol. 23, Amer. Math. Soc., Provi-

dence, R.I., 1967.I7] D. VERE-JONES, Finite bivariate distributions and semigroups of nonnegative matrices, Quart. J.

Math. Oxford (2), 22 (1971), pp. 247-270.[8] A. WELL, L’intOgration dans les Groupes Topologiques et ses Applications, Hermann, Paris, 1938.

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krawtchouk polynomials and the symmetrization of …people.virginia.edu/~der/pdf/der33.pdfsection 5...

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