Introduction The classical sequence of Hermite polynomials form one of the best known sys- tems of orthogonal polynomials in literature

ISSN: 1821-1291, URL: http://www.bmathaa.org Volume 6 Issue 4 (2014), Pages 16-43

GENERALIZED q-HERMITE POLYNOMIALS AND THE q-DUNKL HEAT EQUATION

(COMMUNICATED BY FRANCISCO MARCELLAN )

M. SALEH JAZMATI & KAMEL MEZLINI & N ´EJI BETTAIBI

Abstract. Two classes of generalized discreteq-Hermite polynomials are con- structed. Several properties of these polynomials, and an explicit relations con- necting them with littleq-Laguerre andq-Laguerre polynomials are obtained.

A relationship with theq-Dunkl heat polynomials and theq-Dunkl associated functions are established.

1. Introduction

The classical sequence of Hermite polynomials form one of the best known sys- tems of orthogonal polynomials in literature. Their applications cover many do- mains in applied and pure mathematics. For instance, the Hermite polynomials play central role in the study of the polynomial solutions of the classical heat equa- tion, which is a partial differential equation involving classical derivatives. It is therefore natural that generalizations of the heat equation for generalized oper- ators lead to generalizations of the Hermite polynomials. For instance, Fitouhi introduced and studied in [6] the generalized Hermite polynomials associated with a Sturm-Liouville operator by studying the corresponding heat equation. In [16], R¨osler studied the generalized Hermite polynomials and the heat equation for the Dunkl operator in several variables. In [15], Rosenblum associated Chihara’s gener- alized Hermite polynomials with the Dunkl operator in one variable and used them to study the Bose-like oscillator.

During the mid 1970’s, G. E. Andrews started a period of very fruitful collaboration with R. Askey (see [1, 2]). Thanks to these two mathematicians, basic hypergeomet- ric series is an active field of research today. Since Askey’s primary area of interest is orthogonal polynomials,q-series suddenly provided him and his co-workers, who include W. A. Al-Salam, M. E. H. Ismail, T. H. Koornwinder, W. G. Morris, D.

Stanton, and J. A. Wilson, with a very rich environment for derivingq-extensions of the classical orthogonal polynomials of Jacobi, Gegenbauer, Legendre, Laguerre

2000Mathematics Subject Classification. 33D45, 33D50, 33D90 .

Key words and phrases. Generalizedq-derivative operators. Generalized discrete q-Hermite polynomials. q-Dunkl heat equation.

c

2014. Universiteti i Prishtin¨es, Prishtin¨e, Kosov¨e.

Submitted September 23, 2014.

16

and Hermite. It is in this context that this paper is built around the construction of a new generalization of the Hermite polynomials, using newq-operators.

In this paper, we introduce some generalizedq-derivative operators with param- eterα, which, with theq-Dunkl intertwining operator (see [3]) allow us to introduce and study two families ofq-discrete orthogonal polynomials that generalize the two classes of discreteq-Hermite polynomials given in [11].

Next, we consider a generalized q-Dunkl heat equation and we show that it is re- lated to the generalized discrete q-Hermite polynomials in the same way as the classical ones in [15] and in [16].

This paper is organized as follows: in Section 2, we recall some notations and useful results. In Section 3, we introduce generalized q-shifted factorials and we use them to construct some generalizedq-exponential functions. In Section 4, we introduce a new class ofq-derivative operators. By these operators and theq-Dunkl intertwining operator two classes of discreteq-Hermite polynomials are introduced and analyzed in Section 5. In Section 6, we introduce a q-Dunkl heat equation, and we construct two basic sets of solutions of theq-Dunkl heat equation: the set of generalizedq-heat polynomials and the set of generalizedq-associated functions.

In particular, we show that these classes of solutions are closely related to the generalized discreteq-Hermite I polynomials and the generalized discreteq-Hermite II polynomials, respectively.

2. Notations and Preliminaries

2.1. Basic symbols. We refer to the general reference [9] for the definitions, notations and properties of the q-shifted factorials and the basic hypergeometric series.

Throughout this paper, we fix q ∈ (0,1) an we write Rq = {±qⁿ, n ∈Z}. For a complex numbera, theq-shifted factorials are defined by:

(a;q)0= 1; (a;q)n=

n−1

Y

k=0

(1−aq^k), n= 1,2, ...; (a;q)_∞=

∞

Y

k=0

(1−aq^k).

(a;q)n= (a;q)∞

(aqⁿ;q)_∞. (2.1)

If we changeq byq⁻¹, we obtain

(a;q⁻¹)_n = (a⁻¹;q)_n(−a)ⁿq^−n(n−1)2 , a6= 0. (2.2) We also denote

[x]q = 1−q^x

1−q , x∈C and n!q = (q;q)n

(1−q)ⁿ, n∈N. The basic hypergeometric series is defined by

rφs(a1, ..., ar;b1, ..., bs;q, z) =

∞

X

k=0

(a₁;q)_k...(a_r;q)_k

(−1)^kq^{k(k−1)2 1+s−r} (b₁;q)_k...(b_s;q)_k(q;q)_k z^k.

The two Euler’sq-analogues of the exponential function are given by ( see [9]) Eq(z) =

∞

X

k=0

q^k(k−1)2 (q;q)k

z^k= (−z;q)_∞, (2.3)

e_q(z) =

∞

X

k=0

1

(q;q)_kz^k = 1

(z;q)_∞, |z|<1. (2.4) Note that the function Eq(z) is entire on C. But for the convergence of the sec- ond series, we need |z|<1; however, because of its product representation, eq is continuable to a meromorphic function on C with simple poles at z = q⁻ⁿ, n non-negative integer.

We denote by

exp_q(z) :=e_q((1−q)z) =

∞

X

n=0

zⁿ n!q

, (2.5)

Expq(z) :=Eq((1−q)z) =

∞

X

n=0

q^n(n−1)2 zⁿ n!q

. (2.6)

We have lim

q→1⁻expq(z) = lim

q→1⁻Expq(z) = e^z, wheree^z is the classical exponential function.

The Rubin’sq-exponential function is defined by (see [13]) e(z;q²) =

∞

X

n=0

bn(z;q²), (2.7)

where

bn(z;q²) =q^[n2]([n2]+1) n!q

zⁿ (2.8)

and [x] is the integer part ofx∈R. e(z;q²) is entire onCand we have lim

q→1⁻e(z;q²) =e^z. Theq-Gamma function is given by (see [5, 9] )

Γ_q(z) = (q;q)_∞

(q^z;q)_∞(1−q)^1−z, z6= 0,−1,−2, . . . (2.9) and tends to Γ(z) whenqtends to 1⁻.

We shall need the Jacksonq-integrals defined by (see [9, 10]) Z a

0

f(x)d_qx= (1−q)a

∞

X

n=0

f(aqⁿ)qⁿ, Z b

a

f(x)d_qx= Z b

0

f(x)d_qx− Z a

0

f(x)d_qx, Z ∞

0

f(x)dqx= (1−q)

∞

X

n=−∞

f(qⁿ)qⁿ, (2.10) Z ∞

−∞

f(x)dqx= (1−q)

∞

X

n=−∞

qⁿf(qⁿ) + (1−q)

∞

X

n=−∞

qⁿf(−qⁿ).

We denote byL¹_α,q the space of complex-valued functionsf onRq such that Z ∞

−∞

|f(x)||x|^2α+1dqx <∞.

Theq-Gamma function has theq-integral representation (see [5, 8]) Γq(z) =

Z (1−q)⁻¹ 0

x^z−1Expq(−qx)dqx, (2.11) and satisfies the relation

Z 1 0

t^2x+1 (q²t²;q²)_∞

(q^2yt²;q²)_∞dqt = Γ_q2(x+ 1)Γ_q2(y)

(1 +q)Γ_q2(x+y+ 1), x >−1, y >0.(2.12) 2.2. The q-derivatives.

The Jackson’sq-derivativeDq (see [9, 10]) is defined by : Dqf(z) = f(z)−f(qz)

(1−q)z . (2.13)

We also need a variantD_q⁺, called forward q-derivative, given by D_q⁺f(z) =f(q⁻¹z)−f(z)

(1−q)z . (2.14)

Note that lim

q→1⁻Dqf(z) = lim

q→1⁻D⁺_qf(z) =f⁰(z) wheneverf is differentiable atz.

Recently, R. L. Rubin introduced in [13, 14] aq-derivative operator∂_q as follows

∂qf =D_q⁺fe+Dqfo, (2.15) wherefeandfo are respectively the even and the odd parts off.

We note that iff is differentiable atxthen∂qf(x) tends asq→1⁻ to f⁰(x).

3. The generalized q-exponential functions 3.1. The generalizedq-shifted factorials.

Forα >−1, we define the generalizedq-shifted factorials for non-negative integers nby

(2n)!q,α := (1 +q)²ⁿΓq²(α+n+ 1)Γq²(n+ 1)

Γ_q2(α+ 1) = (q²;q²)n(q^2α+2;q²)n

(1−q)²ⁿ , (2n+ 1)!q,α := (1 +q)²ⁿ⁺¹Γ_q2(α+n+ 2)Γ_q2(n+ 1)

Γ_q2(α+ 1) = (q²;q²)n(q^2α+2;q²)n+1

(1−q)²ⁿ⁺¹ . We denote

(q;q)2n,α= (q²;q²)n(q^2α+2;q²)n and (q;q)2n+1,α = (q²;q²)n(q^2α+2;q²)n+1. Remarks.

(1) Ifα=−1

2, then we get (q;q)_n,−1

2 = (q;q)_n and n!_q,−1

2 =n!_q. (3.1)

(2) We have the recursion relations

(n+ 1)!q,α= [n+ 1 +θn(2α+ 1)]_qn!q,α

and

(q;q)n+1,α= (1−q) [n+ 1 +θn(2α+ 1)]_q(q;q)n,α,

(3.2)

where

θ_n=







1 if nis even 0 if nis odd.

(3.3) (3) It is easy to prove the following limits

lim

q→1⁻(2n)!_q,α = 2²ⁿn!Γ(α+n+ 1)

Γ(α+ 1) =γ_α+1 2(2n), lim

q→1⁻(2n+ 1)!_q,α = 2²ⁿ⁺¹n!Γ(α+n+ 2)

Γ(α+ 1) =γ_α+1

2(2n+ 1),

(3.4)

whereγµ is the Rosenblum’s generalized factorial (see [15]).

3.2. The generalizedq-exponential functions.

By means of the generalized q-shifted factorials, we construct three generalized q-exponential functions as follows:

Definition 3.1. Forz∈C, the generalized q-exponential functions are defined by E_q,α(z) :=

∞

X

k=0

q^k(k−1)2 z^k (q;q)k,α

. (3.5)

e_q,α(z) :=

∞

X

k=0

z^k (q;q)k,α

, |z|<1. (3.6)

ψ_λ^α,q(z) =

∞

X

n=0

b_n,α(iλz;q²), λ∈C, (3.7) where

bn,α(z;q²) = q^[n2]([n2]+1)zⁿ

n!_q,α . (3.8)

Note thatψ_λ^α,q(z) is exactly theq-Dunkl kernel introduced in [3].

For α = −1

2, it follows from (3.1) that E_q,−1

2(z) = E_q(z), e_q,−1

2(z) = e_q(z), b_n,−¹

2(x;q²) =bn(x;q²) and we haveψ⁻

1 2,q

λ (x) =e(iλx;q²) the Rubin’sq-exponential function (2.7). By (3.4), theq-Dunkl kernelψ_λ^α,q(x) tends toe_α+1

2(iλx) asq→1⁻, whereeµ is the Rosenblum’s generalized exponential function (see [15, 16]).

The generalizedq-exponential functionψ_λ^α,q(x) gives rise to aq-integral transform, called theq-Dunkl transform on the real line, which was introduced and studied in [3]:

F_D^α,q(f)(λ) =Kα

Z +∞

−∞

f(x)ψ_−λ^α,q(x)|x|^2α+1dqx, f ∈L¹_α,q, where

Kα= (1−q)^α q^2α+2;q²

∞

2 (q²;q²)_∞ .

4. The generalizedq-derivatives

In this section we introduce a new class ofq-derivatives operators which play an important role in the construction of a generalizedq-Hermite polynomials.

4.1. The generalizedq-derivative operators.

Definition 4.1. The generalized backward and forwardq-derivative operatorsDq,α

andD_q,α⁺ are defined as

Dq,αf(z) = f(z)−q^2α+1f(qz)

(1−q)z , (4.1)

D⁺_q,αf(z) = f(q⁻¹z)−q^2α+1f(z)

(1−q)z . (4.2)

The operators given by

∆_α,qf =D_qf_e+D_q,αf_o, (4.3)

∆⁺_α,qf =D_q⁺fe+D_q,α⁺ fo (4.4) are called the generalizedq-derivatives operators.

Remark that forα=−1

2, we have:

D_q,−1

2 =Dq, D⁺_q,−1 2

=D_q⁺,∆_q,−1

2 =Dq and∆⁺_q,−1 2

=D⁺_q. The following elementary result is useful in the sequel.

Lemma 4.1.

∆⁺_α,qxⁿ =q⁻ⁿ[n+θn+1(2α+ 1)]qxⁿ⁻¹, n= 1,2,3, .... (4.5)

∆⁺_α,q,xe_q,α(qxt) = t

1−qe_q,α(tx), (4.6)

∆⁺_α,q,x

E_q2(−q²x²)e_q,α(qxt)

= t−x

1−qE_q2(−q²x²)e_q,α(tx), (4.7) where the operator ∆⁺_α,q,x acts with respect toxandθis given by (3.3).

Proof. By elementary calculus, we get (4.5).

On the one hand, using the definition of the operator∆⁺_α,q and (4.5), we have

∆⁺_α,q,xeq,α(qxt) =

∞

X

n=0

(qt)ⁿ∆⁺_α,qxⁿ (q;q)n,α

=

∞

X

n=1

tⁿ[n+θn+1(2α+ 1)]qxⁿ⁻¹ (q;q)n,α

. On the other hand, using the second recursion relation in (3.2) and changing the index in the second sum in the previous equation, we obtain

∆⁺_α,q,xeq,α(qxt) = t 1−q

∞

X

n=0

tⁿxⁿ

(q;q)_n,α = t

1−qeq,α(xt).

Finally, using the infinite product representation (2.3) ofE_q2(−q²x²), we get

∆⁺_α,q,x

Eq²(−q²x²)eq,α(qxt)

=Eq²(−q²x²)

∆⁺_α,qeq,α(qxt)− x

1−qeq,α(xt)

,

and (4.7) follows from (4.6).

4.2. The q-Dunkl operator.

We can rewrite theq-Dunkl operator introduced in [3] by means of the generalized q-derivative operators introduced in Definition 4.1 as

Λα,qf =∆⁺_α,qfe+∆α,qfo. (4.8) Indeed.

We have

∂qf =D_q⁺fe+Dqfo, and

Λα,q(f)(x) =∂q[Hα,q(f)] (x) + [2α+ 1]q

f(x)−f(−x)

2x ,

where

Hα,q:f =fe+fo7−→fe+q^2α+1fo. We can write, then,

Λα,qf(x) = ∂qfe(x) +q^2α+1∂qfo(x) +1−q^2α+1 (1−q)x fo(x)

= D_q⁺f_e(x) +q^2α+1D_qf_o(x) +1−q^2α+1 (1−q)x f_o(x)

= D_q⁺fe(x) +Dq,αfo(x)

= ∆⁺_α,qf_e(x) +∆_α,qf_o(x).

It is noteworthy that in the caseα=−1

2, Λ_α,qreduces to the Rubin’sq-derivative operator ∂_q defined in [13] and that for a differentiable function f, the q-Dunkl operator Λ_α,qf tends to the classical Dunkl operator Λ_αf asq tends to 1.

By induction, we prove the following results : Proposition 4.1.

(1) A repeated application of theq-Dunkl operator to the monomialbn,α(x;q²) gives

Λ^k_α,qbn,α(x;q²) =b_n−k,α(x;q²), k= 0,1, ...n. (4.9) (2) If f is an even function, then

Λ²ⁿ_α,qf(x) =q^−n(n+1)∆²ⁿ_α,qf(q⁻ⁿx), n= 0,1,2, ..., (4.10) Λ²ⁿ⁺¹_α,q f(x) =q^−(n+1)2∆²ⁿ⁺¹_α,q f(q⁻ⁿ⁻¹x); n= 0,1,2, .... (4.11) 4.3. The q-Dunkl intertwining operator.

For our further development, we need to extend the notion of the q-Dunkl inter- twining operator introduced in [3] to the space C(R) of continuous functions on R.

Definition 4.2. The q-Dunkl intertwining operatorVα,q is defined on C(R) by V_α,qf(x) = C(α;q)

2 Z 1

−1

W_α(t;q²)(1 +t)f(xt)d_qt, x∈R, (4.12) where

C(α;q) = (1 +q)Γ_q2(α+ 1)

Γ_q2(1/2)Γ_q2(α+ 1/2) (4.13) and

W_α(t;q²) = (t²q²;q²)_∞ (t²q^2α+1;q²)∞

. (4.14)

Since the integrand is continuous, theq-integral (4.12) is well-defined.

In the following proposition we shall show that Vα,q is the intertwining operator between the generalized q-derivatives and the usual q-derivatives which generalize the transmutation relation (53) in [3].

Proposition 4.2. Suppose that the functionf and itsq-derivativesD_qf,D⁺_qf and

∂qf are in C(R), then

∆α,qVα,q(f) =Vα,q(Dqf); (4.15)

∆⁺_α,qVα,q(f) =Vα,q(D_q⁺f); (4.16) Λα,qVα,q(f) =Vα,q(∂qf). (4.17) Proof. By splittingf into its even and odd partsf =fe+fo, and using the fact thatDq changes the parity of the function and theq-integral of an odd function on [−1,1] is equal zero, we obtain

V_α,q(D_qf)(x) = C(α;q) 2

Z 1

−1

W_α(t;q²)tD_qf_e(xt)d_qt +C(α;q)

2 Z 1

−1

Wα(t;q²)Dqf0(xt)dqt.

(4.18)

On the other hand, from the definition of∆_α,q (4.3) we have

∆_α,qV_α,q(f)(x) = C(α;q) 2

Z 1

−1

W_α(t;q²)tD_qf_e(xt)d_qt +C(α;q)

2 Z 1

−1

Wα(t;q²)t²Dq,αf0(xt)dqt.

(4.19)

Then, using the fact that (1−q^2α+1t²)Wα(t;q²) = (1−q²t²)Wα(qt;q²), we get Vα,q(Dqf)(x)−∆α,qVα,q(f)(x) =

Z 1

−1

Wα(t;q²)

Dqf0(xt)−t²Dq,αf0(xt) dqt

= Z 1

−1

Wα(t;q²)(1−t²)f0(xt) (1−q)xt dqt

− Z 1

−1

W_α(t;q²)(1−q^2α+1t²)f0(qxt) (1−q)xt d_qt

= Z 1

−1

W_α(t;q²)(1−t²)f₀(xt) (1−q)xt d_qt

− Z 1

−1

Wα(qt;q²)(1−q²t²)f0(qxt) (1−q)xt dqt.

Since the integrand in the last Jacksonq-integral vanishes at the points−1 and 1, the change of variableu=qtin thisq-integral leads to the relation (4.15).

In a similar way, one can obtain (4.16).

Now by using (4.8), (4.15) and (4.16), we obtain

Λ_α,qV_α,q(f) =∆⁺_α,qV_α,qf_e+∆_α,qV_α,qf_o=V_α,q D_q⁺f_e+D_qf_o

=V_α,q(∂_qf). The following result shows the effect of theq-Dunkl intertwining operator on mono- mial functions and on theq-exponential functions.

Proposition 4.3. The following relations hold:

V_α,qzⁿ= (q;q)_n (q;q)n,α

zⁿ, n= 0,1,2, ...; (4.20) V_α,qb_n(z;q²) =b_n,α(z;q²), n= 0,1,2, ...; (4.21) Vα,qeq(z) =eq,α(z), |z|<1; (4.22)

V_α,qE_q(z) =E_q,α(z); (4.23)

Vα,qe(λz;q²) =ψ_−iλ^α,q(z), λ∈C. (4.24) Proof. If we takex=n−1

2 andy=α+1

2 in (2.12), we get by using (4.13) Vα,qz²ⁿ = C(α;q)z²ⁿ

Z 1 0

(t²q²;q²)_∞

(t²q^2α+1;q²)_∞t²ⁿdqt=C(α;q)Γq²(n+¹₂)Γq²(α+¹₂) (1 +q)Γ_q2(n+α+ 1)z²ⁿ

= Γ_q2(α+ 1)Γ_q2(n+¹₂)

Γ_q2(¹₂)Γ_q2(n+α+ 1)z²ⁿ= (q;q²)n

(q^2α+2;q²)_nz²ⁿ. Similarly, we prove that

Vα,qz²ⁿ⁺¹=C(α;q)z²ⁿ⁺¹ Z 1

0

(t²q²;q²)_∞

(t²q^2α+1;q²)_∞t²⁽ⁿ⁺¹⁾dqt= (q;q²)n+1

(q^2α+2;q²)_n+1z²ⁿ⁺¹. Then (4.20 ) follows from the two following facts

(q;q)_2n (q;q)2n,α

= (q;q²)_n (q^2α+2;q²)n

, (q;q)_2n+1 (q;q)2n+1,α

= (q;q²)_n+1 (q^2α+2;q²)n+1

. (4.25) For|z|<1, we have by using (4.20)

Vα,qeq(z) =

∞

X

k=0

V_α,qz^k (q;q)k

=

∞

X

k=0

z^k (q;q)k,α

=eq,α(z).

The same techniques produce the relations (4.23) and (4.24).

5. The generalized discreteq-Hermite polynomials We begin this section by the following useful lemma.

Lemma 5.1.

(1) Fors≥0, we have Fq(s) :=

Z 1 0

t^sE_q2(−q²t²)dqt= (1−q) (q²;q²)_∞

(q^s+1;q²)_∞. (5.1) (2) Forλ >0andn non-negative integer, we have

G^α_q,n(λ) :=

Z ∞ 0

e_q2(−λy²)y^2n+2α+1dqy=cq,α(λ)q^{−n2−(2α+1)n}

λⁿ q^2α+2;q²

n, (5.2) where

cq,α(λ) = (1−q)(−q^2α+2λ,−q^−2α/λ, q²;q²)_∞

(−λ,−q²/λ, q^2α+2;q²)_∞ . (5.3)

Proof. (1) Lets≥0. From (2.11), we have by doing the two changes of variables u= (1−q²)tandu=t²,

Γ_q2(s) = 1 (1−q²)^s

Z 1 0

t^s−1E_q2(−q²t)d_q2t= 1 +q (1−q²)^s

Z 1 0

u^2s−1E_q2(−q²u²)d_qu.

The relation (5.1) follows then by replacingsby s+ 1

2 in the previous equation.

(2) Letλ >0 andnnon negative integer. Then, from the definition of the Jackson q-integral and the Ramanujan identity (see [9], p. 125), we have

Z ∞ 0

e_q2(−λy²)y^sdqy = (1−q)

∞

X

k=−∞

q^(s+1)k (−λq^2k;q²)_∞

= (1−q) −λq^s+1,−q^−s+1/λ, q²;q²

∞

(−λ, q^s+1,−q²/λ;q²)_∞ . In particular fors= 2n+ 2α+ 1, we have

Z ∞ 0

eq²(−λy²)y^2n+2α+1dqy= (1−q) −λq^2α+2n+2,−q^−2α−2n/λ, q²;q²

∞

(−λ, q^2α+2n+2,−q²/λ;q²)_∞ . We conclude (5.2) by using the following equalities:

−λq^2α+2n+2;q²

∞= −q^2α+2λ;q²

∞

(−q^2α+2λ;q²)_n , q^2α+2n+2;q²

∞= q^2α+2;q²

∞

(q^2α+2;q²)_n and

−q^−2α−2n/λ;q²

∞=q^−n(n+1)(q^2αλ)⁻ⁿ −q^2α+2λ;q²

n −(q^2αλ)⁻¹;q²

∞.

5.1. The generalized discreteq-Hermite I polynomials.

The discreteq-Hermite I polynomials{h_n(x;q)}^∞_n=0 are defined in [11] by h_n(x;q) := xⁿ ₂φ₀(q⁻ⁿ, q⁻ⁿ⁺¹;−;q², q²ⁿ⁻¹x⁻²)

= (q;q)n [ⁿ₂]

X

k=0

(−1)^kq^k(k−1)x^n−2k

(q²;q²)_k(q;q)_n−2k . (5.4) Definition 5.1. The generalized discreteq-Hermite I polynomials {hn,α(x;q)}^∞_n=0 are defined by:

hn,α(x;q) := (q;q)n [ⁿ₂]

X

k=0

(−1)^kq^k(k−1)x^n−2k

(q²;q²)_k(q;q)_n−2k,α. (5.5) Remarks.

(1) For α=−1

2, we geth_n,−¹

2(x;q) =hn(x;q).

(2) Observe that hn,α(p

1−q²x;q)

(1−q²)ⁿ² = n!q

(1 +q)ⁿ

[ⁿ₂]

X

k=0

(−1)^kq^k(k−1)((1 +q)x)^n−2k k!_q2(n−2k)!_q,α .

So by (3.4), we obtain lim

q→1⁻

hn,α(p

1−q²x;q)

(1−q²)ⁿ² =H^α+

1

n 2(x) 2ⁿ , where H^α+

1

n 2(x) is the Rosenblum’s generalized Hermite polynomial (see [15] ).

(3) Each polynomialhn,α(.;q) has the same parity of its degreen.

Lemma 5.2. The generalized discrete q-Hermite I polynomials can be written in terms of basic hypergeometric functions as:























h2n,α(x;q) = (q;q)2n

(q;q)_2n,αx²ⁿ 2φ0(q⁻²ⁿ, q^−2n−2α;−;q², q^4n+2αx⁻²)

= (−1)ⁿqⁿ⁽ⁿ⁻¹⁾(q;q²)_{n 2}φ₁(q⁻²ⁿ,0;q^2α+2;q², q²x²), h_2n+1,α(x;q) = (q;q)2n+1

(q;q)_2n+1,αx²ⁿ⁺¹ ₂φ₀(q⁻²ⁿ, q^{−2n−2α−2};−;q², q^4n+2α+2x⁻²)

= (−1)ⁿqⁿ⁽ⁿ⁻¹⁾(q;q²)n+1

1−q^2α+2x2φ1(q⁻²ⁿ,0;q^2α+4;q², q²x²).

(5.6) Proof. We have

h_2n,α(x;q) = (q;q)_2n

n

X

k=0

(−1)^kq^k(k−1)x^2n−2k

(q²;q²)k(q²;q²)_n−k(q^2α+2;q²)_n−k. Using the identity

(a;q²)_n−k= (a;q²)n

(a⁻¹q²⁻²ⁿ;q²)_k −q²a^−1k

q^{k(k−1)−2nk}, (5.7) we get

(q²;q²)_n−k= (q²;q²)n

(q⁻²ⁿ;q²)_k(−1)^kq^{k(k−1)−2nk} (5.8) and

(q^2α+2;q²)_n−k= (q^2α+2;q²)n

(q^−2n−2α;q²)_k −q^−2αk

q^{k(k−1)−2nk}. (5.9) It follows, then, that

h2n,α(x;q) = (q;q)2n

(q;q)_2n,αx²ⁿ

×

n

X

k=0

(−1)^kq^−k(k−1)(q⁻²ⁿ;q²)k(q^−2n−2α;q²)k q^4n+2αx^−2k (q²;q²)_k

= (q;q)2n

(q;q)2n,α

x²ⁿ 2φ0(q⁻²ⁿ, q^−2n−2α;−;q², q^4n+2αx⁻²).

Now, we have

h2n+1,α(x;q) = (q;q)2n+1 n

X

k=0

(−1)^kq^k(k−1)x^2n+1−2k

(q²;q²)_k(q²;q²)_n−k(q^2α+2;q²)_n+1−k.

Then, using (5.8) and replacingnbyn+ 1 in (5.9), we obtain (q^2α+2;q²)_n+1−k= (q^2α+2;q²)_n+1

(q^{−2n−2α−2};q²)k

−q^−2αk

qk(k−1)−2nk−2k

and

h2n+1,α(x;q) = (q;q)_2n+1 (q;q)2n+1,α

x²ⁿ⁺¹

×

n

X

k=0

(−1)^kq^−k(k−1)(q⁻²ⁿ;q²)_k(q^{−2n−2α−2};q²)_k q^4n+2α+2x^−2k (q²;q²)k

= (q;q)2n+1

(q;q)_2n+1,αx²ⁿ⁺¹ ₂φ₀(q⁻²ⁿ, q^{−2n−2α−2};−;q², q^4n+2α+2x⁻²).

On the other hand, taking b=q^−2n−2α andz=q^2n+2αx⁻²in the following trans- formation formula (see [11], p.19)

2φ0(q⁻²ⁿ, b;−;q², q²ⁿz) = (b;q²)nzⁿ 2φ1(q⁻²ⁿ,0;b⁻¹q²⁻²ⁿ;q²,q²

bz), (5.10) we obtain

2φ0(q⁻²ⁿ, q^−2n−2α;−;q², q^4n+2αx⁻²) = (q^−2n−2α;q²)nq^2n2+2αnx⁻²ⁿ

×2φ1(q⁻²ⁿ,0;q^2α+2;q², q²x²).

By the identity (see [11], p.9)

(a;q²)n= (a⁻¹q²⁻²ⁿ;q²)n(−a)ⁿqⁿ⁽ⁿ⁻¹⁾, a6= 0, (5.11) we get

2φ0(q⁻²ⁿ, q^−2n−2α;−;q², q^4n+2αx⁻²) = (q^2α+2;q²)n(−1)ⁿqⁿ⁽ⁿ⁻¹⁾x⁻²ⁿ

×2φ1(q⁻²ⁿ,0;q^2α+2;q², q²x²).

So, it follows from (4.25) that

h2n,α(x;q) = (−1)ⁿqⁿ⁽ⁿ⁻¹⁾(q;q²)n 2φ1(q⁻²ⁿ,0;q^2α+2;q², q²x²).

Now, takeb=q^{−2n−2α−2} andz=q^2n+2α+2x⁻²in (5.10), to obtain

2φ0(q⁻²ⁿ, q^{−2n−2α−2};−;q², q^4n+2α+2x⁻²) = (q^{−2n−2α−2};q²)nq^2n2+2αn+2nx⁻²ⁿ

×2φ1(q⁻²ⁿ,0;q^2α+4;q², q²x²) By identity (5.11), we have

2φ0(q⁻²ⁿ, q^{−2n−2α−2};−;q², q^4n+2α+2x⁻²) = (q^2α+4;q²)n(−1)ⁿqⁿ⁽ⁿ⁻¹⁾x⁻²ⁿ

×2φ1(q⁻²ⁿ,0;q^2α+4;q², q²x²), and by (4.25), we get

h2n+1,α(x;q) = (−1)ⁿqⁿ⁽ⁿ⁻¹⁾(q;q²)n+1

1−q^2α+2x2φ1(q⁻²ⁿ,0;q^2α+4;q², q²x²).

The littleq-Laguerre polynomials{p_n(x;a|q)}^∞_n=0 are defined in [11] by

p_n(x;a|q) = 2φ₁(q⁻ⁿ,0;aq;q, qx). (5.12)

Using (5.39 ) and (5.12), the generalized discreteq-Hermite I polynomialshn,α(x;q) can be expressed in terms of the littleq-Laguerre polynomialspn(x;a|q) as follows:







h_2n,α(x;q) = (−1)ⁿqⁿ⁽ⁿ⁻¹⁾(q;q²)_np_n(x²;q^2α|q²), h2n+1,α(x;q) = (−1)ⁿqⁿ⁽ⁿ⁻¹⁾(q;q²)n+1

1−q^2α+2xpn(x²;q^2α+2|q²).

Proposition 5.1. The following relations hold:

(1)q-Integral representation of Mehler type

h_n,α(x;q) =V_α,qh_n(x;q). (5.13) (2) Generating function

E_q2(−z²)eq,α(xz) =

∞

X

n=0

hn,α(x;q) zⁿ

(q;q)_n, (|xz|<1). (5.14) (3) Inversion formula

xⁿ= (q;q)n,α [ⁿ₂]

X

k=0

h_n−2k,α(x;q)

(q²;q²)k(q;q)_n−2k. (5.15) (4) Three terms recursion formula

xh_n,α(x;q)−qn−1+(2α+1)θn(1−qⁿ)h_n−1,α(x;q) = 1−qn+1+(2α+1)θ_n

1−qⁿ⁺¹ h_n+1,α(x;q).

(5.16) Proof. (1) The relation (5.13) follows by application of (4.20 ) to each term in (5.4).

(2) The generating function for the discrete q-Hermite I polynomials is given by (see [11])

E_q2(−z²)e_q(xz) =

∞

X

n=0

h_n(x;q) zⁿ

(q;q)_n. (5.17)

So, by applying the operator V_α,q, with respect to x, to both sides of equation (5.17) and using (5.13) and (4.22), we obtain (5.14).

(3) follows by application of the operator V_α,q to both sides of the well-known relation (see [12]):

xⁿ= (q;q)n [ⁿ₂]

X

k=0

hn−2k(x;q) (q²;q²)k(q;q)_n−2k, (4.20) and (5.13).

(4) To prove (5.16), we consider, separately, the even and the odd cases in the expression

xhn,α(x;q)−qn−1+(2α+1)θn(1−qⁿ)hn−1,α(x;q).

We have

xh_2n,α(x;q) − q^2n+2α(1−q²ⁿ)h_2n−1,α(x;q)

= (q;q)2n n

X

k=0

(−1)^kq^k(k−1)x^2n−2k+1 (q²;q²)k(q;q)_2n−2k,α

−q^2n+2α(q;q)2n n−1

X

k=0

(−1)^kq^k(k−1)x2n−2(k+1)+1

(q²;q²)_k(q;q)2n−2(k+1)+1,α

.

Changektok−1 in the second sum, then combine with the first to get (q;q)2nx²ⁿ⁺¹

(q;q)2n,α

+ (q;q)2n n

X

k=1

(−1)^kq^k(k−1)x^2n−2k+1 (q²;q²)k(q;q)_2n−2k+1,α

×

(1−q^{2n−2k+2α+2}) +q^2n+2αq^−2k+2(1−q^2k) Simplify to obtain

(q;q)2nx²ⁿ⁺¹ (q;q)2n,α

+ (1−q^2n+2α+2)(q;q)2n n

X

k=1

(−1)^kq^k(k−1)x^2n−2k+1 (q²;q²)k(q;q)_2n−2k+1,α. The last expression can be written as

1−q^2n+2α+2

1−q²ⁿ⁺¹ h_2n+1,α(x;q).

In the odd case, the proof follows the same steps as the even case.

The following result states the affect of the operators ∆α,q and ∆⁺_α,q on the generalizedq-Hermite I polynomials.

Proposition 5.2.

(1) The forward shift operator:

∆α,qhn,α(x;q) = 1−qⁿ

1−q h_n−1,α(x;q), n= 1,2, ... (5.18) or equivalently

hn,α(x;q)−q^(2α+1)θn+1hn,α(qx;q) = (1−qⁿ)xh_n−1,α(x;q). (5.19) (2) The backward shift operator:

∆⁺_α,q

E_q2(−q²x²)hn,α(x;q)

=−q⁻ⁿ[θn(2α+ 1) +n+ 1]q

1−qⁿ⁺¹ E_q2(−q²x²)hn+1,α(x;q), (5.20) or equivalently

q^θn+1(2α+1)hn,α(x;q)−(1−x²)hn,α(q⁻¹x;q) = q⁻ⁿ[θn(2α+ 1) +n+ 1]q

[n+ 1]_q xhn+1,α(x;q).

Proof. (1) It is well known (see [11]) that Dqhn(x;q) =1−qⁿ

1−q hn−1(x;q).

So, by application of theq-Dunkl intertwining operator to both sides of this result, we obtain (5.18), by the use of (4.15) and (5.13).

(2) Put

g(x, t) =E_q2(−q²x²)E_q2(−q²t²)eq,α(qxt).

From (4.7), we have

∆⁺_α,q,xg(x, t) = E_q2(−q²t²)t−x

1−qE_q2(−q²x²)e_q,α(xt)

= −Eq²(−q²x²)x−t

1−qEq²(−q²t²)eq,α(xt)

= −∆⁺_α,q,tg(x, t).

(5.21)

But, using the generating function (5.17), we obtain

∆⁺_α,q,xg(x, t) =

∞

X

n=0

∆⁺_α,q,x

E_q2(−q²x²)hn,α(x;q) (qt)ⁿ

(q;q)_n (5.22) and

∆⁺_α,q,xg(x, t) =−

∞

X

n=0

E_q2(−q²x²)hn,α(x;q) qⁿ

(q;q)_n∆⁺_α,qtⁿ. It follows from (4.5 ) and (5.21) that

∆⁺_α,q,xg(x, t) =−

∞

X

n=1

E_q2(−q²x²)hn,α(x;q) (q;q)n

[θ_n+1(2α+ 1) +n]_qtⁿ⁻¹. (5.23) Therefore, by comparing the coefficients of tⁿ in the two series (5.22) and (5.23),

we obtain (5.20).

The following formula can now be proved by induction.

Proposition 5.3. The Rodrigues-formula for {hn,α(x;q)}^∞_n=0 is given by E_q2(−q²x²)h_n,α(x;q) = (q−1)ⁿq^n(n−1)2 (q;q)n

(q;q)_n,α

∆⁺_α,qⁿ

E_q2(−q²x²)

. (5.24) Proof. Sinceh0,α(x;q) = 1, the formula is clearly true forn= 0. Assume that it is true for an integern. Then, using (5.20) and (3.2), the application of the operator

∆⁺_α,q to the both sides of (5.24) completes the induction proof.

Proposition 5.4. The polynomials{hn,α(x;q)}^∞_n=0satisfy the followingq-difference equations:

q^2α+1h_2n,α(qx;q)−(q+q^2α+1−q¹⁻²ⁿx²)h_2n,α(x;q) +q(1−x²)h_2n,α(q⁻¹x;q) = 0.

(5.25) q^2α+1h2n+1,α(qx;q)−(1+q^2α+2−q⁻²ⁿx²)h2n+1,α(x;q)+q(1−x²)h2n+1,α(q⁻¹x;q) = 0.

(5.26) Proof. By (5.18), we have

∆²_α,qh_2n,α(x;q) = (1−q²ⁿ)(1−q²ⁿ⁻¹)

(1−q)² h_2n−2,α(x;q), n= 1,2,3, ....

But, from the three recursion term formulas (5.16), we obtain

xh_2n−1,α(x;q)−q²ⁿ⁻²(1−q²ⁿ⁻¹)h_2n−2,α(x;q) =h_2n,α(x;q) (5.27) and from (5.19), we get

h_2n,α(x;q)−h_2n,α(qx;q) = (1−q²ⁿ)xh_2n−1,α(x;q). (5.28) It follows, then, from (5.27) and (5.28) that

∆²_α,qh2n,α(x;q) = 1 (1−q)²

q²h2n,α(x;q)−q²⁻²ⁿh2n,α(qx;q) or equivalently,

(1−q²x²)h_2n,α(x;q)−(1+q^2α−q²⁻²ⁿx²)h_2n,α(qx;q)+q^2αh_2n,α(q²x;q) = 0. (5.29)

Replacexbyq⁻¹xin (5.29) and multiply the result byq, we obtain (5.25).

Following the same steps, we prove (5.26).

To prove the orthogonality property ofhn,α(x;q), we need the following lemma:

Lemma 5.3. The polynomials{hn,α(x;q)}^∞_n=0 satisfy Z 1

−1

x^phn,α(x;q)E_q2(−q²x²)|x|^2α+1dqx

=







2(1−q)(q²;q²)_∞

(q^2α+2;q²)_∞ (q;q)nq^[n2](^2[n+12 ]+2α) if p=n,

0 if p= 0,1, ..., n−1.

(5.30)

Proof. Since the parity of theq-polynomials{hn,α(x;q)}is the parity of thier de- grees and theq-integral in (5.30) of odd function is zero. It is, then, sufficient to consider the cases wherepandnare both even or odd.

From the definition (5.4), we can write Z 1

−1

x^2ph_2n,α(x;q)E_q2(−q²x²)|x|^2α+1d_qx

= 2(q;q)2n n

X

k=0

(−1)^n−kq(n−k)(n−k−1)Fq(2k+ 2p+ 2α+ 1) (q²;q²)_n−k(q²;q²)k(q^2α+2;q²)k

,

(5.31)

whereF_q is the function defined by (5.1), then the above sum becomes 2(1−q)(q²;q²)_∞(q;q)2n

n

X

k=0

(−1)^n−kq(n−k)(n−k−1)

(q²;q²)_n−k(q²;q²)_k(q^2α+2;q²)_k(q^2k+2p+2α+2;q²)_∞. Using (2.1), it is possible to rewrite the sum in (5.31) in the form

2(1−q)(q²;q²)∞(q;q)2n

(q^2α+2;q²)_∞

n

X

k=0

(−1)^n−kq(n−k)(n−k−1)

(q²;q²)_n−k(q²;q²)k

(q^2k+2α+2;q²)p. By using the transformation formulas (5.8) and the fact that

(q^2k+2α+2;q²)p= (q^2α+2;q²)p(q^2p+2α+2;q²)k

(q^2α+2;q²)_k , (5.32) the sum in (5.31) can be written as

(−1)ⁿqⁿ⁽ⁿ⁻¹⁾2(1−q)(q²;q²)_∞(q;q)2n(q^2α+2;q²)p

(q^2α+2;q²)_∞(q²;q²)_n

×

n

X

k=0

(q⁻²ⁿ;q²)k(q^2p+2α+2;q²)k

(q²;q²)k(q^2α+2;q²)k

q^2k

= (−1)ⁿqⁿ⁽ⁿ⁻¹⁾2(1−q)(q²;q²)_∞(q;q)_2n(q^2α+2;q²)_p (q^2α+2;q²)∞(q²;q²)n

×₂φ₁(q⁻²ⁿ, q^2p+2α+2;q^2α+2;q², q²).

By the summation formula (see [11], p.15)

2φ₁(q⁻²ⁿ, b;c;q², q²) = (b⁻¹c;q²)_n (c;q²)n

bⁿ,