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 = {±q^{n}, 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)∞

(aq^{n};q)_{∞}. (2.1)

If we changeq byq^{−1}, we obtain

(a;q^{−1})_{n} = (a^{−1};q)_{n}(−a)^{n}q^{−}^{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

(a_{1};q)_{k}...(a_{r};q)_{k}

(−1)^{k}q^{k(k−1)}^{2} ^{1+s−r}
(b_{1};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)_{k}z^{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

z^{n}
n!q

, (2.5)

Expq(z) :=Eq((1−q)z) =

∞

X

n=0

q^{n(n−1)}^{2} z^{n}
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;q^{2}) =

∞

X

n=0

bn(z;q^{2}), (2.7)

where

bn(z;q^{2}) =q^{[}^{n}^{2}^{]([}^{n}^{2}^{]+1)}
n!q

z^{n} (2.8)

and [x] is the integer part ofx∈R.
e(z;q^{2}) is entire onCand we have lim

q→1^{−}e(z;q^{2}) =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_{q}x= (1−q)a

∞

X

n=0

f(aq^{n})q^{n},
Z b

a

f(x)d_{q}x=
Z b

0

f(x)d_{q}x−
Z a

0

f(x)d_{q}x,
Z ∞

0

f(x)dqx= (1−q)

∞

X

n=−∞

f(q^{n})q^{n}, (2.10)
Z ∞

−∞

f(x)dqx= (1−q)

∞

X

n=−∞

q^{n}f(q^{n}) + (1−q)

∞

X

n=−∞

q^{n}f(−q^{n}).

We denote byL^{1}_{α,q} the space of complex-valued functionsf onRq such that
Z ∞

−∞

|f(x)||x|^{2α+1}dqx <∞.

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

Z (1−q)^{−1}
0

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

Z 1 0

t^{2x+1} (q^{2}t^{2};q^{2})_{∞}

(q^{2y}t^{2};q^{2})_{∞}dqt = Γ_{q}2(x+ 1)Γ_{q}2(y)

(1 +q)Γ_{q}2(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^{−1}z)−f(z)

(1−q)z . (2.14)

Note that lim

q→1^{−}Dqf(z) = lim

q→1^{−}D^{+}_{q}f(z) =f^{0}(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 f^{0}(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)^{2n}Γq^{2}(α+n+ 1)Γq^{2}(n+ 1)

Γ_{q}2(α+ 1) = (q^{2};q^{2})n(q^{2α+2};q^{2})n

(1−q)^{2n} ,
(2n+ 1)!q,α := (1 +q)^{2n+1}Γ_{q}2(α+n+ 2)Γ_{q}2(n+ 1)

Γ_{q}2(α+ 1) = (q^{2};q^{2})n(q^{2α+2};q^{2})n+1

(1−q)^{2n+1} .
We denote

(q;q)2n,α= (q^{2};q^{2})n(q^{2α+2};q^{2})n and (q;q)2n+1,α = (q^{2};q^{2})n(q^{2α+2};q^{2})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)]_{q}n!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} = 2^{2n}n!Γ(α+n+ 1)

Γ(α+ 1) =γ_{α+}1
2(2n),
lim

q→1^{−}(2n+ 1)!_{q,α} = 2^{2n+1}n!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;q^{2}), λ∈C, (3.7)
where

bn,α(z;q^{2}) = q^{[}^{n}^{2}^{]([}^{n}^{2}^{]+1)}z^{n}

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;q^{2}) =bn(x;q^{2}) and we haveψ^{−}

1 2,q

λ (x) =e(iλx;q^{2}) 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+1}dqx, f ∈L^{1}_{α,q},
where

Kα= (1−q)^{α} q^{2α+2};q^{2}

∞

2 (q^{2};q^{2})_{∞} .

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+1}f(qz)

(1−q)z , (4.1)

D^{+}_{q,α}f(z) = f(q^{−1}z)−q^{2α+1}f(z)

(1−q)z . (4.2)

The operators given by

∆_{α,q}f =D_{q}f_{e}+D_{q,α}f_{o}, (4.3)

∆^{+}_{α,q}f =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,q}x^{n} =q^{−n}[n+θn+1(2α+ 1)]qx^{n−1}, n= 1,2,3, .... (4.5)

∆^{+}_{α,q,x}e_{q,α}(qxt) = t

1−qe_{q,α}(tx), (4.6)

∆^{+}_{α,q,x}

E_{q}2(−q^{2}x^{2})e_{q,α}(qxt)

= t−x

1−qE_{q}2(−q^{2}x^{2})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,x}eq,α(qxt) =

∞

X

n=0

(qt)^{n}∆^{+}_{α,q}x^{n}
(q;q)n,α

=

∞

X

n=1

t^{n}[n+θn+1(2α+ 1)]qx^{n−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,x}eq,α(qxt) = t
1−q

∞

X

n=0

t^{n}x^{n}

(q;q)_{n,α} = t

1−qeq,α(xt).

Finally, using the infinite product representation (2.3) ofE_{q}2(−q^{2}x^{2}), we get

∆^{+}_{α,q,x}

Eq^{2}(−q^{2}x^{2})eq,α(qxt)

=Eq^{2}(−q^{2}x^{2})

∆^{+}_{α,q}eq,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,q}fe+∆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+1}fo.
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+1}D_{q}f_{o}(x) +1−q^{2α+1}
(1−q)x f_{o}(x)

= D_{q}^{+}fe(x) +Dq,αfo(x)

= ∆^{+}_{α,q}f_{e}(x) +∆_{α,q}f_{o}(x).

It is noteworthy that in the caseα=−1

2, Λ_{α,q}reduces to the Rubin’sq-derivative
operator ∂_{q} defined in [13] and that for a differentiable function f, the q-Dunkl
operator Λ_{α,q}f tends to the classical Dunkl operator Λ_{α}f 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;q^{2})
gives

Λ^{k}_{α,q}bn,α(x;q^{2}) =b_{n−k,α}(x;q^{2}), k= 0,1, ...n. (4.9)
(2) If f is an even function, then

Λ^{2n}_{α,q}f(x) =q^{−n(n+1)}∆^{2n}_{α,q}f(q^{−n}x), n= 0,1,2, ..., (4.10)
Λ^{2n+1}_{α,q} f(x) =q^{−(n+1)}^{2}∆^{2n+1}_{α,q} f(q^{−n−1}x); 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,q}f(x) = C(α;q)

2 Z 1

−1

W_{α}(t;q^{2})(1 +t)f(xt)d_{q}t, x∈R, (4.12)
where

C(α;q) = (1 +q)Γ_{q}2(α+ 1)

Γ_{q}2(1/2)Γ_{q}2(α+ 1/2) (4.13)
and

W_{α}(t;q^{2}) = (t^{2}q^{2};q^{2})_{∞}
(t^{2}q^{2α+1};q^{2})∞

. (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_{q}f,D^{+}_{q}f and

∂qf are in C(R), then

∆α,qVα,q(f) =Vα,q(Dqf); (4.15)

∆^{+}_{α,q}Vα,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_{q}f)(x) = C(α;q)
2

Z 1

−1

W_{α}(t;q^{2})tD_{q}f_{e}(xt)d_{q}t
+C(α;q)

2 Z 1

−1

Wα(t;q^{2})Dqf0(xt)dqt.

(4.18)

On the other hand, from the definition of∆_{α,q} (4.3) we have

∆_{α,q}V_{α,q}(f)(x) = C(α;q)
2

Z 1

−1

W_{α}(t;q^{2})tD_{q}f_{e}(xt)d_{q}t
+C(α;q)

2 Z 1

−1

Wα(t;q^{2})t^{2}Dq,αf0(xt)dqt.

(4.19)

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

Z 1

−1

Wα(t;q^{2})

Dqf0(xt)−t^{2}Dq,αf0(xt)
dqt

= Z 1

−1

Wα(t;q^{2})(1−t^{2})f0(xt)
(1−q)xt dqt

− Z 1

−1

W_{α}(t;q^{2})(1−q^{2α+1}t^{2})f0(qxt)
(1−q)xt d_{q}t

= Z 1

−1

W_{α}(t;q^{2})(1−t^{2})f_{0}(xt)
(1−q)xt d_{q}t

− Z 1

−1

Wα(qt;q^{2})(1−q^{2}t^{2})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,q}V_{α,q}(f) =∆^{+}_{α,q}V_{α,q}f_{e}+∆_{α,q}V_{α,q}f_{o}=V_{α,q} D_{q}^{+}f_{e}+D_{q}f_{o}

=V_{α,q}(∂_{q}f).
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,q}z^{n}= (q;q)_{n}
(q;q)n,α

z^{n}, n= 0,1,2, ...; (4.20)
V_{α,q}b_{n}(z;q^{2}) =b_{n,α}(z;q^{2}), n= 0,1,2, ...; (4.21)
Vα,qeq(z) =eq,α(z), |z|<1; (4.22)

V_{α,q}E_{q}(z) =E_{q,α}(z); (4.23)

Vα,qe(λz;q^{2}) =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,qz^{2n} = C(α;q)z^{2n}

Z 1 0

(t^{2}q^{2};q^{2})_{∞}

(t^{2}q^{2α+1};q^{2})_{∞}t^{2n}dqt=C(α;q)Γq^{2}(n+^{1}_{2})Γq^{2}(α+^{1}_{2})
(1 +q)Γ_{q}2(n+α+ 1)z^{2n}

= Γ_{q}2(α+ 1)Γ_{q}2(n+^{1}_{2})

Γ_{q}2(^{1}_{2})Γ_{q}2(n+α+ 1)z^{2n}= (q;q^{2})n

(q^{2α+2};q^{2})_{n}z^{2n}.
Similarly, we prove that

Vα,qz^{2n+1}=C(α;q)z^{2n+1}
Z 1

0

(t^{2}q^{2};q^{2})_{∞}

(t^{2}q^{2α+1};q^{2})_{∞}t^{2(n+1)}dqt= (q;q^{2})n+1

(q^{2α+2};q^{2})_{n+1}z^{2n+1}.
Then (4.20 ) follows from the two following facts

(q;q)_{2n}
(q;q)2n,α

= (q;q^{2})_{n}
(q^{2α+2};q^{2})n

, (q;q)_{2n+1}
(q;q)2n+1,α

= (q;q^{2})_{n+1}
(q^{2α+2};q^{2})n+1

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

Vα,qeq(z) =

∞

X

k=0

V_{α,q}z^{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^{s}E_{q}2(−q^{2}t^{2})dqt= (1−q) (q^{2};q^{2})_{∞}

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

G^{α}_{q,n}(λ) :=

Z ∞ 0

e_{q}2(−λy^{2})y^{2n+2α+1}dqy=cq,α(λ)q^{−n}^{2}^{−(2α+1)n}

λ^{n} q^{2α+2};q^{2}

n, (5.2) where

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

(−λ,−q^{2}/λ, q^{2α+2};q^{2})_{∞} . (5.3)

Proof. (1) Lets≥0. From (2.11), we have by doing the two changes of variables
u= (1−q^{2})tandu=t^{2},

Γ_{q}2(s) = 1
(1−q^{2})^{s}

Z 1 0

t^{s−1}E_{q}2(−q^{2}t)d_{q}2t= 1 +q
(1−q^{2})^{s}

Z 1 0

u^{2s−1}E_{q}2(−q^{2}u^{2})d_{q}u.

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_{q}2(−λy^{2})y^{s}dqy = (1−q)

∞

X

k=−∞

q^{(s+1)k}
(−λq^{2k};q^{2})_{∞}

= (1−q) −λq^{s+1},−q^{−s+1}/λ, q^{2};q^{2}

∞

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

Z ∞ 0

eq^{2}(−λy^{2})y^{2n+2α+1}dqy= (1−q) −λq^{2α+2n+2},−q^{−2α−2n}/λ, q^{2};q^{2}

∞

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

−λq^{2α+2n+2};q^{2}

∞= −q^{2α+2}λ;q^{2}

∞

(−q^{2α+2}λ;q^{2})_{n} , q^{2α+2n+2};q^{2}

∞= q^{2α+2};q^{2}

∞

(q^{2α+2};q^{2})_{n}
and

−q^{−2α−2n}/λ;q^{2}

∞=q^{−n(n+1)}(q^{2α}λ)^{−n} −q^{2α+2}λ;q^{2}

n −(q^{2α}λ)^{−1};q^{2}

∞.

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) := x^{n} _{2}φ_{0}(q^{−n}, q^{−n+1};−;q^{2}, q^{2n−1}x^{−2})

= (q;q)n
[^{n}_{2}]

X

k=0

(−1)^{k}q^{k(k−1)}x^{n−2k}

(q^{2};q^{2})_{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
[^{n}_{2}]

X

k=0

(−1)^{k}q^{k(k−1)}x^{n−2k}

(q^{2};q^{2})_{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−q^{2}x;q)

(1−q^{2})^{n}^{2} = n!q

(1 +q)^{n}

[^{n}_{2}]

X

k=0

(−1)^{k}q^{k(k−1)}((1 +q)x)^{n−2k}
k!_{q}2(n−2k)!_{q,α} .

So by (3.4), we obtain lim

q→1^{−}

hn,α(p

1−q^{2}x;q)

(1−q^{2})^{n}^{2} =H^{α+}

1

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

= (−1)^{n}q^{n(n−1)}(q;q^{2})_{n} _{2}φ_{1}(q^{−2n},0;q^{2α+2};q^{2}, q^{2}x^{2}),
h_{2n+1,α}(x;q) = (q;q)2n+1

(q;q)_{2n+1,α}x^{2n+1} _{2}φ_{0}(q^{−2n}, q^{−2n−2α−2};−;q^{2}, q^{4n+2α+2}x^{−2})

= (−1)^{n}q^{n(n−1)}(q;q^{2})n+1

1−q^{2α+2}x2φ1(q^{−2n},0;q^{2α+4};q^{2}, q^{2}x^{2}).

(5.6) Proof. We have

h_{2n,α}(x;q) = (q;q)_{2n}

n

X

k=0

(−1)^{k}q^{k(k−1)}x^{2n−2k}

(q^{2};q^{2})k(q^{2};q^{2})_{n−k}(q^{2α+2};q^{2})_{n−k}.
Using the identity

(a;q^{2})_{n−k}= (a;q^{2})n

(a^{−1}q^{2−2n};q^{2})_{k} −q^{2}a^{−1}^{k}

q^{k(k−1)−2nk}, (5.7)
we get

(q^{2};q^{2})_{n−k}= (q^{2};q^{2})n

(q^{−2n};q^{2})_{k}(−1)^{k}q^{k(k−1)−2nk} (5.8)
and

(q^{2α+2};q^{2})_{n−k}= (q^{2α+2};q^{2})n

(q^{−2n−2α};q^{2})_{k} −q^{−2α}^{k}

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

h2n,α(x;q) = (q;q)2n

(q;q)_{2n,α}x^{2n}

×

n

X

k=0

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

= (q;q)2n

(q;q)2n,α

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

Now, we have

h2n+1,α(x;q) = (q;q)2n+1 n

X

k=0

(−1)^{k}q^{k(k−1)}x^{2n+1−2k}

(q^{2};q^{2})_{k}(q^{2};q^{2})_{n−k}(q^{2α+2};q^{2})_{n+1−k}.

Then, using (5.8) and replacingnbyn+ 1 in (5.9), we obtain
(q^{2α+2};q^{2})_{n+1−k}= (q^{2α+2};q^{2})_{n+1}

(q^{−2n−2α−2};q^{2})k

−q^{−2α}^{k}

qk(k−1)−2nk−2k

and

h2n+1,α(x;q) = (q;q)_{2n+1}
(q;q)2n+1,α

x^{2n+1}

×

n

X

k=0

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

= (q;q)2n+1

(q;q)_{2n+1,α}x^{2n+1} _{2}φ_{0}(q^{−2n}, q^{−2n−2α−2};−;q^{2}, q^{4n+2α+2}x^{−2}).

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

2φ0(q^{−2n}, b;−;q^{2}, q^{2n}z) = (b;q^{2})nz^{n} 2φ1(q^{−2n},0;b^{−1}q^{2−2n};q^{2},q^{2}

bz), (5.10) we obtain

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

×2φ1(q^{−2n},0;q^{2α+2};q^{2}, q^{2}x^{2}).

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

(a;q^{2})n= (a^{−1}q^{2−2n};q^{2})n(−a)^{n}q^{n(n−1)}, a6= 0, (5.11)
we get

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

×2φ1(q^{−2n},0;q^{2α+2};q^{2}, q^{2}x^{2}).

So, it follows from (4.25) that

h2n,α(x;q) = (−1)^{n}q^{n(n−1)}(q;q^{2})n 2φ1(q^{−2n},0;q^{2α+2};q^{2}, q^{2}x^{2}).

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

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

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

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

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

h2n+1,α(x;q) = (−1)^{n}q^{n(n−1)}(q;q^{2})n+1

1−q^{2α+2}x2φ1(q^{−2n},0;q^{2α+4};q^{2}, q^{2}x^{2}).

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

p_{n}(x;a|q) = 2φ_{1}(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)^{n}q^{n(n−1)}(q;q^{2})_{n}p_{n}(x^{2};q^{2α}|q^{2}),
h2n+1,α(x;q) = (−1)^{n}q^{n(n−1)}(q;q^{2})n+1

1−q^{2α+2}xpn(x^{2};q^{2α+2}|q^{2}).

Proposition 5.1. The following relations hold:

(1)q-Integral representation of Mehler type

h_{n,α}(x;q) =V_{α,q}h_{n}(x;q). (5.13)
(2) Generating function

E_{q}2(−z^{2})eq,α(xz) =

∞

X

n=0

hn,α(x;q) z^{n}

(q;q)_{n}, (|xz|<1). (5.14)
(3) Inversion formula

x^{n}= (q;q)n,α
[^{n}_{2}]

X

k=0

h_{n−2k,α}(x;q)

(q^{2};q^{2})k(q;q)_{n−2k}. (5.15)
(4) Three terms recursion formula

xh_{n,α}(x;q)−qn−1+(2α+1)θn(1−q^{n})h_{n−1,α}(x;q) = 1−qn+1+(2α+1)θ_{n}

1−q^{n+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_{q}2(−z^{2})e_{q}(xz) =

∞

X

n=0

h_{n}(x;q) z^{n}

(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]):

x^{n}= (q;q)n
[^{n}_{2}]

X

k=0

hn−2k(x;q)
(q^{2};q^{2})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−q^{n})hn−1,α(x;q).

We have

xh_{2n,α}(x;q) − q^{2n+2α}(1−q^{2n})h_{2n−1,α}(x;q)

= (q;q)2n n

X

k=0

(−1)^{k}q^{k(k−1)}x^{2n−2k+1}
(q^{2};q^{2})k(q;q)_{2n−2k,α}

−q^{2n+2α}(q;q)2n
n−1

X

k=0

(−1)^{k}q^{k(k−1)}x2n−2(k+1)+1

(q^{2};q^{2})_{k}(q;q)2n−2(k+1)+1,α

.

Changektok−1 in the second sum, then combine with the first to get
(q;q)2nx^{2n+1}

(q;q)2n,α

+ (q;q)2n n

X

k=1

(−1)^{k}q^{k(k−1)}x^{2n−2k+1}
(q^{2};q^{2})k(q;q)_{2n−2k+1,α}

×

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

(q;q)2nx^{2n+1}
(q;q)2n,α

+ (1−q^{2n+2α+2})(q;q)2n
n

X

k=1

(−1)^{k}q^{k(k−1)}x^{2n−2k+1}
(q^{2};q^{2})k(q;q)_{2n−2k+1,α}.
The last expression can be written as

1−q^{2n+2α+2}

1−q^{2n+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−q^{n}

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

hn,α(x;q)−q^{(2α+1)θ}^{n+1}hn,α(qx;q) = (1−q^{n})xh_{n−1,α}(x;q). (5.19)
(2) The backward shift operator:

∆^{+}_{α,q}

E_{q}2(−q^{2}x^{2})hn,α(x;q)

=−q^{−n}[θn(2α+ 1) +n+ 1]q

1−q^{n+1} E_{q}2(−q^{2}x^{2})hn+1,α(x;q),
(5.20)
or equivalently

q^{θ}^{n+1}^{(2α+1)}hn,α(x;q)−(1−x^{2})hn,α(q^{−1}x;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−q^{n}

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_{q}2(−q^{2}x^{2})E_{q}2(−q^{2}t^{2})eq,α(qxt).

From (4.7), we have

∆^{+}_{α,q,x}g(x, t) = E_{q}2(−q^{2}t^{2})t−x

1−qE_{q}2(−q^{2}x^{2})e_{q,α}(xt)

= −Eq^{2}(−q^{2}x^{2})x−t

1−qEq^{2}(−q^{2}t^{2})eq,α(xt)

= −∆^{+}_{α,q,t}g(x, t).

(5.21)

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

∆^{+}_{α,q,x}g(x, t) =

∞

X

n=0

∆^{+}_{α,q,x}

E_{q}2(−q^{2}x^{2})hn,α(x;q) (qt)^{n}

(q;q)_{n} (5.22)
and

∆^{+}_{α,q,x}g(x, t) =−

∞

X

n=0

E_{q}2(−q^{2}x^{2})hn,α(x;q) q^{n}

(q;q)_{n}∆^{+}_{α,q}t^{n}.
It follows from (4.5 ) and (5.21) that

∆^{+}_{α,q,x}g(x, t) =−

∞

X

n=1

E_{q}2(−q^{2}x^{2})hn,α(x;q)
(q;q)n

[θ_{n+1}(2α+ 1) +n]_{q}t^{n−1}. (5.23)
Therefore, by comparing the coefficients of t^{n} 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_{q}2(−q^{2}x^{2})h_{n,α}(x;q) = (q−1)^{n}q^{n(n−1)}^{2} (q;q)n

(q;q)_{n,α}

∆^{+}_{α,q}^{n}

E_{q}2(−q^{2}x^{2})

. (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=0}satisfy the followingq-difference
equations:

q^{2α+1}h_{2n,α}(qx;q)−(q+q^{2α+1}−q^{1−2n}x^{2})h_{2n,α}(x;q) +q(1−x^{2})h_{2n,α}(q^{−1}x;q) = 0.

(5.25)
q^{2α+1}h2n+1,α(qx;q)−(1+q^{2α+2}−q^{−2n}x^{2})h2n+1,α(x;q)+q(1−x^{2})h2n+1,α(q^{−1}x;q) = 0.

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

∆^{2}_{α,q}h_{2n,α}(x;q) = (1−q^{2n})(1−q^{2n−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)−q^{2n−2}(1−q^{2n−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−q^{2n})xh_{2n−1,α}(x;q). (5.28)
It follows, then, from (5.27) and (5.28) that

∆^{2}_{α,q}h2n,α(x;q) = 1
(1−q)^{2}

q^{2}h2n,α(x;q)−q^{2−2n}h2n,α(qx;q)
or equivalently,

(1−q^{2}x^{2})h_{2n,α}(x;q)−(1+q^{2α}−q^{2−2n}x^{2})h_{2n,α}(qx;q)+q^{2α}h_{2n,α}(q^{2}x;q) = 0. (5.29)

Replacexbyq^{−1}xin (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^{p}hn,α(x;q)E_{q}2(−q^{2}x^{2})|x|^{2α+1}dqx

=

2(1−q)(q^{2};q^{2})_{∞}

(q^{2α+2};q^{2})_{∞} (q;q)nq^{[}^{n}^{2}^{]}(^{2[}^{n+1}2 ]+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^{2p}h_{2n,α}(x;q)E_{q}2(−q^{2}x^{2})|x|^{2α+1}d_{q}x

= 2(q;q)2n n

X

k=0

(−1)^{n−k}q(n−k)(n−k−1)Fq(2k+ 2p+ 2α+ 1)
(q^{2};q^{2})_{n−k}(q^{2};q^{2})k(q^{2α+2};q^{2})k

,

(5.31)

whereF_{q} is the function defined by (5.1), then the above sum becomes
2(1−q)(q^{2};q^{2})_{∞}(q;q)2n

n

X

k=0

(−1)^{n−k}q(n−k)(n−k−1)

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

2(1−q)(q^{2};q^{2})∞(q;q)2n

(q^{2α+2};q^{2})_{∞}

n

X

k=0

(−1)^{n−k}q(n−k)(n−k−1)

(q^{2};q^{2})_{n−k}(q^{2};q^{2})k

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

(q^{2k+2α+2};q^{2})p= (q^{2α+2};q^{2})p(q^{2p+2α+2};q^{2})k

(q^{2α+2};q^{2})_{k} , (5.32)
the sum in (5.31) can be written as

(−1)^{n}q^{n(n−1)}2(1−q)(q^{2};q^{2})_{∞}(q;q)2n(q^{2α+2};q^{2})p

(q^{2α+2};q^{2})_{∞}(q^{2};q^{2})_{n}

×

n

X

k=0

(q^{−2n};q^{2})k(q^{2p+2α+2};q^{2})k

(q^{2};q^{2})k(q^{2α+2};q^{2})k

q^{2k}

= (−1)^{n}q^{n(n−1)}2(1−q)(q^{2};q^{2})_{∞}(q;q)_{2n}(q^{2α+2};q^{2})_{p}
(q^{2α+2};q^{2})∞(q^{2};q^{2})n

×_{2}φ_{1}(q^{−2n}, q^{2p+2α+2};q^{2α+2};q^{2}, q^{2}).

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

2φ_{1}(q^{−2n}, b;c;q^{2}, q^{2}) = (b^{−1}c;q^{2})_{n}
(c;q^{2})n

b^{n},