On fractional $q$-extensions of some $q$-orthogonal polynomials

Fractional $q$-extensions of some classical $q$-orthogonal polynomials are introduced and some of the main properties of the new defined functions are given. Next, a fractional $q$-difference equation of Gauss type is introduced and solved by means of power series method.


INTRODUCTION
Fractional calculus is the field of mathematical analysis which deals with the investigation and applications of derivatives and integrals of arbitrary (real or complex) order. It is an interesting topic having interconnections with various problems of function theory, integral and differential equations, and other branches of analysis. It has been continually developped, sitmulated by ideas and results in various fields of mathematical analysis. This is demonstrated by the many pubilcations-hundreds of papers in the past years-and by the many conferences devoted to the problems of fractional calculus.
A family {P n (x)}, (n ∈ N := {0, 1, 2 . . . }, k n = 0) of polynomials of degree exactly n is a family of classical q-orthogonal polynomials of the q-Hahn class if it is the solution of a q-differential equation of the type (1) σ(x)D q D 1/q P n (x) + τ (x)D q P n (x) + λ n P n (x) = 0, where σ(x) = ax 2 +bx+c is a polynomial of at most second order and τ (x) = dx+e is a polynomial of first order. Here, The q-difference operator D q is defined by called q-Bessel) polynomials, the q-Charlier polynomials, the Al Salam-Carlitz I polynomials, the Al Salam-Carlitz II polynomials, the Stieltjes-Wigert polynomials, the Discrete q-Hermite I polynomials, and the Discrete q-Hermite II polynomials.
In [8], the authors have defined some fractional extensions of the Jacobi polynomials from their Rodrigues representation and provided several properties of these new functions. Also, they introduced a fractional version of the Gauss hypergeometric differential equation and used the modified power series method to provide some of its solutions.
In [9], the authors defined the C-Laguerre functions from the Rodrigues representation of the Laguerre polynomials by replacing the ordinary derivative by a fractional type derivative, then they gave several properties of the new defined functions.
In this work, we introduce a new fractional q-differential operator and following previous works (see [8,9,13]), we introduce fractional q-extensions of some q-orthogonal polynomials of the q-Hahn class. The hypergeometric representations of the new defined functions are given and in some cases the limit transitions are provided.
The paper is organised as follows: 1. In Section 2, we present the preliminary results and definitions that are useful for a better reading of this manuscript.
2. In Section 3, we introduce the fractional q-calculus 3. In Section 4, we introduce a new fractional q-differential equation D α q −1 and apply it to some functions, 4. In Section 5, fractional q-extension of some q-orthogonal polynomials are given and their basic hypergeometric representation provided. We prove for some of these new defined functions some limit transitions.
5. In Section 6, we introduce a fractional q-extension of the q-hypergeometric q-difference equation and provide some of its solution. are given.

PRELIMINARY DEFINITIONS AND RESULTS
This section contains some preliminary definitions and results that are useful for a better reading of the manuscript. The q-hypergeometric series, a fractional qderivative and fractional q-integral are defined. The reader will consult the references [11,14] for more informations about these concepts.
The notation (a; q) n is the so-called q-Pochhammer symbol.
Proposition 1. [11,Page 16] The basic hypergeometric series fulfil the following identities Relation (2) is the so-called q-binomial theorem.
The next proposition gives some important Heine transformation formulas for basic hypergeometric series.
Definition 2. For any complex number λ, where the principal value of q λ is taken.
We will also use the following common notations called the q-bracket and the q-binomial coefficients, respectively.
Definition 3 (see [10]). Suppose 0 < a < b. The definite q-integral is defined as Definition 4. The q-Gamma function is defined by Remark 1. From Definition 2, the q-Gamma function is also represented by Note also that the q-Gamma function satisfies the functional equation Note that for arbitrary complex α, .
The exponential function has two different natural q-extensions, denoted by e q (z) and E q (z), which can be defined by These q-analogues of the exponential function are clearly related by e q (z)E q (−z) = 1.
Theorem 1. The following expansions hold true.
Proof. From the expansion of e q (z), we have respectively e q (αz) = ∞ n=0 α n z n (q; q) n and e q (βz) = ∞ n=0 β n z n (q; q) n .
Proof. For the first relation we use the equation e q (αz) e q (βz) = e q (αz)E q (−βz) and for the second relation we use Proposition 4. The following relations are valid.

A BRIEF REVIEW OF FRACTIONAL q-CALCULUS
The usual starting point for a definition of fractional operators in q-calculus taken in [2,3,5,15,16], is the q-analogue of the Riemann-Liouville fractional integral This q-integral was motivated from the q-analogue of the Cauchy formula for a repeated q-integral The reduction of the multiple q-integral to a single one was considered by Al-Salam in [4]. In [16], the authors allow the lower parameter in (17) to be any real number a ∈ (0, z). There are several definitions of the fractional q-integral and fractional q-derivatives. We adopt in this work the definition of the fractional q-integral given in [15].
The following lemma is of great importance in the sequel.
Mahmoud Annaby and Zeinab Mansour [2, P. 148] prove that the Riemann Liouville fractional operator D α q,0 coincides with a q-analogue of the Grünwald Letnikov fractional operator defined by In the sequel of this work, we will most of the time make use of this definition of the fractional q-derivative of order α.
[15] For 0 < c < x, the operators I α q,c and D α q,c satisfy the following property The following lemma is of great importance in the sequel.

MORE FRACTIONAL q-OPERATOR
Since many Rodrigues-type formulas for some of the orthogonal polynomials of the q-Hahn class are expressed in terms of the q-operator D q −1 instead of D q , and since our new functions are defined using the Rodrigues-type formula of each family, there is a need to develop a fractional calculus for D q −1 . The more natural way to do it is to start by the power derivative of D q −1 . The following proposition (see [6]) gives the result.
Proposition 5. Let n ∈ N 0 and f a given function defined on {q k , k ∈ Z}. Then the following power derivative rule for D q −1 applies Proof. This proof is also from [6]. The result is clear for n = 0. Assume the assertion is true for n ≥ 0, then: Note that, using the obvious relation n k q = n n − k q , and reversing the order of summation, (23) reads Next, using the fact that Note that replacing q by q −1 in (25), it follows that Taking care that and so we get Now, we can write This is exactely another way to write the result (23) obtained in [6] thanks to (24).
We are about to define a fractional extension of D q −1 . Since n k q = 0 for k > n, we can write (24) as Equation (26) suggests that for any arbitrary complex number α, D α q −1 could be defined by provided that the infinite series of the right hand side converges. Now, using equation (11), we obtain We then set the following definition.
Definition 7. For any complex number α, we define the fractional operator D α provided that the right hand side of (27) converges.
Note also that we could use directly (23) to write which may be looked as another fractional extension of D q −1 .

FRACTIONAL q-EXTENSIONS OF SOME q-ORTHOGONAL POLYNOMIALS
In this section we introduce some fractional q-extensions of some orthogonal polynomials of the q-Hahn class. The families that are of interest here are those which use D q and D q −1 in their Rodrigues representations.
They can also be represented by the Rodrigues-type formula [11, P. 438] w(x; a, b, c; q)P n (x; a, b, c; q) = a n c n q n(n+1) (1 − q) n (aq, cq; q) n D n q [w(x; aq n , bq n , cq n ; q)], Definition 8. Let λ ∈ R, we define the fractional Big q-Jacobi functions by Proposition 7. The fractional Big q-Jacobi functions defined by relation (29) have the following basic hypergeometric representation Proof. Using the definitions of the fractional Big q-Jacobi functions and the fractional q-derivative (21), we have: Using the properties of the q-pochhammer symbol, we have: Using the fact that (aq λ ; q) ∞ = (a; q) ∞ (a; q) λ then, we have
They can also be represented by the Rodrigues-type formula [11, P. 480] w(x; a, b; q)P n (x; a, b; q) = a n b n q n(n+1) (1 − q) n (aq, bq; q) n D n q [w(x; aq n , bq n ; q)] Definition 9. Let λ ∈ R, we define the fractional Big q-Laguerre functions by Proposition 8. The fractional Big q-Laguerre functions defined by relation (30) have the following basic hypergeometric representation Proof. Using the definitions of the fractional Big q-Laguerre functions and the fractional q-derivative (21), we have: Using the properties of the q-pochhammer symbol, we have: Using the fact that (aq λ ; q) ∞ = (a; q) ∞ (a; q) λ then, we have
They can also be represented by the Redrigues-type formula Definition 10. Let λ ∈ R, we define the fractional Little q-Jacobi functions by Proposition 9. The fractional Little q-Jacobi functions defined by relation (31) have the following basic hypergeometric representation Proof. Applying the q-binomial theorem (2), we have Thus, Hence, Using the fact that Now, applying the Heine-Euler transformation formula (5), we have So we have We finally obtain P λ (x; q α , q β ; q) = (−1) λ q λα+ λ(λ+1) Proposition 10. The following limit transition holds: where n is a nonnegative integer.
Proof. It is not difficult to see that so, lim λ→n P λ (x; q α ; q) = p n (x; q α ; q).
They can also be represented by the Rodrigues-type formula [11, P. 520] w(x; α; q)p n (x; Definition 11. Let λ ∈ R, we define the fractional Little q-Laguerre functions by Proposition 11. The fractional Little q-Laguerre functions defined by relation (32) have the following representation Proof. It is easy to see that Thus using the definitions of the fractional Little q-Laguerre functions and the fractional q-derivative (27), combined with the transformations (4) and (6) we have: Proposition 12. Let n be a nonnegative integer, then the following limit transition holds true lim λ→n P λ (x; q α ; q) = p n (x; q α ; q).
It is not difficult to see that w(q λ+n x; q) = (q λ x) n q ( n 2 ) w(q λ x; q).
Proposition 17. Let n be a nonnegative integer. Then the following transition limit holds.