Some results on a fractional q-integral operator involving generalized basic hypergeometric function

Leda Galué

Centro de Investigación de Matemática Aplicada (CIMA), Facultad de Ingeniería, Universidad del Zulia. Maracaibo, Venezuela. lgalue@hotmail.com

Abstract

In this paper the operator L(.) of the basic multiple hypergeometric function given by Yadav et al. is used in order to obtain the fractional q-integral operator L(.) of the generalized basic hypergeometric function _rΦ_s(.), also the q-Mellin transformfor the operator L(.) is presented. Various interesting special cases, involving q-special functions, have been derived as application of the main result.

Keywords: fractional q-integral operators, generalized basic hypergeometric function, q-Mellin transform, q-special functions.

AMS Subject Classification: 33D60, 33D90, 26A33.

Algunos resultados sobre un operador q-integral fraccional que incluye la función hipergeométrica básica generalizada

Resumen

En este trabajo el operador L(.) de la función hipergeométrica múltiple básica dado por Yadav et al. es usado con el fin de obtener el operador q-integral fraccional L(.) de la función hipergeométrica básica generalizada _rΦ_s(.), además se presenta la transformada q-Mellin del operador L(.). Varios casos especiales interesantes, que incluyen funciones q-especiales, han sido derivados como aplicación del resultado principal.

Palabras clave: operadores q-integrales fraccionales, función hipergeométrica básica generalizada, transformada q-Mellin, funciones q-especiales.

Recibido el 23 de Febrero de 2012

En forma revisada el 10 de Septiembre de 2012

1. Introduction

Nowadays, the fractional calculus theory is applied in almost all the areas of science and engineering. Operators of fractional calculus and their q-analogues have many applications, for example, they can used to solve dual integral and series equations which arise in crack problems in elasticity [1]. They find applications also in control systems, signal processing, bio-medical engineering, radars, sonars, etc. [2-4].

The concept of differ-integral of complex order n, which is a generalization of the ordinary n–th derivative and n times integral to any complex number, can be introduced in several ways. The most widely used definition of an integral of fractional order is via an integral transform, called the Riemann-Liouville operator of fractional integration: [5, p. 146]

Many authors [5-19] have defined and studied operators of fractional integration through an integral transform. Some of these operators are:

1.1. Erdélyi-Kober Operator:

[10, p. 4, No. (20)]

where n is the minor integer major that a.

1.2. Saxena operator: [17, p. 288, No. (1)]

where F(a,b;c;x) denotes the Gauss hypergeometric function, and the parameters involved are complex numbers.

1.3. The operator L(.)

We consider the operator L(.) introduced by Delgado and Galué [8] in the following form:

As particular cases of this operator we have:

where   g ÎC, l,m₁ are non-negative integers, b₁ ≠0,-1,-2,..., with Á[-l,b₁, g, m₁; f(x)] as defined in (3).

with g ÎC, l is non-negative integer, b₁ ≠ 0,-1,-2,..., and I_g,l+1 f(x) as in (2).

In this paper the operator L(.) of the basic multiple hypergeometric function given by Yadav et al. is used in order to obtain the fractional q-integral operator L(.) of the generalized basic hypergeometric function _rf_s(.), also the q-Mellin transform for the operator L(.) is presented. Various interesting special cases, involving q-special functions, have been derived as application of the main result.

2. Basic hypergeometric series

In this section we present some definitions necessary for the development of the next sections.

2.1. The q-shifted factorial

The q-shifted factorial is defined as: [20]

Also,

which converges for |q|<1 and diverges for a ≠ 0 and |q|≥1, and

2.2. Identities

We recall here the following q-identities [20, p. 233, No. (I.13); p. 235, No. (I.35)]:

2.3. Generalized basic hypergeometric series

A generalization of the basic hypergeometric series ₂f₁ is given by: [20]

Some special cases of the _rf_s(.) are:

i) The two q-exponential functions, [20, p. 9, Nos. (1.3.15), (1.3.16)]

ii) q-analogues of Bessel functions, [20, p. 25]

iii) The q-Laguerre polynomials defined by [20, p. 194]

iv) The little q-Jacobi polynomials: [21]

v) The Wall polynomials: [20, p. 196]

vi) The generalized Stieltjes-Wigert polynomials: [20, p. 196]

2.4. q-analogue of the Karlsson-Minton summation formula

Gasper (1981) derived a q-analogue of the Karlsson-Minton summation formula, which is given by: [20, p. 16, No. (1.9.10)]

where m₁, m₂,...,m_r are non-negative integers, n ≥ m₁ +...+ m_r.

2.5. The basic multiple hypergeometric function

It was introduced by H.M. Srivastava [22] and is given by

where the arguments x₁,..., x_n, the complex parameters

and the associated coefficients

are so constrained that the multiple serie (23) converges.

As particular case of (23) for n = 1, A = C = 0, f'_j = 1, j = 1,...,B', d'_j = 1, j = 1,...,D' we obtain

which is a general basic hypergeometric series [23].

2.6. The q-derivative operator

It is denoted by D_q and defined for fixed q by [20, p. 22]

2.7. q-Beta function

It is defined by [20, p. 18, No. (1.10.13), p. 19, No. (1.11.7)]

From these results we have

now applying (12), when x = u + n, Re(u) > 0, n Î Z,

3. Fractional q-integral operator L(.) of the generalized basic hypergeometric series

R.K. Yadav, S.L. Kalla and G. Kaur [24] applied the operator L(.) to the basic multiple hypergeometric function and established the following result:

where g ÎC, l, m₁,...,m_r are non-negative integers, b₁,...,b_r ≠0,-1,-2,..., and l_i(i = 1,...,n) are arbitrary quantities.

From (30) and using (24) we get

where g ÎC, l, m₁,...,m_r are non-negative integers, b₁,...,b_r ≠0,-1,-2,...

In the rest of the paper for convenience we will use the following notation:

Lemma: Let be a fractional q-integral operator and _rf_s(.) the generalized basic hypergeometric series, as defined in (4) and (13) respectively, then

with l, m₁,...,m_r non-negative integers, l ≥ m₁+...+m_r; gÎC, Re(g + l + 1) > 0; b_j, g + l + 2 - b_j - m_j ≠ 0,-1,-2,..., j = 1,2,...,r; c₁, c₂,...,c_n ≠ q^-n for n = 0,1,2,...

Proof: From (5) we get

now, taking in (31) μ₁ = r, x₁ = x with _rf_s(.) as defined in (13), we have

Now, using the results (11) and (10) joint with (12) we obtain respectively

Then the substitution of (34) and (35) in (33) leads us to (32).

Particular cases: By replacing a₁,a₂,...,a_u,c₁,c₂,...,c_n in (32) by , respectively, and letting q ®1^- we obtain according to (6) and (7)

l, m₁ non-negative integres, l ≥ m₁; g Î C, Re(g + l + 1) > 0; b₁, g + l + 2 - b₁ - m₁ ≠ 0,-1,-2,...; c₁, c₂,...,c_n ≠ q^-n for n = 0,1,2,...

l non-negative integer; g Î C, Re(g + l + 1) > 0; b₁, g + l + 2 - b₁ ≠ 0,-1,-2,...; c₁, c₂,...,c_n ≠ q^-n for n = 0,1,2,...

Interestingly, by making a suitable change to the parameters a₁,a₂,...,a_u,c₁,c₂,...,c_n and the argument x in conjunction with definitions given in (14)-(21), we obtain the following results given in Table 1.

Where

4. q-Mellin Transform of L(.)

In this section we establish the q-Mellin transform of the fractional q-integral operador

The q-Mellin Transform of f(x) is defined by [25]

Theorem 1. Let  be a fractional q-integral operator, as defined in (4), then

where M_q{f(x);s} denote the q-Mellin transform of f(x); l, m₁,...,m_r non-negative integers, l ≥ m₁+...+m_r; gÎC, Re(g - r - s + 1) > 0; b₁,...,b_r ≠0,-1,-2,...

Proof: From (5) and (46) we obtain,

where we have interchanged the order of integration.

Making a change of variable in the inner integral and using (25) we have,

The substitution of this expression in (48) yields

Now, applying (13),

and from (29)

Then, (50) and (51) lead us to

which using (13) can be written as

and by virtue of (22)

On the other hand, from (10) and (12)

and using newly (12)

Finally, from the results (52) and (53) we get (47).

As particular case from this theorem we get

where l,m non-negative integers, l ≥ m; g Î C, Re(g - r - s + 1) > 0; b ≠ 0,-1,-2,...

Acknowledgement

The author would like to thanks to CONDES-Universidad del Zulia for financial support.

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