On transformations involving generalized basic hypergeometric function of two variables* 

R.K. Yadav¹, S.D. Purohit², V.K. Vyas¹ 

¹Department of Mathematics and Statistics, J. N. Vyas University. Jodhpur-342005, India.
rkmdyadav@gmail.com, vyas.vijay01@rediffmail.com.

²Department of Basic Science (Mathematics), College of Technology and Engineering,
M.P University of Agriculture and Technology. Udaipur, India. sunil_a_purohit@yahoo.com 

Abstract 

In the present paper, transformations for basic analogue of the Fox’s H-function of one and two variables have been derived by the application of the q-Leibniz rule for the product of two basic functions. Some special cases involving a basic analogue of Meijer’s G-function are also derived. 

Key words:  Fractional q-derivative operator, q-Leibniz rule, basic analogue of Fox’s H-function and basic Meijer’s G-function. 

Transformaciones de la función hipergeométrica básica generalizada de dos variables 

Resumen 

En el presente trabajo han sido derivadas transformaciones para la análoga básica de la función H de Fox de una y dos variables, aplicando la regla de Leibniz-q para el producto de dos funciones básicas. Algunos casos especiales que incluyen una análoga básica de la función G de Meijer son también derivados. 

Palabras clave:  Operador derivada-q fraccional, regla de Leibniz-q, análoga básica de la función H de Fox y la función G de Meijer básca. 

Recibido el 27 de Julio de 2009 

En forma revisada el 12 de Abril de 2010 

* 2010: Mathematics Subject Classification: Primary 33D60, secondary 26A33.

1. Introduction 

Recently in a couple of papers Yadav and Purohit [1, 2] have used the q-Leibniz rule for the fractional q-derivatives of the product of various basic hypergeometric function full stop. This has resulted in deduction of several transformations and expansion formulae involving the basic hypergeometric functions. Earlier Denis [3] and Shukla [4] have used the q-Leibniz rule to derive certain transformations for basic hypergeometric functions. 

Recently Yadav et al. [5] have investigated the fractional q-calculus operators involving the basic analogue of Fox’s H-function and basic analogue of Meijer’s G-function of two variables. 

Motivated by the aforementioned work, we investigate the applications of the q-Leibniz rule to a product involving the basic analogue of Fox’s H-function of two variables. This shall further be used to derive transformations involving the above mentioned functions. 

The fractional q-differential operator of arbitrary order µ, cf. Al-Salam [6], is defined as: 

,    (1) 

2. Transformations involving a basic analogue of Fox’s H-function of two variables 

In this section, we shall establish certain theorems involving some transformations associated with the basic analogue of the Fox’s H-function and Meijer’s G-function of two variables. 

3. Applications 

The q-extension of the H-function of two variables defined by (10) in terms of the Mellin-Barnes type of basic contour integrals, possess the advantage that a number of q-special functions (including Fox’s H-function of one variable) happen to be the particular cases of this function. The transformations deduced in the previous section can find many applications giving rise to the transformations for various q-special functions, which are special cases of the Fox’s H-function. 

Acknowledgement 

The authors are thankful to the referees for their valuable comments, which have helped in improvement of the paper 

References 

1.  Yadav, R.K. and Purohit, S.D.: Fractional q-derivatives and certain basic hypergeometric transformations. South-East Asian J. Math. & Math. Sc., 2(2) (2004), 37-46. 

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