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The purpose of these notes is to give a short survey of an interesting connection between partition functions of supersymmetric gauge theories and hypergeometric functions and to present the recent progress in this direction.

On Generalized Weyl Fractional q-Integral Operator Involving Generalized Basic Hypergeometric Functions

Yadav, R., Purohit, S., Kalla, S. (2008)

Fractional Calculus and Applied Analysis

Mathematics Subject Classification: 33D60, 33D90, 26A33Fractional q-integral operators of generalized Weyl type, involving generalized basic hypergeometric functions and a basic analogue of Fox’s H-function have been investigated. A number of integrals involving various q-functions have been evaluated as applications of the main results.

On q–Analogues of Caputo Derivative and Mittag–Leffler Function

Rajkovic, Predrag, Marinkovic, Sladjana, Stankovic, Miomir (2007)

Fractional Calculus and Applied Analysis

Mathematics Subject Classification: 33D60, 33E12, 26A33Based on the fractional q–integral with the parametric lower limit of integration, we consider the fractional q–derivative of Caputo type. Especially, its applications to q-exponential functions allow us to introduce q–analogues of the Mittag–Leffler function. Vice versa, those functions can be used for defining generalized operators in fractional q–calculus.

On q-Laplace Transforms of the q-Bessel Functions

Purohit, S., Kalla, S. (2007)

Fractional Calculus and Applied Analysis

Mathematics Subject Classification: 33D15, 44A10, 44A20The present paper deals with the evaluation of the q-Laplace transforms of a product of basic analogues of the Bessel functions. As applications, several useful special cases have been deduced.

On some operational representations of $q$ -polynomials

Mumtaz Ahmad Khan (1995)

Czechoslovak Mathematical Journal

On the Riemann-Liouville Fractional q-Integral Operator Involving a Basic Analogue of Fox H-Function

Kalla, S., Yadav, R., Purohit, S. (2005)

Fractional Calculus and Applied Analysis

2000 Mathematics Subject Classification: 33D60, 26A33, 33C60The present paper envisages the applications of Riemann-Liouville fractional q-integral operator to a basic analogue of Fox H-function. Results involving the basic hypergeometric functions like Gq(.), Jv(x; q), Yv(x; q),Kv(x; q), Hv(x; q) and various other q-elementary functions associated with the Riemann-Liouville fractional q-integral operator have been deduced as special cases of the main result.

Polynomial Expansions for Solutions of Higher-Order Bessel Heat Equation in Quantum Calculus

Ben Hammouda, M.S., Nemri, Akram (2007)

Fractional Calculus and Applied Analysis

Mathematics Subject Class.: 33C10,33D60,26D15,33D05,33D15,33D90In this paper we give the q-analogue of the higher-order Bessel operators studied by I. Dimovski [3],[4], I. Dimovski and V. Kiryakova [5],[6], M. I. Klyuchantsev [17], V. Kiryakova [15], [16], A. Fitouhi, N. H. Mahmoud and S. A. Ould Ahmed Mahmoud [8], and recently by many other authors. Our objective is twofold. First, using the q-Jackson integral and the q-derivative, we aim at establishing some properties of this function with proofs...

Several $q$ -series identities from the Euler expansions of ${(a; q)}_{\infty}$ and $\frac{1}{{(a; q)}_{\infty}}$

Zhizheng Zhang, Yang, Jizhen (2009)

Archivum Mathematicum

In this paper, we first give several operator identities which extend the results of Chen and Liu, then make use of them to two $q$ -series identities obtained by the Euler expansions of ${(a; q)}_{\infty}$ and $\frac{1}{{(a; q)}_{\infty}}$ . Several $q$ -series identities are obtained involving a $q$ -series identity in Ramanujan’s Lost Notebook.

Some recurrence relations for the generalized basic hypergeometric functions.

Purohit, S.D. (2009)

Bulletin of Mathematical Analysis and Applications [electronic only]

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