q–Special Function and Integral Transform

International Journal of Research and Scientific Innovation (IJRSI) | Volume VII, Issue VI, June 2020 | ISSN 2321–2705

q–Special Function and Integral Transform

Ritu Sharma¹, Abha Tenguria^2
1Barkatullah Vishwavidhyalaya, Bhopal (M.P), India
²Govt. Maharani Laxmi Bai Girl’s P.G. (Autonomous) College, Bhopal (M.P.), India

Abstract:-In this paper we discuss about special function with q-analog and find out relation of q-Gamma function into Laplace Transform and Fourier Transform. We find out some new property and relations.

Keywords – Special Function, Laplace Transform, Fourier Transform, q-analog.

I. INTRODUCTION

Quantum calculus or q- calculus is widely used in Mathematics. It is considered to be one of the most difficult subject to engage in mathematics. Quantum calculus and its application use in various fields of Physics, Mechanics and Mathematical Science. In previous years q-analogy play important role in Mathematics like q-Gamma function, q-Beta function and q-Integral Transform etc.
In this paper we present the definition of q – beta function and generalized q –gamma function and their relation and properties on q-integral.
We give notation and preliminaries of q analog in second section and discuss about q-pochhammer symbol. In third section we will define generalized q-gamma function and q beta function and their relation to integral transform and obtain some auxiliary result.

Notations and Preliminaries

q –Pochhammer symbol-First we define how to apply q-notation in factorial n! , we now that by definition of limits, for q tends to 1.

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