A study of integral transform of convolution type integrals involving k-hypergeometric functions

Abstract

Convolution type integrals are of immense importance in the field of probability, statistics, computer vision and many more. It gives us a way to deal with transforms of fairly arbitrary product of functions. Various authors have discussed and obtained several Laplace transforms of convolution type integrals such as those containing generalized hypergeometric function $_p F_q$ and Kummer's function $_1 F_1$. The main purpose of our paper is to use the extended form of generalized hypergeometric and Kummer's function defined as $k$-analogue hypergeometric function or generalized $k$-hypergeometric function. We will derive the Laplace transform of convolution type integral involving generalized $k$-hypergeometric function and reduce it to some already known results existing in literature. We will also discuss various new and intriguing Laplace transforms of convolution type integrals involving product of two special $k$-analogue hypergeometric function using various summation formulas. To illustrate our results, some graphical representation of the results have also been mentioned.