Results Concerning the Analysis of Multi-Index Whittaker Function

Nabiullah Khan,Talha Usman,Serkan Araci,Saddam Husain,Clemente Cesarano

doi:10.1155/2022/3828104

Nabiullah Khan, Talha Usman + Show 3 more

Open Access

https://doi.org/10.1155/2022/3828104

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Abstract

A variety of functions, their extensions, and variants have been extensively investigated, mainly due to their potential applications in diverse research areas. In this paper, we aim to introduce a new extension of Whittaker function in terms of multi-index confluent hypergeometric function of first kind. We discuss multifarious properties of newly defined multi-index Whittaker function such as integral representation, integral transform (i.e., Mellin transform and Hankel transform), and derivative formula. The results presented here, being very general, are pointed out to reduce to yield some known or new formulas and identities for relatively functions.

Highlights

Generalized and multivariable forms of the special functions of mathematical physics have witnessed a significant evolution during recent years
The Whittaker function is a solution of Whittaker equation, which is a modified form of confluent hypergeometric function of first kind, and it has various applications in multifarious area such as mathematical physics and many research areas, which are studied by various mathematicians
Inspired by the abovementioned work, in this paper, we introduced a new extension of Whittaker function in terms of multi-index Mittag–Leffler function by using an extended confluent hypergeometric function and studied their various properties such as integral transform, integral representation, and derivative formula of it

Summary

Introduction

Generalized and multivariable forms of the special functions of mathematical physics have witnessed a significant evolution during recent years. Ghayasuddin et al [7] defined a new type of confluent hypergeometric function by using extended beta function in terms of multi-index Mittag–Leffler function. Inspired by the abovementioned work, in this paper, we introduced a new extension of Whittaker function in terms of multi-index Mittag–Leffler function by using an extended confluent hypergeometric function and studied their various properties such as integral transform, integral representation, and derivative formula of it. In 2004, Chaudhary et al [6] introduced the extended hypergeometric and confluent hypergeometric functions in terms of extended beta function (6) as follows: Fρ(u, v; w;. Ghayasuddin et al [7] introduced an extension of beta function using multi-index Mittag–Leffler function as follows: Baρ1,...,as,b1,...bs (u, v) 􏽚 tu− 1(1 −. In 2013, Nagar et al [3] generalized the Whittaker function by using extended confluent hypergeometric function Φρ which is defined as

Multi-Index Whittaker Function

Integral Transform of Multi-Index Whittaker Function

Conclusion