Investigation of the k-Analogue of Gauss Hypergeometric Functions Constructed by the Hadamard Product

Muajebah Hidan,Mohamed Abdalla

doi:10.3390/sym13040714

Muajebah Hidan, Mohamed Abdalla

Open Access

https://doi.org/10.3390/sym13040714

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Journal: Symmetry	Publication Date: Apr 18, 2021
Citations: 5	License type: CC BY 4.0

Affiliation: King Khalid University, South Valley University

Abstract

Traditionally, the special function theory has many applications in various areas of mathematical physics, economics, statistics, engineering, and many other branches of science. Inspired by certain recent extensions of the k-analogue of gamma, the Pochhammer symbol, and hypergeometric functions, this work is devoted to the study of the k-analogue of Gauss hypergeometric functions by the Hadamard product. We give a definition of the Hadamard product of k-Gauss hypergeometric functions (HPkGHF) associated with the fourth numerator and two denominator parameters. In addition, convergence properties are derived from this function. We also discuss interesting properties such as derivative formulae, integral representations, and integral transforms including beta transform and Laplace transform. Furthermore, we investigate some contiguous function relations and differential equations connecting the HPkGHF. The current results are more general than previous ones. Moreover, the proposed results are useful in the theory of k-special functions where the hypergeometric function naturally occurs.

Highlights

The hypergeometric function of parameters consisting of two numerators and one denominator has become of increasing significance due to its applications in many fields of biophysics, quantum mechanics, spectroscopy, engineering, and other scientific areas
Some mathematical properties, such as integrals’ formulae, inequalities, differential equations, contiguous function relations, generating functions and fractional calculus related to the k-hypergeometric function of parameters consisting of two numerators and one denominator have been presented in many references, for instance, Mubeen et al [19,20,21,22], Rahman et al [23], Chinra et al [24], Korkmaz-Duzgun and ErkusDuman [25], Nisar et al [26], Li and Dong [27], Kiryakova [28], Naz and Naeem [28] and Yilmazer and Ali [29]
We introduce certain interesting k-Beta transforms, Laplace transforms, and inverse Laplace transforms associated with the Hadamard product of k-Gauss hypergeometric functions H(k; η )

Summary

Introduction

The hypergeometric function of parameters consisting of two numerators and one denominator has become of increasing significance due to its applications in many fields of biophysics, quantum mechanics, spectroscopy, engineering, and other scientific areas (see, e.g., [1,2,3,4] and the references therein). Numerous attempts have been made to generalize and improve on hypergeometric function by using an extension of the Pochhammer symbol. One class of extensions of hypergeometric functions comprises the so-called khypergeometric functions in terms of the k-analogue of gamma and the Pochhammer symbol, which can be found in the work of Diaz and Pariguan [18]. Diaz and Pariguan [18] defined the k-analogue of gamma, beta and hypergeometric functions as follows.

Hadamard Product of k-Gauss Hypergeometric Functions

Convergence Property

Derivative Formulae

Integral Representations

Integral Transforms

Contiguous Function Relations and Differential Equations η

Concluding Remarks