Relation of Some Known Functions in terms of Generalized Meijer  G -Functions

Xiaolong Qin,Syed Ali Haider Shah,Shahid Mubeen,Gauhar Rahman,Jihad Younis

doi:10.1155/2021/7032459

Relation of Some Known Functions in terms of Generalized Meijer G -Functions

Xiaolong Qin, Syed Ali Haider Shah + Show 3 more

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https://doi.org/10.1155/2021/7032459

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Abstract

The aim of this paper is to prove some identities in the form of generalized Meijer G -function. We prove the relation of some known functions such as exponential functions, sine and cosine functions, product of exponential and trigonometric functions, product of exponential and hyperbolic functions, binomial expansion, logarithmic function, and sine integral, with the generalized Meijer G -function. We also prove the product of modified Bessel function of first and second kind in the form of generalized Meijer G -function and solve an integral involving the product of modified Bessel functions.

Highlights

E exponential functions, sine and cosine functions, and binomial expansion in the form of hypergeometric functions are given in [2, 3] as, respectively, eα
Sarivastava et al [7] worked on Mittag-Leffler type functions and estimated the Faber polynomial coefficient of biclose-to-convex functions connected with the Borel distribution of the Mittag-Leffler type. ey considered the Fekete-Szegotype inequalities for biclose
E concept of k-symbol was introduced by Diaz [11, 12]. e k-theory gave boost to the field of special functions. e researchers [13,14,15] started to work on this particular k-symbol and proved many properties and identities

Summary

Introduction

E exponential functions, sine and cosine functions, and binomial expansion in the form of hypergeometric functions are given in [2, 3] as, respectively, eα. E definitions of Meijer G-function in the form of hypergeometric function are given [31], respectively, as We first define the generalized form of Meijer G-function in the integral and hypergeometric forms and obtain some known special functions such as the Bessel function, exponential function, sine function, cosine function, sine and cosine hyperbolic functions, product of exponential and trigonometric functions, product of exponential and hyperbolic functions, binomial expansions, and logarithmic function by using the definitions of generalized Meijer G-functions for different choice of parameters.

Results

Conclusion