• Bulletin of the Korean Mathematical Society
• /
• v.60 no.3
• /
• pp.575-591
• /
• 2023

Motivated by several generalizations of the Pochhammer symbol and their associated families of hypergeometric functions and hypergeometric polynomials, by choosing to use a very generalized Pochhammer symbol, we aim to introduce certain extensions of the generalized Lauricella function F⁽ⁿ⁾_A and the Humbert's confluent hypergeometric function Ψ⁽ⁿ⁾ of n variables with, as their respective particular cases, the second Appell hypergeometric function F₂ and the generalized Humbert's confluent hypergeometric functions Ψ₂ and investigate their several properties including, for example, various integral representations, finite summation formulas with an s-fold sum and integral representations involving the Laguerre polynomials, the incomplete gamma functions, and the Bessel and modified Bessel functions. Also, pertinent links between the major identities discussed in this article and different (existing or novel) findings are revealed.

• Honam Mathematical Journal
• /
• v.37 no.1
• /
• pp.113-126
• /
• 2015

Recently, Liu and Wang generalized Appell's and Lauricella's hypergeometric functions. Motivated by the work of Liu and Wang, the main object of this paper is to present new generalizations of Appell's and Lauricella's hypergeometric functions. Some integral representations, transformation formulae, differential formulae and recurrence relations are obtained for these new generalized Appell's and Lauricella's functions.

• Honam Mathematical Journal
• /
• v.36 no.2
• /
• pp.357-385
• /
• 2014

Recently several authors have extended the Gamma function, Beta function, the hypergeometric function, and the confluent hypergeometric function by using their integral representations and provided many interesting properties of their extended functions. Here we aim at giving further extensions of the abovementioned extended functions and investigating various formulas for the further extended functions in a systematic manner. Moreover, our extension of the Beta function is shown to be applied to Statistics and also our extensions find some connections with other special functions and polynomials such as Laguerre polynomials, Macdonald and Whittaker functions.

• Journal of the Korean Mathematical Society
• /
• v.53 no.2
• /
• pp.363-379
• /
• 2016

Motivated mainly by certain interesting recent extensions of the generalized hypergeometric function [Integral Transforms Spec. Funct. 23 (2012), 659-683] and the second Appell function [Appl. Math. Comput. 219 (2013), 8332-8337] by means of the incomplete Pochhammer symbols $({\lambda};{\kappa})_{\nu}$ and $[{\lambda};{\kappa}]_{\nu}$, we introduce here the family of the incomplete generalized ${\tau}$-hypergeometric functions $2{\gamma}_1^{\tau}(z)$ and $2{\Gamma}_1^{\tau}(z)$. The main object of this paper is to study these extensions and investigate their several properties including, for example, their integral representations, derivative formulas, Euler-Beta transform and associated with certain fractional calculus operators. Further, we introduce and investigate the family of incomplete second ${\tau}$-Appell hypergeometric functions ${\Gamma}_2^{{\tau}_1,{\tau}_2}$ and ${\gamma}_2^{{\tau}_1,{\tau}_2}$ of two variables. Relevant connections of certain special cases of the main results presented here with some known identities are also pointed out.

• Kyungpook Mathematical Journal
• /
• v.55 no.3
• /
• pp.695-703
• /
• 2015

Several interesting and useful extensions of some familiar special functions such as Beta and Gauss hypergeometric functions and their properties have, recently, been investigated by many authors. Motivated mainly by those earlier works, we establish some fractional integral formulas involving the extended generalized Gauss hypergeometric function by using certain general pair of fractional integral operators involving Gauss hypergeometric function $_2F_1$, Some interesting special cases of our main results are also considered.

• Korean Journal of Mathematics
• /
• v.32 no.1
• /
• pp.173-182
• /
• 2024

In this paper, we derive the necessary and sufficient conditions for the Gaussian hypergeometric function to be in some subclasses of analytic functions.

• Communications of the Korean Mathematical Society
• /
• v.32 no.2
• /
• pp.321-332
• /
• 2017

In this paper, we generalize the extended Appell's and Lauricella's hypergeometric functions which have recently been introduced by Liu [9] and Khan [7]. Also, we aim at establishing some (presumbly) new integral representations and transforms for the extended generalized Appell's and Lauricella's hypergeometric functions.

• Honam Mathematical Journal
• /
• v.33 no.2
• /
• pp.129-142
• /
• 2011

A (presumably) new class of generalized triple hyper-geometric functions is presented. We also give integral representations of Laplace type for certain special cases of the new class of functions.

• Journal of the Chungcheong Mathematical Society
• /
• v.24 no.4
• /
• pp.745-758
• /
• 2011

Generalizing the Burchnall-Chaundy operator method, the authors are aiming at presenting certain decomposition formulas for the chosen six Exton functions expressed in terms of Appell's functions $F_3$ and $F_4$, Horn's functions $H_3$ and $H_4$, and Gauss's hypergeometric function F. We also give some integral representations for the Exton functions $X_i$ (i = 6, 8, 14) each of whose kernels contains the Horn's function $H_4$.

• Nonlinear Functional Analysis and Applications
• /
• v.27 no.1
• /
• pp.169-187
• /
• 2022

There have been provided a surprisingly large number of summation formulae for generalized hypergeometric functions and series incorporating a variety of elementary and special functions in their various combinations. In this paper, we aim to consider certain generalized hypergeometric function ₃F₂ with particular arguments, through which a number of summation formulas for _p+1F_p(1) are provided. We then establish a power series whose coefficients are involved in generalized hypergeometric functions with unit argument. Also, we demonstrate that the generalized hypergeometric functions with unit argument mentioned before may be expressed in terms of Bell polynomials. Further, we explore several special instances of our primary identities, among numerous others, and raise a problem that naturally emerges throughout the course of this investigation.