On several new Laplace transforms of generalized hypergeometric functions 2F2(x)

By employing generalizations of Gauss's second, Bailey's and Kummer's summation theorems obtained earlier by Rakha and Rathie, we aim to establish unknown Laplace transform of six rather general formulas of generalized hypergeometric function 2F2[a,b;c,d;x]. The results obtained in this paper are simple, interesting, easily established and may be useful in theoretical physics, engineering and mathematics. Results obtained earlier by Kim et al. and Choi and Rathie follow special cases of our main findings.

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Extended Mittag-Leffler function and associated fractional calculus operators

Georgian Mathematical Journal ◽

10.1515/gmj-2019-2030 ◽

2020 ◽

Vol 27 (2) ◽

pp. 199-209 ◽

Cited By ~ 2

Author(s):

Junesang Choi ◽

Rakesh K. Parmar ◽

Purnima Chopra

Keyword(s):

Fractional Calculus ◽

Hypergeometric Function ◽

Hypergeometric Functions ◽

Fractional Integrals ◽

Generalized Hypergeometric Function ◽

Pochhammer Symbol ◽

Special Cases ◽

Fractional Calculus Operators ◽

Fractional Integrals And Derivatives ◽

Appl Math

AbstractMotivated mainly by certain interesting recent extensions of the generalized hypergeometric function [H. M. Srivastava, A. Çetinkaya and I. Onur Kıymaz, A certain generalized Pochhammer symbol and its applications to hypergeometric functions, Appl. Math. Comput. 226 2014, 484–491] by means of the generalized Pochhammer symbol, we introduce here a new extension of the generalized Mittag-Leffler function. We then systematically investigate several properties of the extended Mittag-Leffler function including some basic properties, Mellin, Euler-Beta, Laplace and Whittaker transforms. Furthermore, certain properties of the Riemann–Liouville fractional integrals and derivatives associated with the extended Mittag-Leffler function are also investigated. Some interesting special cases of our main results are pointed out.

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Application of the E-operator to evaluate an infinite integral

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s030500410000356x ◽

1969 ◽

Vol 65 (3) ◽

pp. 725-730 ◽

Cited By ~ 1

Author(s):

F. Singh

Keyword(s):

Finite Difference ◽

Hypergeometric Function ◽

Hypergeometric Functions ◽

General Nature ◽

Difference Operators ◽

Generalized Hypergeometric Function ◽

Generalized Hypergeometric Functions ◽

Confluent Hypergeometric Functions ◽

Infinite Integral

1. The object of this paper is to evaluate an infinite integral, involving the product of H-functions, generalized hypergeometric functions and confluent hypergeometric functions by means of finite difference operators E. As the generalized hypergeometric function and H-function are of a very general nature, the integral, on specializing the parameters, leads to a generalization of many results some of which are known and others are believed to be new.

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A new expansion formula for hypergeometric functions of two variables

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004100042973 ◽

1968 ◽

Vol 64 (2) ◽

pp. 413-416

Author(s):

B. L. Sharma

Keyword(s):

Hypergeometric Function ◽

Hypergeometric Functions ◽

Generalized Hypergeometric Function ◽

Expansion Formula ◽

Generalized Hypergeometric Functions ◽

Functions Of Two Variables ◽

Function Of Two Variables

The main object of this paper is to derive an expansion formula for a generalized hypergeometric function of two variables in a series of products of generalized hypergeometric functions of two variables and a Meijer's G-function. The result established in this paper is the extension of the results recently given by Srivastava (5) and Verma (6). It is interesting to note that some interesting expansions can be derived from the result by specializing the parameters.

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A NOTE ON THE ASYMPTOTIC EXPANSION OF GENERALIZED HYPERGEOMETRIC FUNCTIONS

Analysis and Applications ◽

10.1142/s0219530513500346 ◽

2013 ◽

Vol 12 (01) ◽

pp. 107-115 ◽

Cited By ~ 2

Author(s):

HANS VOLKMER ◽

JOHN J. WOOD

Keyword(s):

Probability Theory ◽

Asymptotic Expansion ◽

Hypergeometric Function ◽

Explicit Expression ◽

Hypergeometric Functions ◽

Generalized Hypergeometric Function ◽

Generalized Hypergeometric Functions

We use arguments from probability theory to derive an explicit expression for the asymptotic expansion of the generalized hypergeometric function for p = q and show that the coefficients in the expansion are polynomials in the parameters of the hypergeometric function. The idea behind this paper originates from the second author's Ph.D. thesis, where the case p = 2 is treated.

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The integration of generalized hypergeometric functions

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004100003376 ◽

1969 ◽

Vol 65 (3) ◽

pp. 591-595 ◽

Cited By ~ 2

Author(s):

G. E. Barr

Keyword(s):

Hypergeometric Function ◽

Hypergeometric Functions ◽

Generalized Hypergeometric Function ◽

Pochhammer Symbol ◽

Generalized Hypergeometric Functions ◽

Image Position

Let the generalized hypergeometric function of one variable be denoted bywhere (a)m is the Pochhammer symbol ((1, 3)).

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Expansions of Generalized Hypergeometric Functions in Series of Products of Generalized Whittaker Functions

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004100036471 ◽

1962 ◽

Vol 58 (2) ◽

pp. 239-243 ◽

Cited By ~ 1

Author(s):

F. M. Ragab

Keyword(s):

Hypergeometric Function ◽

Hypergeometric Functions ◽

Whittaker Function ◽

Generalized Hypergeometric Function ◽

Whittaker Functions ◽

Generalized Hypergeometric Functions ◽

In Series ◽

Image Position

In a previous paper (l) in this journal L. J. Slater gave expansions of the generalized Whittaker functions . She gave this name to the generalized hypergeometric function in the sense that it is a generalization of the well-known Whittaker function . In this paper series of products of generalized Whittaker functions will be evaluated in terms of such functions or in terms of generalized hypergeometric functions pFp(x). These expansions are These formulae will be proved in § 2 and particular cases will be given in § 3.

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On Values Approximation of Some Special Type Hypergeometric Functions with Irrational Parameters

Mathematics and Mathematical Modeling ◽

10.24108/mathm.0319.0000191 ◽

2019 ◽

pp. 36-44

Author(s):

P. L. Ivankov

Keyword(s):

Power Series ◽

Rational Function ◽

Hypergeometric Function ◽

Hypergeometric Functions ◽

Opposite Sign ◽

High Order ◽

Pigeonhole Principle ◽

Generalized Hypergeometric Function ◽

Generalized Hypergeometric Functions ◽

Identical Unit

In this paper we investigate arithmetic nature of the values of generalized hypergeometric functions and their derivatives. To solve the problem one often makes use of Siegel’s method. The first step in corresponding reasoning is, using the pigeonhole principle, to construct a functional linear approximating form, which has high order of zero at the origin of the coordinates.A hypergeometric function is defined as a sum of a power series whose coefficients are the products of the values of some rational function. Taken with the opposite sign, the zeroes of a numerator and a denominator of this rational function are called parameters of the corresponding generalized hypergeometric function. If the parameters are irrational it is impossible, as a rule, to employ Siegel’s method. In this case one applies the method based on the effective construction of the linear approximating form.Additional difficulties arise in case the rational function numerator involved in the formation of the coefficients of the hypergeometric function under consideration is different from the identical unit. In this situation even the availability of the effective construction of approximating form does not enable achieving an arithmetic result yet. In this paper we consider just such a case. To overcome difficulties arisen here we consider the values of the corresponding hypergeometric function and its derivatives at small points only and impose additional restrictions on parameters of the function.

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An extension of the multiple Erdélyi-Kober operator and representations of the generalized hypergeometric functions

Fractional Calculus and Applied Analysis ◽

10.1515/fca-2018-0071 ◽

2018 ◽

Vol 21 (5) ◽

pp. 1360-1376

Author(s):

Dmitrii B. Karp ◽

José L. López

Keyword(s):

Integral Operator ◽

Hypergeometric Function ◽

Hypergeometric Functions ◽

Fractional Integral ◽

Fractional Integral Operator ◽

Generalized Hypergeometric Function ◽

Generalized Hypergeometric Functions ◽

Arbitrary Complex ◽

Derivatives Of ◽

Special Case

Abstract In this paper we investigate the extension of the multiple Erdélyi-Kober fractional integral operator of Kiryakova to arbitrary complex values of parameters by the way of regularization. The regularization involves derivatives of the function in question and the integration with respect to a kernel expressed in terms of special case of Meijer’s G-function. An action of the regularized multiple Erdélyi-Kober operator on some simple kernels leads to decomposition formulas for the generalized hypergeometric functions. In the ultimate section, we define an alternative regularization better suited for representing the Bessel type generalized hypergeometric function p−1Fp. A particular case of this regularization is then used to identify some new facts about the positivity and reality of zeros of this function.

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Integrals involving products of generalized Whittaker functions

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004100003996 ◽

1965 ◽

Vol 61 (2) ◽

pp. 429-432 ◽

Cited By ~ 4

Author(s):

F. M. Ragab ◽

M. A. Simary

Keyword(s):

Hypergeometric Function ◽

Hypergeometric Functions ◽

Whittaker Function ◽

Generalized Hypergeometric Function ◽

Whittaker Functions ◽

Generalized Hypergeometric Functions

In ((4)) Slater gave expansions of the generalized Whittaker functons pFp(x). (She gave this name to the generalized hypergeometric function pFp(x), since it is a generalization of the well-known Whittaker function 1F1(x).) In another paper ((2)), Ragab gave a series of products of generalized Whittaker functions in terms of such functions or in terms of generalized hypergeometric functions pFq(x). In the present paper, integrals involving products of generalized Whittaker functions will be evaluated in terms of such functions or of other generalized hypergeometric functions.

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INTEGRAL TRANSFORM OF K4-FUNCTION

Journal of Science and Arts ◽

10.46939/j.sci.arts-21.2-a10 ◽

2021 ◽

Vol 21 (2) ◽

pp. 429-436

Author(s):

SEEMA KABRA ◽

HARISH NAGAR

Keyword(s):

Laplace Transform ◽

Hypergeometric Function ◽

Integral Transform ◽

Integral Transforms ◽

Gauss Hypergeometric Function ◽

Wright Function ◽

Generalized Wright Function ◽

Special Cases ◽

Euler Transform

In this present work we derived integral transforms such as Euler transform, Laplace transform, and Whittaker transform of K4-function. The results are given in generalized Wright function. Some special cases of the main result are also presented here with new and interesting results. We further extended integral transforms derived here in terms of Gauss Hypergeometric function.

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