A new expansion formula for hypergeometric functions of two variables

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.

Application of the E-operator to evaluate an infinite integral

1969 ◽  
Vol 65 (3) ◽  
pp. 725-730 ◽  
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.

A NOTE ON THE ASYMPTOTIC EXPANSION OF GENERALIZED HYPERGEOMETRIC FUNCTIONS

2013 ◽  
Vol 12 (01) ◽  
pp. 107-115 ◽  
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.

The integration of generalized hypergeometric functions

1969 ◽  
Vol 65 (3) ◽  
pp. 591-595 ◽  
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)).

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

1962 ◽  
Vol 58 (2) ◽  
pp. 239-243 ◽  
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.

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

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.

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

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.

Integrals involving products of generalized Whittaker functions

1965 ◽  
Vol 61 (2) ◽  
pp. 429-432 ◽  
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.

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

2021 ◽  
Vol 39 (4) ◽  
pp. 97-109
Author(s):  
Asmaa Orabi Mohammed ◽  
Medhat A. Rakha ◽  
Mohammed M. Awad ◽  
Arjun K. Rathie
Keyword(s):  
Laplace Transform ◽  
Hypergeometric Function ◽  
Hypergeometric Functions ◽  
Laplace Transforms ◽  
Theoretical Physics ◽  
Generalized Hypergeometric Function ◽  
Generalized Hypergeometric Functions ◽  
Special Cases ◽  
And Mathematics

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.

An integral involving the G-function and Kampé de Fériet function

1968 ◽  
Vol 64 (4) ◽  
pp. 1041-1044 ◽  
Author(s):  
O Shanker
Keyword(s):  
Hypergeometric Function ◽  
Hypergeometric Functions ◽  
Hypergeometric Series ◽  
Special Functions ◽  
General Function ◽  
Generalized Hypergeometric Functions ◽  
General Integral ◽  
Infinite Integral ◽  
Function Of Two Variables ◽  
General Hypergeometric Function

The object of this paper is to evaluate an infinite integral involving the product of Meijer's G-function (5) and Kampé de Fériet function (1) in terms of Kampé de Fériet function. A number of papers of Bailey (3,4), Ragab (7,8), Slater (9), and Srivastava (10) have appeared, evaluating an integral in terms of a hypergeometric function of two variables or in terms of an E-function. Their results are obviously the particular cases of my result. Since Meijer's G-function is the most general function of one variable which can be expressed in terms of special functions (5) and Kampé de Fériet's function being the most general hypergeometric function of two variables, the integral given by me is the most general integral ever obtained and generalizes most of the results obtained so far for the integral of Mellin type in terms of generalized hypergeometric series. This is because the Kampé de Fériet function reduces to the product of two generalized hypergeometric functions by choosing parameters suitably.

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