Some properties of the hypergeometric functions of one variable (original) (raw)

New Generalized Hypergeometric Functions

Ikonion Journal of Mathematics

The classical Gauss hypergeometric function and the Kumar confluent hypergeometric function are defined using a classical Pochammer symbol , and a factorial function. This research paper will present a two-parameter Pochhammer symbol, and discuss some of its properties such as recursive formulae and integral representation. In addition, the generalized Gauss and Kumar confluent hypergeometric functions are defined using a two-parameter Pochhammer symbol and two-parameter factorial function and some of the properties of the new generalized hypergeometric functions were also discussed.

Further results on generalized hypergeometric functions

Applied Mathematics and Computation, 2003

Recently, Virchenko et al. [Integral Transform. and Spec. Funct. 12 (1) (2001) 89-100] have defined and studied a generalized hypergeometric function of the form 2 R s 1 ða; b; c; s; zÞ ¼ CðcÞ CðbÞCðc À bÞ Z 1 0 t bÀ1 ð1 À tÞ cÀbÀ1 ð1 À zt s Þ dt:

Properties of a Newly Defined Hypergeometric Power Series Function

2009

Ahmad (2007b) has recently defined a generalized hypergeometric series function and referred to it as a hypergeometric power series function or rs H -function which is an alternative notation for the rs F -function, the rs H notation has advantages when the arguments are large and parameters are repeated and discussed the some basic properties. In this paper further properties of the hypergeometric power series functions have been developed.

On a new hypergeometric transformation

The aim of this article is to establish a general transformation for generalized hypergeometric function involving hypergeometric polynomials, by the method of elementary manipulation of series representation and to derive certain Chaundy’s formulae [3] by another method. Two applications are presented; Watson’s theorem on the sum of 3F2 and their contiguous summation formulae are deduced by means of the generalized Gauss’ second summation theorem.

Certain Properties of the Generalized Gauss Hypergeometric Functions

Applied Mathematics & Information Sciences, 2014

In this paper, we aim at establishing certain integral transform and fractional integral formulas for the generalized Gauss hypergeometric functions, which was introduced and studied byÖzergin et al. [J. Comput. Appl. Math. 235(2011), 4601-4610]. All the results derived here are of general character and can yield a number of (known and new) results in the theory of integral transforms and fractional integrals.

Some properties of generalized hypergeometric coefficients

Advances in Applied Mathematics, 1988

Let s and z be complex variables,   s  be the Gamma function, and       differentiation and integration, representation in terms of p q F and in terms of Mellin-Barnes type integral. Euler (Beta) transforms, Laplace transform, Mellin transform, Whittaker transform have also been obtained; along with its relationship with Fox H-function and Wright hypergeometric function.   .      When r  and t  are equal to 1, Equation (1.3) differs from the generalized hypergeometric function p q F by a constant multiplier only. The generalized form of the hypergeometric function has been investigated by Dotsenko [5], Malovichko [6] and one of the special cases considered by Dotsenko [5] as

A GENERALIZATION OF HYPERGEOMETRIC AND GAMMA FUNCTIONS

Al-Saqabi et al. [4] defined a gamma-type function and its probability density function involving a confluent hypergeometirc function. We then define gamma-type function involving newly defined hypergeometric function of two variables and discuss its probability density function along with some of its associated statistical functions. We use inverse Mellon transform technique to derive closed form of gamma-type function and moment generating function.

Note on generating relations associated with the generalized Gauss hypergeometric function

Applied Mathematical Sciences, 2016

The objective of this paper is to establish some new generating relations involving the generalized hypergeometric function and the generalized confluent hypergeometric function by mainly applying Taylor's theorem. Due to their very general nature, the main results can be shown to be specialized to yield a large number of new, interesting and useful generating relations involving the Gauss hypergeometric function and its related functions.

Extended Generalized Τ -Gauss’ Hypergeometric Functions and Their Applications

South East Asian J. of Mathematics and Mathematical Sciences

In this article, by means of the extended beta function, we introduce new extension of the generalized τ -Gauss’ hypergeometric functions and present some new integral and series representations (including the one obtained by adopt- ing the well-known Ramanujan’s Master Theorem). We also consider some new and known results as consequences of our proposed extension of the generalized τ -Gauss hypergeometric function.

Properties and Applications of Extended Hypergeometric Functions

2014

In this article, we study several properties of extended Gauss hypergeometric and extended confluent hypergeometric functions. We derive several integrals, inequalities and establish relationship between these and other special functions. We also show that these functions occur naturally in statistical distribution theory. MSC:33C90