On an extension of the generalized Pochhammer symbol and its applications to hypergeometric functions

2016 ◽  
Vol 09 (03) ◽  
pp. 1650064 ◽  
Author(s):  
Vivek Sahai ◽  
Ashish Verma
Keyword(s):  
Hypergeometric Function ◽  
Generating Functions ◽  
Hypergeometric Functions ◽  
Generalized Hypergeometric Function ◽  
Pochhammer Symbol ◽  
Contiguous Relations

The main object of this paper is to present a generalization of the Pochhammer symbol. We present some contiguous relations of this generalized Pochhammer symbol and use it to give an extension of the generalized hypergeometric function [Formula: see text]. Finally, we present some properties and generating functions of this extended generalized hypergeometric function.

Extended Mittag-Leffler function and associated fractional calculus operators

2020 ◽  
Vol 27 (2) ◽  
pp. 199-209 ◽  
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.

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)).

scholarly journals Some properties of generalized hypergeometric Appell polynomials

2020 ◽  
Vol 12 (1) ◽  
pp. 129-137 ◽  
Author(s):  
L. Bedratyuk ◽  
N. Luno
Keyword(s):  
Differential Operator ◽  
Power Series ◽  
Hypergeometric Function ◽  
Explicit Formula ◽  
Series Representation ◽  
Generalized Hypergeometric Function ◽  
Appell Polynomials ◽  
Pochhammer Symbol ◽  
Formal Power ◽  
Polynomial Family

Let $x^{(n)}$ denotes the Pochhammer symbol (rising factorial) defined by the formulas $x^{(0)}=1$ and $x^{(n)}=x(x+1)(x+2)\cdots (x+n-1)$ for $n\geq 1$. In this paper, we present a new real-valued Appell-type polynomial family $A_n^{(k)}(m,x)$, $n, m \in {\mathbb{N}}_0$, $k \in {\mathbb{N}},$ every member of which is expressed by mean of the generalized hypergeometric function ${}_{p} F_q \begin{bmatrix} \begin{matrix} a_1, a_2, \ldots, a_p \:\\ b_1, b_2, \ldots, b_q \end{matrix} \: \Bigg| \:z \end{bmatrix}= \sum\limits_{k=0}^{\infty} \frac{a_1^{(k)} a_2^{(k)} \ldots a_p^{(k)}}{b_1^{(k)} b_2^{(k)} \ldots b_q^{(k)}} \frac{z^k}{k!}$ as follows $$ A_n^{(k)}(m,x)= x^n{}_{k+p} F_q \begin{bmatrix} \begin{matrix} {a_1}, {a_2}, {\ldots}, {a_p}, {\displaystyle -\frac{n}{k}}, {\displaystyle -\frac{n-1}{k}}, {\ldots}, {\displaystyle-\frac{n-k+1}{k}}\:\\ {b_1}, {b_2}, {\ldots}, {b_q} \end{matrix} \: \Bigg| \: \displaystyle \frac{m}{x^k} \end{bmatrix} $$ and, at the same time, the polynomials from this family are Appell-type polynomials. The generating exponential function of this type of polynomials is firstly discovered and the proof that they are of Appell-type ones is given. We present the differential operator formal power series representation as well as an explicit formula over the standard basis, and establish a new identity for the generalized hypergeometric function. Besides, we derive the addition, the multiplication and some other formulas for this polynomial family.

scholarly journals Expansions of the Solutions of the General Heun Equation Governed by Two-Term Recurrence Relations for Coefficients

2018 ◽  
Vol 2018 ◽  
pp. 1-9 ◽  
Author(s):  
T. A. Ishkhanyan ◽  
T. A. Shahverdyan ◽  
A. M. Ishkhanyan
Keyword(s):  
Power Series ◽  
Hypergeometric Function ◽  
Hypergeometric Functions ◽  
Recurrence Relations ◽  
Power Series Expansion ◽  
Heun Equation ◽  
Generalized Hypergeometric Function ◽  
Gamma Functions ◽  
Heun Function ◽  
Finite Sum

We examine the expansions of the solutions of the general Heun equation in terms of the Gauss hypergeometric functions. We present several expansions using functions, the forms of which differ from those applied before. In general, the coefficients of the expansions obey three-term recurrence relations. However, there exist certain choices of the parameters for which the recurrence relations become two-term. The coefficients of the expansions are then explicitly expressed in terms of the gamma functions. Discussing the termination of the presented series, we show that the finite-sum solutions of the general Heun equation in terms of generally irreducible hypergeometric functions have a representation through a single generalized hypergeometric function. Consequently, the power-series expansion of the Heun function for any such case is governed by a two-term recurrence relation.

scholarly journals Integral representations of generalized Lauricella hypergeometric functions

1992 ◽  
Vol 15 (4) ◽  
pp. 653-657 ◽  
Author(s):  
Vu Kim Tuan ◽  
R. G. Buschman
Keyword(s):  
Integral Representation ◽  
Hypergeometric Function ◽  
Hypergeometric Functions ◽  
Integral Representations ◽  
Generalized Hypergeometric Function ◽  
Appell Function

The generalized hypergeometric function was introduced by Srivastava and Daoust. In the present paper a new integral representation is derived. Similarly new integral representations of Lauricella and Appell function are obtained.

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.

Some Results on the Extended Hypergeometric Function

2020 ◽  
Vol 87 (1-2) ◽  
pp. 70
Author(s):  
Ranjan Kumar Jana ◽  
Bhumika Maheshwari ◽  
Ajay Kumar Shukla
Keyword(s):  
Differential Equations ◽  
Hypergeometric Function ◽  
Hypergeometric Functions ◽  
Pochhammer Symbol

An attempt is made to define the extended Pochhammer symbol <em>(λ)n,α</em> which leads to an extension of the classical hypergeometric functions. Differential equations and some properties have also been discussed.

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.

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.

Generalized Neumann expansions involving hypergeometric functions

1967 ◽  
Vol 63 (2) ◽  
pp. 425-429 ◽  
Author(s):  
H. M. Srivastava
Keyword(s):  
Hypergeometric Function ◽  
Hypergeometric Functions ◽  
Generalized Hypergeometric Function ◽  
Image Position

1. Making use of the familiar abbreviationlet us adopt a contracted notation for the generalized hypergeometric function AFB[x] and writewhere (a) denotes the sequence of parameters

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