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

Opposite Sign ◽

High Order ◽

Pigeonhole Principle ◽

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.

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

Advances in High Energy Physics ◽

10.1155/2018/4263678 ◽

2018 ◽

Vol 2018 ◽

pp. 1-9 ◽

Cited By ~ 6

Author(s):

T. A. Ishkhanyan ◽

T. A. Shahverdyan ◽

A. M. Ishkhanyan

Keyword(s):

Power Series ◽

Hypergeometric Function ◽

Recurrence Relations ◽

Power Series Expansion ◽

Heun Equation ◽

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.

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 ◽

General Nature ◽

Difference Operators ◽

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 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 ◽

Expansion Formula ◽

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

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 ◽

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

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 ◽

Pochhammer Symbol ◽

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

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 ◽

Whittaker Function ◽

Whittaker 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.

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 ◽

Fractional Integral ◽

Fractional Integral Operator ◽

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

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 ◽

Whittaker Function ◽

Generalized Hypergeometric Functions

Whittaker 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.

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

Boletim da Sociedade Paranaense de Matemática ◽

10.5269/bspm.42207 ◽

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 ◽

Laplace Transforms ◽

Theoretical Physics ◽

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.

Some properties of generalized hypergeometric Appell polynomials

Carpathian Mathematical Publications ◽

10.15330/cmp.12.1.129-137 ◽

2020 ◽

Vol 12 (1) ◽

pp. 129-137 ◽

Cited By ~ 1

Author(s):

L. Bedratyuk ◽

N. Luno

Keyword(s):

Differential Operator ◽

Power Series ◽

Hypergeometric Function ◽

Explicit Formula ◽

Series Representation ◽

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.