scholarly journals The behaviour of the fourth type of Lauricella's hypergeometric series in n variables near the boundaries of its convergence region

1994 ◽  
Vol 57 (3) ◽  
pp. 281-294 ◽  
Author(s):  
Megumi Saigo ◽  
H. M. Srivastava
Keyword(s):  
Hypergeometric Function ◽  
Hypergeometric Functions ◽  
Hypergeometric Series ◽  
Convergence Region ◽  
Multiple Series ◽  
Fourth Type ◽  
Region Of Convergence

AbstractFor Lauricella's hypergeometric function F(n)D of n variables, we prove two formulas exhibiting its behaviour near the boundaries of the n-dimensional region of convergence of the multiple series defining it. Each of these results can be applied to deduce the corresponding properties of several simpler hypergeometric functions of one, two, and more variables.

Recursion formulas for the Srivastava–Daoust and related multivariable hypergeometric functions

2016 ◽  
Vol 09 (04) ◽  
pp. 1650081 ◽  
Author(s):  
Vivek Sahai ◽  
Ashish Verma
Keyword(s):  
Hypergeometric Function ◽  
Hypergeometric Functions ◽  
Hypergeometric Series ◽  
Generalized Hypergeometric Function ◽  
Recursion Formulas

This paper concludes the study of recursion formulas of multivariable hypergeometric functions. Earlier in [V. Sahai and A. Verma, Recursion formulas for multivariable hypergeometric functions, Asian–Eur. J. Math. 8 (2015) 50, 1550082], the authors have given the recursion formulas for three variable Lauricella functions, Srivastava’s triple hypergeometric functions and [Formula: see text]-variable Lauricella functions. Further, in [V. Sahai and A. Verma, Recursion formulas for Recursion formulas for Srivastava’s general triple hypergeometric functions, Asian–Eur. J. Math. 9 (2016) 17, 1650063], we have obtained recursion formulas for Srivastava general triple hypergeometric function [Formula: see text]. We present here the recursion formulas for generalized Kampé de Fériet series and Srivastava and Daoust multivariable hypergeometric function. Certain particular cases leading to recursion formulas of certain generalized hypergeometric function of one variable, certain Horn series, Humbert’s confluent hypergeometric series and some confluent forms of Lauricella series in [Formula: see text]-variables are also presented.

The hypergeometric system for one-loop triangle integral

2019 ◽  
Vol 34 (35) ◽  
pp. 1950232
Author(s):  
Xiu-Yi Yang ◽  
Hong-Na Li
Keyword(s):  
Parameter Space ◽  
Gröbner Basis ◽  
Hypergeometric Series ◽  
Polynomial Ideal ◽  
Convergence Theory ◽  
Grobner Basis ◽  
Convergence Region ◽  
Multiple Series

We derive the holonomic hypergeometric system for the [Formula: see text] function with two equal virtual masses, and present the expression of [Formula: see text] in hypergeometric series in corresponding convergent region. Combining the Horn’s convergence theory with Gröbner basis of polynomial ideal, one can calculate the convergence region of the corresponding multiple series concretely. Using the system given here, one can analytically continue [Formula: see text] to whole parameter space.

XIII.—Expressions for Generalised Hypergeometric Functions in Multiple Series

1940 ◽  
Vol 59 ◽  
pp. 141-144
Author(s):  
T. M. MacRobert
Keyword(s):  
Hypergeometric Function ◽  
Hypergeometric Functions ◽  
Multiple Series ◽  
Image Position

In a recent paper (Proc. Roy. Soc. Edin., vol. lix, 1939, pp. 49–54) expressions in multiple series with argument 1 – z were found for Generalised Hypergeometric Functions of the typewhere p > 1. These formulæ are generalisations of known formulæ for the ordinary hypergeometric function F(α, β; ρ; z), and they are established by induction. The same method will be here employed to obtain generalisations of other known formulæ, these generalisations being in the form of multiple series.

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.

Recursion formulas for multivariable hypergeometric functions

2015 ◽  
Vol 08 (04) ◽  
pp. 1550082 ◽  
Author(s):  
Vivek Sahai ◽  
Ashish Verma
Keyword(s):  
Hypergeometric Function ◽  
Radiation Field ◽  
Hypergeometric Functions ◽  
Integral Transforms ◽  
Recursion Formulas ◽  
Appl Math ◽  
Appell Functions

Recently, Opps, Saad and Srivastava [Recursion formulas for Appell’s hypergeometric function [Formula: see text] with some applications to radiation field problems, Appl. Math. Comput. 207 (2009) 545–558] presented the recursion formulas for Appell’s function [Formula: see text] and also gave its applications to radiation field problems. Then Wang [Recursion formulas for Appell functions, Integral Transforms Spec. Funct. 23(6) (2012) 421–433] obtained the recursion formulas for Appell functions [Formula: see text] and [Formula: see text]. In our investigation here, we derive the recursion formulas for 14 three-variable Lauricella functions, three Srivastava’s triple hypergeometric functions and four [Formula: see text]-variable Lauricella functions.

scholarly journals Задача Холмгрена для эллиптического уравнения с сингулярными коэффициентами

2020 ◽  
pp. 114-126
Author(s):  
T.G. Ergashev ◽  
A. Hasanov
Keyword(s):  
Elliptic Equation ◽  
Fundamental Solution ◽  
Hypergeometric Function ◽  
Explicit Form ◽  
Green's Function ◽  
Explicit Solution ◽  
Hypergeometric Functions ◽  
Summation Formulae ◽  
Singular Coefficients ◽  
Abc Method

In the present work, we investigate the Holmgren problem for an multidimensional elliptic equation with several singular coefficients. We use a fundamental solution of the equation, containing Lauricella’s hypergeometric function in many variables. Then using an «abc» method, the uniqueness for the solution of the Holmgren problem is proved. Applying a method of Green’s function, we are able to find the solution of the problem in an explicit form. Moreover, decomposition and summation formulae, formulae of differentiation and some adjacent relations for Lauricella’s hypergeometric functions in many variables were used in order to find the explicit solution for the formulated problem. В данной работе мы исследуем задачу Холмгрена для многомерного эллиптического уравнения с несколькими сингулярными коэффициентами. Мы используем фундаментальное решение уравнения, содержащее гипергеометрическую функцию Лауричеллы от многих переменных. Затем методом «abc» доказывается единственность решения проблемы Холмгрена. Применяя метод функции Грина, мы можем найти решение задачи в явном виде. Более того, формулы разложения и суммирования, формулы дифференцирования и некоторые смежные соотношения для гипергеометрических функций Лауричеллы от многих переменных были использованы для нахождения явного решения поставленной задачи.

New subclasses of analytic functions defined by convolution involving the hypergeometric function and the Owa–Srivastava operator

2019 ◽  
Vol 26 (3) ◽  
pp. 449-458
Author(s):  
Khalida Inayat Noor ◽  
Rashid Murtaza ◽  
Janusz Sokół
Keyword(s):  
Hypergeometric Function ◽  
Convolution Operator ◽  
Analytic Functions ◽  
Hypergeometric Functions ◽  
Integral Operators ◽  
Generalized Hypergeometric Functions ◽  
Convolution Properties ◽  
Appl Math ◽  
Inclusion Results

Abstract In the present paper we introduce a new convolution operator on the class of all normalized analytic functions in {|z|<1} , by using the hypergeometric function and the Owa–Srivastava operator {\Omega^{\alpha}} defined in [S. Owa and H. M. Srivastava, Univalent and starlike generalized hypergeometric functions, Canad. J. Math. 39 1987, 5, 1057–1077]. This operator is a generalization of the operators defined in [S. K. Lee and K. M. Khairnar, A new subclass of analytic functions defined by convolution, Korean J. Math. 19 2011, 4, 351–365] and [K. I. Noor, Integral operators defined by convolution with hypergeometric functions, Appl. Math. Comput. 182 2006, 2, 1872–1881]. Also we introduce some new subclasses of analytic functions using this operator and we discuss some interesting results, such as inclusion results and convolution properties. Our results generalize the results of [S. K. Lee and K. M. Khairnar, A new subclass of analytic functions defined by convolution, Korean J. Math. 19 2011, 4, 351–365].

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 The monodromy representation and twisted period relations for Appell’s hypergeometric function F 4

2015 ◽  
Vol 217 ◽  
pp. 61-94
Author(s):  
Yoshiaki Goto ◽  
Keiji Matsumoto
Keyword(s):  
Differential Equations ◽  
Hypergeometric Function ◽  
Hypergeometric Series ◽  
Integral Representations ◽  
Monodromy Representation ◽  
Homology Groups ◽  
Linearly Independent ◽  
Fundamental Systems

AbstractWe consider the systemF4(a, b, c)of differential equations annihilating Appell's hypergeometric seriesF4(a,b,c;x). We find the integral representations for four linearly independent solutions expressed by the hypergeometric seriesF4. By using the intersection forms of twisted (co)homology groups associated with them, we provide the monodromy representation ofF4(a, b, c)and the twisted period relations for the fundamental systems of solutions ofF4.

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.

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