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.

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 Fractional operators with generalized Mittag-Leffler k-function

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Shahid Mubeen ◽  
Rana Safdar Ali
Keyword(s):  
Hypergeometric Function ◽  
Hypergeometric Functions ◽  
Integral Transforms ◽  
Fractional Integrals ◽  
Fractional Operators ◽  
Euler Integral ◽  
K Function ◽  
Generalized Wright Hypergeometric Function

AbstractIn this paper, our main aim is to deal with two integral transforms involving the Gauss hypergeometric functions as their kernels. We prove some composition formulas for such generalized fractional integrals with Mittag-Leffler k-function. The results are established in terms of the generalized Wright hypergeometric function. The Euler integral k-transformation for Mittag-Leffler k-functions has also been developed.

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.

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.

scholarly journals A (p,ν)-extension of Srivastava's triple hypergeometric function H_ B and its properties

Analysis ◽  
2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Showkat Ahmad Dar ◽  
R. B. Paris
Keyword(s):  
Hypergeometric Function ◽  
Hypergeometric Functions ◽  
Mellin Transform ◽  
Beta Function ◽  
Integral Representations ◽  
Recursion Formulas ◽  
Extended Beta Function ◽  
Extended Function

Abstract In this paper, we obtain a ( p , ν ) {(p,\nu)} -extension of Srivastava’s triple hypergeometric function H B ⁢ ( ⋅ ) {H_{B}(\,\cdot\,)} , by using the extended beta function B p , ν ⁢ ( x , y ) {B_{p,\nu}(x,y)} introduced in [R. K. Parmar, P. Chopra and R. B. Paris, On an extension of extended beta and hypergeometric functions, J. Class. Anal. 11 2017, 2, 91–106]. We give some of the main properties of this extended function, which include several integral representations involving Exton’s hypergeometric function, the Mellin transform, a differential formula, recursion formulas and a bounded inequality.

Recursion formulas for Srivastava’s general triple hypergeometric function

2016 ◽  
Vol 09 (03) ◽  
pp. 1650063 ◽  
Author(s):  
Vivek Sahai ◽  
Ashish Verma
Keyword(s):  
Hypergeometric Function ◽  
Hypergeometric Functions ◽  
Recursion Formulas

This paper continues the study of recursion formulas of multivariable hypergeometric functions. In [Recursion formulas for multivariable hypergeometric functions, Asian-Eur. J. Math. 8(4) (2015) Article ID: 1550082, 50 pp.], the authors have given the recursion formulas for three-variable Lauricella functions, three Srivastava’s triple hypergeometric functions and four [Formula: see text]-variable Lauricella functions. We present here the recursion formulas for the general triple hypergeometric function.

INTEGRAL TRANSFORM OF K4-FUNCTION

2021 ◽  
Vol 21 (2) ◽  
pp. 429-436
Author(s):  
SEEMA KABRA ◽  
HARISH NAGAR
Keyword(s):  
Laplace Transform ◽  
Hypergeometric Function ◽  
Integral Transform ◽  
Integral Transforms ◽  
Gauss Hypergeometric Function ◽  
Wright Function ◽  
Generalized Wright Function ◽  
Special Cases ◽  
Euler Transform

In this present work we derived integral transforms such as Euler transform, Laplace transform, and Whittaker transform of K4-function. The results are given in generalized Wright function. Some special cases of the main result are also presented here with new and interesting results. We further extended integral transforms derived here in terms of Gauss Hypergeometric function.

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.

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.

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