On Some Formulas for the Lauricella Function

Lauricella, G. in 1893 defined four multidimensional hypergeometric functions FA, FB, FC and FD. These functions depended on three variables but were later generalized to many variables. Lauricella’s functions are infinite sums of products of variables and corresponding parameters, each of them has...

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Vydáno v:	Mathematics (Basel) Ročník 11; číslo 24; s. 4978
Hlavní autoři:	Ryskan, Ainur, Ergashev, Tuhtasin
Médium:	Journal Article
Jazyk:	angličtina
Vydáno:	Basel MDPI AG 01.12.2023
Témata:	Analysis Appell functions Boundary value problems Decomposition differentiation formulas Equality expansion formulas Formulae Functions, Hypergeometric Hypergeometric functions integral representation Lauricella functions Mathematical analysis Mathematics Parameters Partial differential equations Representations summation formulas Variables Variables (Mathematics) Kazakhstan
ISSN:	2227-7390, 2227-7390
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Abstract	Lauricella, G. in 1893 defined four multidimensional hypergeometric functions FA, FB, FC and FD. These functions depended on three variables but were later generalized to many variables. Lauricella’s functions are infinite sums of products of variables and corresponding parameters, each of them has its own parameters. In the present work for Lauricella’s function FA(n), the limit formulas are established, some expansion formulas are obtained that are used to write recurrence relations, and new integral representations and a number of differentiation formulas are obtained that are used to obtain the finite and infinite sums. In the presentation and proof of the obtained formulas, already known expansions and integral representations of the considered FA(n) function, definitions of gamma and beta functions, and the Gaussian hypergeometric function of one variable are used.
AbstractList	Lauricella, G. in 1893 defined four multidimensional hypergeometric functions FA, FB, FC and FD. These functions depended on three variables but were later generalized to many variables. Lauricella’s functions are infinite sums of products of variables and corresponding parameters, each of them has its own parameters. In the present work for Lauricella’s function FA(n), the limit formulas are established, some expansion formulas are obtained that are used to write recurrence relations, and new integral representations and a number of differentiation formulas are obtained that are used to obtain the finite and infinite sums. In the presentation and proof of the obtained formulas, already known expansions and integral representations of the considered FA(n) function, definitions of gamma and beta functions, and the Gaussian hypergeometric function of one variable are used. Lauricella, G. in 1893 defined four multidimensional hypergeometric functions F[sub.A] , F[sub.B] , F[sub.C] and F[sub.D] . These functions depended on three variables but were later generalized to many variables. Lauricella’s functions are infinite sums of products of variables and corresponding parameters, each of them has its own parameters. In the present work for Lauricella’s function F[sub.A] [sup.(n)] , the limit formulas are established, some expansion formulas are obtained that are used to write recurrence relations, and new integral representations and a number of differentiation formulas are obtained that are used to obtain the finite and infinite sums. In the presentation and proof of the obtained formulas, already known expansions and integral representations of the considered F[sub.A] [sup.(n)] function, definitions of gamma and beta functions, and the Gaussian hypergeometric function of one variable are used.
Audience	Academic
Author	Ryskan, Ainur Ergashev, Tuhtasin
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Cites_doi	10.1080/10652469.2013.822207 10.1093/qmath/os-11.1.249 10.1134/S0012266120070046 10.3390/math10071094 10.1093/qmath/os-12.1.112 10.1080/17476933.2019.1606803 10.1088/0305-4470/16/9/007 10.1007/BF03012437 10.1016/j.aml.2005.03.009 10.1016/j.camwa.2006.07.007
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Snippet	Lauricella, G. in 1893 defined four multidimensional hypergeometric functions FA, FB, FC and FD. These functions depended on three variables but were later... Lauricella, G. in 1893 defined four multidimensional hypergeometric functions F[sub.A] , F[sub.B] , F[sub.C] and F[sub.D] . These functions depended on three... Lauricella, G. in 1893 defined four multidimensional hypergeometric functions FA , FB , FC and FD . These functions depended on three variables but were later...
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SubjectTerms	Analysis Appell functions Boundary value problems Decomposition differentiation formulas Equality expansion formulas Formulae Functions, Hypergeometric Hypergeometric functions integral representation Lauricella functions Mathematical analysis Mathematics Parameters Partial differential equations Representations summation formulas Variables Variables (Mathematics)
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