A class of generalized hypergeometric functions in several variables.
Item
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Title
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A class of generalized hypergeometric functions in several variables.
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Identifier
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AAI9108193
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identifier
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9108193
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Creator
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Yan, Zhimin.
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Contributor
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Adviser: Adam Koranyi
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Date
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1990
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Language
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English
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Publisher
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City University of New York.
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Subject
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Mathematics
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Abstract
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We study a class of generalized hypergeometric functions in several variables, {dollar}\sb{lcub}p{rcub}F\sbsp{lcub}q{rcub}{lcub}(d){rcub}{dollar}, introduced by A. Koranyi. We prove that {dollar}\sb2 F\sbsp{lcub}1{rcub}{lcub}(d){rcub}{dollar} is the unique solution of a system of partial differential equations, and, as an application, we obtain analogues of such classical results as Kummer relations. Euler integral representations for generalized hypergeometric functions are gotten, in particular, in the case of two variables, {dollar}\sb1 F\sbsp{lcub}0{rcub}{lcub}(d){rcub}{dollar} and {dollar}\sb2 F\sbsp{lcub}1{rcub}{lcub}(d){rcub}{dollar} are expressed in terms of classical hypergeometric functions. Some integral formulas about Jack polynomials in two variables are given, which have many applications. It is shown that in two variables, {dollar}\sb2 F\sbsp{lcub}1{rcub}{lcub}(d){rcub}{dollar} is a hypergeometric function in the sense of Heckman and Opdam. It follows that for some special parameters, {dollar}\sb2 F\sbsp{lcub}1{rcub}{lcub}(d){rcub}{dollar} is a spherical function of a symmetric space of root system {dollar}BC\sb2{dollar}. We obtain the asymptotic behavior of {dollar}\sb{lcub}p + 1{rcub} F\sbsp{lcub}p{rcub}{lcub}(d){rcub}{dollar}. As an application, we get the generalized Rudin-Forelli inequalities in function theory on a bounded symmetric domain, which are due to J. Faraut and A. Koranyi for {dollar}\sb2 F\sbsp{lcub}1{rcub}{lcub}(d){rcub}{dollar} with some special parameters. Our results also include, in a unified way, some estimates obtained for the classical Cartan domains by J. Mitchell and G. Sampson. In the case of two variables, we introduced the generalized Laplace transform and prove the injectivity of the generalized Laplace transform. It is shown that the Laplace transform of {dollar}\sb{lcub}p{rcub} F\sbsp{lcub}q{rcub}{lcub}(d){rcub}{dollar} is a {dollar}\sb{lcub}p + 1{rcub} F\sbsp{lcub}q{rcub}{lcub}(d){rcub}{dollar} function as in the classical case. We also introduce the generalized Laguerre polynomials {dollar}L\sbsp{lcub}\kappa{rcub}{lcub}\gamma{rcub}{dollar}. We find the generating function and integral representation for {dollar}L\sbsp{lcub}\kappa{rcub}{lcub}\gamma{rcub}{dollar}. We also establish the orthogonality relations for {dollar}L\sbsp{lcub}\kappa{rcub}{lcub}\gamma{rcub}{dollar}. Finally, we define the generalized Hankel transform. Generalizations of some classical results about Hankel transform are obtained. A generalized Tricomi Theorem is given.
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Type
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dissertation
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Source
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PQT Legacy CUNY.xlsx
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degree
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Ph.D.
