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Softcover ISBN:  9780821815724 
Product Code:  MMONO/22 
List Price:  $165.00 
MAA Member Price:  $148.50 
AMS Member Price:  $132.00 
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Product Code:  MMONO/22.E 
List Price:  $155.00 
MAA Member Price:  $139.50 
AMS Member Price:  $124.00 
Softcover ISBN:  9780821815724 
eBook ISBN:  9781470444396 
Product Code:  MMONO/22.B 
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Book DetailsTranslations of Mathematical MonographsVolume: 22; 1968; 613 ppMSC: Primary 33; Secondary 22
A standard scheme for a relation between special functions and group representation theory is the following: certain classes of special functions are interpreted as matrix elements of irreducible representations of a certain Lie group, and then properties of special functions are related to (and derived from) simple wellknown facts of representation theory. The book combines the majority of known results in this direction. In particular, the author describes connections between the exponential functions and the additive group of real numbers (Fourier analysis), Legendre and Jacobi polynomials and representations of the group \(SU(2)\), and the hypergeometric function and representations of the group \(SL(2,R)\), as well as many other classes of special functions.

Table of Contents

Chapters

Introduction

Group representations

The additive group of real numbers and the exponential function. Fourier series and integrals

The group of second order unitary matrices and the polynomials of Legendre and Jacobi

Representations of the group of motions of the plane and Bessel functions

Representations of the group of motions of the pseudoeuclidean plane and the functions of Bessel and Macdonald

Representations of the group $QU(2)$ of unimodular quasiunitary matrices of the second order and the functions of Legendre and Jacobi

Representations of the group of real unimodular matrices and the hypergeometric function

Representations of the group of third order triangular matrices and the Whittaker functions

The group of rotations of $n$dimensional euclidean space and Gegenbauer functions

Representations of the group of hyperbolic rotations of $n$dimensional space and Legendre functions

The group of motions of the $n$dimensional euclidean space and Bessel functions


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A standard scheme for a relation between special functions and group representation theory is the following: certain classes of special functions are interpreted as matrix elements of irreducible representations of a certain Lie group, and then properties of special functions are related to (and derived from) simple wellknown facts of representation theory. The book combines the majority of known results in this direction. In particular, the author describes connections between the exponential functions and the additive group of real numbers (Fourier analysis), Legendre and Jacobi polynomials and representations of the group \(SU(2)\), and the hypergeometric function and representations of the group \(SL(2,R)\), as well as many other classes of special functions.

Chapters

Introduction

Group representations

The additive group of real numbers and the exponential function. Fourier series and integrals

The group of second order unitary matrices and the polynomials of Legendre and Jacobi

Representations of the group of motions of the plane and Bessel functions

Representations of the group of motions of the pseudoeuclidean plane and the functions of Bessel and Macdonald

Representations of the group $QU(2)$ of unimodular quasiunitary matrices of the second order and the functions of Legendre and Jacobi

Representations of the group of real unimodular matrices and the hypergeometric function

Representations of the group of third order triangular matrices and the Whittaker functions

The group of rotations of $n$dimensional euclidean space and Gegenbauer functions

Representations of the group of hyperbolic rotations of $n$dimensional space and Legendre functions

The group of motions of the $n$dimensional euclidean space and Bessel functions