**Translations of Mathematical Monographs**

1968;
613 pp;
Softcover

MSC: Primary 33;
Secondary 22

**Print ISBN: 978-0-8218-1572-4
Product Code: MMONO/22**

List Price: $135.00

AMS Member Price: $108.00

MAA Member Price: $121.50

**Electronic ISBN: 978-1-4704-4439-6
Product Code: MMONO/22.E**

List Price: $135.00

AMS Member Price: $108.00

MAA Member Price: $121.50

# Special Functions and the Theory of Group Representations

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*N. Ja. Vilenkin*

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 well-known 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

# Table of Contents

## Special Functions and the Theory of Group Representations

- Cover Cover11
- Title page i2
- Contents vii8
- Introduction 112
- Group representations 718
- The additive group of real numbers and the exponential function. Fourier series and integrals 7081
- The group of second order unitary matrices and the polynomials of Legendre and Jacobi 97108
- Representations of the group of motions of the plane and Bessel functions 195206
- Representations of the group of motions of the pseudo-euclidean plane and the functions of Bessel and Macdonald 231242
- Representations of the group ๐๐(2) of unimodular quasi-unitary matrices of the second order and the functions of Legendre and Jacobi 288299
- Representations of the group of real unimodular matrices and the hypergeometric function 344355
- Representations of the group of third order triangular matrices and the Whittaker functions 400411
- The group of rotations of ๐-dimensional euclidean space and Gegenbauer functions 435446
- Representations of the group of hyperbolic rotations of ๐-dimensional space and Legendre functions 503514
- The group of motions of the ๐-dimensional euclidean space and Bessel functions 547558
- Remarks and bibliographical notes 569580
- Bibliography 575586
- Subject Index 601612
- Notation Index 611622
- Back Cover Back Cover1626