Table Of ContentAnalytical Methods Orthogonal
and
Special Functions
Polynomials
An International Series of
Monographs in Mathematics
in Two Variables
P.K. Suetin
Gordon and Breach Science Publishers
Digitized by the Internet Archive
in 2019 with funding from
Kahle/Austin Foundation
https://archive.org/details/orthogonalpolynoOOOOsuet
Orthogonal Polynomials in Two Variables
ANALYTICAL METHODS AND SPECIAL FUNCTIONS
An International Series of Monographs in Mathematics
EDITOR IN CHIEF: A.P. Prudnikov (Russia)
ASSOCIATE EDITORS: C.F. Dunkl (USA), H.-J. Glaeske (Germany) and M. Saigo (Japan)
Volume 1
Series of Faber Polynomials
P.K. Suetin
Volume 2
Inverse Problems for Differential Operators
V. A. Yurko
Volume 3
Orthogonal Polynomials in Two Variables
P.K. Suetin
Additional Volumes in Preparation
Bessel Functions and Their Applications
B.G. Korenev
Hypersingular Integrals and Their Applications
S.G. Samko
Fourier Transforms and Approximations
AM. Sedletskii
This book is part of a series. The publisher will accept continuation orders which may be cancelled
at any time and which provide for automatic billing and shipping of each title in the series upon
publication. Please write for details.
ebomus j. bate Library
fRENI UNIVERSITY
PETERBOROUGH, ONTARIO
Orthogonal Polynomials in Two Variables
P.K. Suetin
Technical University of Communication and Informatics
Moscow, Russia
Translated from the Russian by E.V. Pankratiev
GORDON AND BREACH SCIENCE PUBLISHERS
Australia • Canada • China • France • Germany • India • Japan • Luxembourg
Malaysia • The Netherlands • Russia • Singapore • Switzerland
Qf\ I
Copyright © 1999 OPA (Overseas Publishers Association) N.V. Published by license under
the Gordon and Breach Science Publishers imprint.
All rights reserved.
Originally published in Russian in 1988 by Nauka, Moscow
© 1988
No part of this book may be reproduced or utilized in any form or by any means, electronic or
mechanical, including photocopying and recording, or by any information storage or retrieval
system, without permission in writing from the publisher. Printed in Singapore.
Amsteldijk 166
1st Floor
1079 LH Amsterdam
The Netherlands
The camera-ready copy for this book was typeset using the AMS-TEX macro software
package.
British Library Cataloguing in Publication Data
Suetin, P. K.
Orthogonal polynomials in two variables. - (Analytical methods and
special functions; v. 3)
1. Orthogonal polynomials
I. Title
515.5’5
ISBN: 90-5699-167-1
ISSN: 1027-0264
CONTENTS
Preface ix
Preface to the English Edition xv
Notation xix
Chapter I. General properties of polynomials orthogonal over a
domain 1
§ 1. Polynomials in two variables orthogonal over a domain 1
§ 2. The existence theorem and criteria of orthogonality 6
§ 3. Algebraic properties 10
§ 4. Monic orthogonal polynomials 18
§ 5. Normal biorthogonal systems 24
§ 6. Fourier series of orthogonal polynomials in two variables 28
§ 7. Fourier series for differentiable functions 31
Chapter II. Some typical examples and special cases of
orthogonality over a domain 37
§ 1. Different products of classical orthogonal polynomials 37
§ 2. Various cases of connection between orthogonality over a domain
and orthogonality on an interval 42
§ 3. Some theorems in the case of a weight function with separating
variables 48
§ 4. Conditions of interconnection between the weight
function and the domain of orthogonality 52
§ 5. Examples of computations of weight function moments 57
Chapter III. Classical Appell’s orthogonal polynomials 63
§ 1. Rodrigues formula for Appell’s polynomials 63
§ 2. Representation of the Appell polynomials via the hypergeometric
function of two variables 69
§ 3. Differential equation for the Appell polynomials 72
§ 4. Orthogonality of eigenfunctions of the Appell equation 75
§ 5. Normal biorthogonal Appell system 79
§ 6. Series in the Appell polynomials 83
Chapter IV. Admissible differential equation for polynomials
orthogonal over a domain 87
§ 1. The main differential operator and a theorem on orthogonality 87
§ 2. Admissibility conditions for the main differential equation 92
v
CONTENTS
VI
§ 3. Some examples and properties of admissible differential equations 97
§ 4. Affine transformations of the arguments of the main differential
equation 101
§ 5. Transformation of the coefficients of the characteristic polynomial 105
§ 6. Normal forms of the admissible differential equation 115
§ 7. Normal forms when reducing the degree of the characteristic
polynomial 123
Chapter V. Potentially self-adjoint equation and Rodrigues
formula 131
§ 1. Potentially self-adjoint operators 131
§ 2. Admissible and potentially self-adjoint equations 135
§ 3. Rodrigues formula for polynomials orthogonal over a domain 146
§ 4. Weight functions and the Rodrigues formula in the most typical
cases 153
Chapter VI. Harmonic polynomials orthogonal over a domain 163
§ 1. Homogeneous harmonic polynomials 163
§ 2. An analogue of the Christoffel-Darboux formula 169
§ 3. Harmonic polynomials orthogonal in the unit disk 173
§ 4. Harmonic polynomials orthogonal over a domain in the general
case 176
§ 5. Harmonic polynomials superorthogonal over a domain 179
Chapter VII. Polynomials in two variables orthogonal on a
contour 187
§ 1. Alain definitions and the simplest properties 187
§ 2. Polynomials in two variables orthogonal on an algebraic curve 191
§ 3. Harmonic polynomials orthogonal on a contour 196
§ 4. Fourier series in harmonic polynomials orthogonal on a contour 200
§ 5. Harmonic polynomials superorthogonal on a contour 206
§ 6. Examples of superorthogonal systems of harmonic polynomials 213
Chapter VIII. Generalized orthogonal polynomials in two
variables 223
§ 1. Alain definitions and the simplest properties 223
§ 2. The existence theorem in the most general form 228
§ 3. Fourier series in generalized orthogonal polynomials in two
variables 233
§ 4. Atonic orthogonal polynomials under minimal conditions 241
§ 5. Generalized generating functions for monic orthogonal
polynomials 247
Chapter IX. Other results concerning the connection between
orthogonal polynomials and differential equations 253
§ 1. Canonical admissible differential equation and monic
orthogonal polynomials 253
§ 2. Necessary consistency conditions of the canonical operator and
the functional 258
