Table Of ContentSpringer Series in Computational Physics
Editors: C. A. J. Fletcher R. Glowinski W. Hillebrandt M. Holt P. Hut
H. B. Keller J. Killeen S. A. Orszag V. V. Rusanov
Springer Series in Computational Physics
Editors: C. A. 1. Fletcher R. Glowinski W. Hillebrandt M. Holt P. Hut
H. B. Keller 1. Killeen S. A. Orszag V. V. Rusanov
A Computational Method in Plasma Physics
F. Bauer, O. Betancourt, P. Garabedian
Implementation of Finite Element Methods for Navie .... Stokes Equations
F. Thomasset
Finite-Difference Techniques for Vectorized Fluid Dynamics Calculations
Edited by D. Book
Unsteady Viscous Flows. D. P. Telionis
Computational Methods for Fluid Flow. R. Peyret, T. D. Taylor
Computational Methods in Bifurcation Theory and Dissipative Structures
M. Kubicek, M. Marek
Optimal Shape Design for Elliptic Systems. O. Pironneau
The Method of Differential Approximation. Yu. I. Shokin
Computational Galerkin Methods. C. A. 1. Fletcher
Numerical Methods for Nonlinear Variational Problems
R. Glowinski
Numerical Methods in Fluid Dynamics. Second Edition
M. Holt
Computer Studies of Phase Transitions and Critical Phenomena
O. G. Mouritsen
Finite Element Methods in Linear Ideal Magnetohydrodynamics
R. Gruber, 1. Rappaz
Numerical Simulation of Plasmas. Y. N. Dnestrovskii, D. P. Kostomarov
Computational Methods for Kinetic Models of Magnetically Confined Plasmas
1. Killeen, G. D. Kerbel, M. C. McCoy, A. A. Mirin
Spectral Methods in Fluid Dynamics. Second Edition
C. Canuto, M. Y. Hussaini, A. Quarteroni, T. A. Zang
Computational Techniques for Fluid Dynamics 1. Second Edition
Fundamental and General Techniques. C. A. 1. Fletcher
Computational Techniques for Fluid Dynamics 2. Second Edition
Specific Techniques for Different Flow Categories. C. A. 1. Fletcher
Methods for the Localization of Singularities in Numerical Solutions
of Gas Dynamics Problems. E. V. Vorozhtsov, N. N. Yanenko
Classical Orthogonal Polynomials of a Discrete Variable
A. F. Nikiforov, S. K. Suslov, V B. Uvarov
Flux Coordinates and Magnetic Field Structure:
A Guide to a Fundamental Tool of Plasma Theory
W. D. D'haeseleer, W. N. G. Hitchon, 1. D. Callen, 1. L. Shohet
Monte Carlo Methods in Boundary Value Problems
K. K. Sabelfeld
A.F. Nikiforov S.K. Suslov
V.B. Uvarov
Classical Orthogonal
Polynomials
of a Discrete Variable
With 26 Figures
Springer-Verlag
Berlin Heidelberg New York
London Paris Tokyo
Hong Kong Barcelona
Budapest
Professor Dr. Arnold F. Nikiforov
Professor Dr. Vasilii B. Uvarov
M.V. Keldysh Institute of Applied Mathematics, Academy of Sciences
of the USSR, Miusskaya pI. 4, SU-125047 Moscow, USSR
Sergei K. Suslov
Kurchatov Institute of Atomic Energy, SU-123182 Moscow, USSR
Editors P. Hut
The Institute for Advanced Study
C. A. J. Fletcher School of Natural Sciences
Department of Mechanical Engineering Princeton, NJ 08540, USA
The University of Sydney
H.B. Keller
New South Wales, 2006
Australia Applied Mathematics 101-50
Firestone Laboratory
California Institute of Technology
R. Glowinski
Pasadena, CA 91125, USA
Institut de Recherche d'Informatique
et d'Automatique (INRIA) 1. Killeen
Domaine de Voluceau Lawrence Livermore Laboratory
Rocquencourt, B. P. 105 PO. Box 808
F-78150 Le Chesnay, France Livermore, CA 94551, USA
S.A.Orszag
W. Hillebrandt
Max-Planck-Institut fUr Astrophysik Department of Mechanical and
Karl-Schwarzschild-StraBe 1 Aerospace Engineering
W-8046 Garching, Fed. Rep. of Germany Princeton University
Princeton, NJ 08544, USA
M. Holt VV Rusanov
College of Engineering and M. v. Keldysh Institute
Mechanical Engineering of Applied Mathematics
University of California Miusskaya pI. 4
Berkeley, CA 94720, USA SU-125047 Moscow, USSR
ISBN-13: 978-3-642-74750-2 e-ISBN-13: 978-3-642-74748-9
DOl: 10.1007/978-3-642-74748-9
Library of Congress Cataloging-in-Publication Data. Nikiforov, A. F. [Klassicheskie ortogonal 'nye poli
nomy diskretnoi peremennoi. English). Classical orthogonal polynomials of a discrete variable I A. F.
Nikiforov, S.K. Suslov, v.B. Uvarov. p. cm. - (Springer series in computational physics). Translation
of: Klassicheskie ortogonal 'nye polinomy diskretnoi peremennoi. Includes bibliographical references
and index. ISBN 0-387-51123-7 (U.S. : alk. paper). 1. Orthogonal polynomials. 2. Functions, Special.
3. Multivariate analysis. 4. Mathematical physics. 1. Suslov, S.K. (Sergei Konstantinovich). II. Uvarov,
v.B. (Vasilii Borisovich). III. Title. IV. Series. QC20.7.075N5513 1990 515'.55---dc20 90-9793
This work is subject to copyright. All rights are reserved, whether the whole or part of the material is con
cerned, specifically the· rights of translation, reprinting, reuse of illustrations, recitation, broadcasting,
reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or
parts thereof is permitted only under the provisions of the German Copyright Law of September 9,1965, in
its current version, and permission for use must always be obtained from Springer-Verlag. Violations are
liable for prosecution under the German Copyright Law.
© Springer-Verlag Berlin Heidelberg 1991
Softcover reprint of the hardcover 15t edition 1991
The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply,
even in the absence of a specific statement, that such names are exempt from the relevant protective laws and
regulations and therefore free for general use.
Typesetting: SpringerTEX inhouse system
57/3140-543210 - Printed on acid-free paper
Preface
Mathematical modelling of many physical processes involves rather complex dif
ferential, integral, and integro-differential equations which can be solved directly
only in a number of cases. Therefore, as a first step, an original problem has to
be considerably simplified in order to get a preliminary knowledge of the most
important qualitative features of the process under investigation and to estimate
the effect of various factors. Sometimes a solution of the simplified problem
can be obtained in the analytical form convenient for further investigation. At
this stage of the mathematical modelling it is useful to apply various special
functions.
Many model problems of atomic, molecular, and nuclear physics, electrody
namics, and acoustics may be reduced to equations of hypergeometric type,
a(x)y" + r(x)y' + AY = 0 , (0.1)
where a(x) and r(x) are polynomials of at most the second and first degree re
spectively and A is a constant [E7, AI, N18]. Some solutions of (0.1) are functions
extensively used in mathematical physics such as classical orthogonal polyno
mials (the Jacobi, Laguerre, and Hermite polynomials) and hypergeometric and
confluent hypergeometric functions.
On the other hand, along with the special functions which are the solu
tions of (0.1), in various fields of physics and mathematics wide use is made
of quantities that are determined on a discrete set of argument values. Examples
are the Clebsch-Gordan and Racah coefficients (the 3j- and 6j-symbols), which
have long been used in quantum mechanics and in the theory of group repre
sentations. Moreover, in probability theory (specifically, in problems of queuing
theory [F1a] and birth and death processes [F1a, K6-9]), in coding theory [L6],
etc. classical orthogonal polynomials of a discrete variable (the Hahn, Meixner,
Kravchuk, and Charlier polynomials) are extensively used. It has been proved
that all these functions may be described by means of a unified treatment in terms
of polynomials which are solutions of the difference equation of hypergeometric
type, proposed in [N17]. This equation may be obtained by approximating the
differential equation (0.1) on lattices of certain classes. It has been found that the
polynomial solutions of the above difference equation include the polynomials
earlier introduced from various considerations by Markov [M5], Chebyshev [1'3],
Rogers [R21-R23], Stieltjes and Wigert [S24, W6], Pollaczek [PlO], and Karlin
and MacGregor [K6], and also the Askey-Wilson polynomials [A27, W8, A29] ,
introduced by means of the basic hypergeometric series.
v
A general theory of polynomial solutions for the difference equation of hyper
geometric type was first constructed in preprint [NI7] by generalizing the methcxl
applied earlier to the differential equation (0.1) [NI6, NI6a]. The present mono
graph represents an appreciably revised and extended version of the book [NI4]
earlier published in Russian (see also [NI8]).
In the monograph the reader will find a systematic, concise presentation of
the theory of polynomial solutions of the hypergeometric-type difference equa
tion. The book contains methods for solving a wide class of difference equations
and recursion relations as well as applications of the classical orthogonal poly
nomials of a discrete variable in computational mathematics, probability theory,
and coding theory, for information compression and storage. It is shown that the
basic quantities of the representation theory of the rotation group - generalized
spherical harmonics and the Clebsch-Gordan and Racah coefficients - can be
expressed in terms of the classical orthogonal polynomials of a discrete variable.
Moreover, a general methcxl for obtaining particular solutions of arbitrary
difference equations of hypergeometric type in the form of generalized q
hypergeometric series has been constructed.
The book has been written according to the following scheme. The first three
chapters form the basis of the book. Chapter 1 gives a concise review of the
theory of classical orthogonal polynomials - the Jacobi, Laguerre, and Hermite
polynomials which satisfy the differential equation of hypergeometric type (0.1).
Polynomial solutions of Eq. (0.1) are studied, using the fact that derivatives of the
solutions of (0.1) also satisfy the equation of type (0.1) [N5,NI8]. This enables
us to obtain readily explicit expressions for classical orthogonal polynomials in
the form of the Rodrigues formula. Then the orthogonality property is proved
and differentiation formulas, recursion relations, and some other properties are
deduced.
Chapter 2 considers the difference equation of hypergeometric type
a(x) [y(X + h) - y(x) _ y(x) - y(x - h)]
h h h
(0.2)
r(x) [y(X + h) - y(x) y(x) - y(x - h)] \ ( ) = 0
+ 2 h + h + Ay X ,
which approximates the differential second-order equation (0.1) on a lattice with
constant mesh h up to the second order of accuracy [S 1, G 17]1. For some values
of A = An (n = 0, 1, ...) the particular solutions of Eq. (0.2) are classical or
thogonal polynomials of a discrete variable - the Hahn, Meixner, Kravchuk, and
Charlier polynomials2• The theory of these polynomials is developed following
1 The difference operator Lh approximates the differential operator L at point z to the order of
accuracy m with respect to mesh h if LhY(Z) - Ly(z) = O(hm) when h ..... O.
2 The origin of the term "polynomials of a discrete variable" may be explained as follows: for the
Hahn, Meixner, Kravchuk, and Charlier polynomials the orthogonality property is written in the
form of a sum with a certain weight over a discrete set of lattice points (instead of the integral
form for the Jacobi, Lquerre, and Hermite polynomials).
VI
the same logical scheme as in the theory of the Jacobi, Laguerre, and Hennite
polynomials, but the derivatives have to be replaced by appropriate difference
quotients. For the Hahn, Meixner, Kravchuk:, and Charlier polynomials an analog
of the Rodrigues fonnula is obtained, the orthogonality property is established,
and the "difference differentiation" formula, asymptotic representations, etc. are
derived.3
Chapter 3, which is fundamental to the whole book, gives a generalization
of the difference equation of hypergeometric type (0.2) to the case of a lattice
with a varying mesh. In order to obtain the respective difference equation for
nonuniform lattices it is convenient to pass from the variable x to the variable s
assuming x = x(s) and to use the values of x(s) on an s-unifonn lattice s = Si
(i = 0, 1, ... ), where Lls = Si+l-Si = h. Then the mesh LlX(Si) = X(Si+l) -X(Si)
will be variable. After the replacement of x by x(s) we obtain the difference
equation
a[x(s)] [y(S + h) - y(s) y(s) - y(s - h)]
x(s + hj2) - x(s - hj2) x(s + h) - x(s) - x(s) - x(s - h)
(0.3)
r[x(s)] [y(S + h) - y(s) y(s) - y(s - h)] ). () 0
= ,
+ -2- x(s + h) - x(s) + x(s) - x(s _ h) + y s
which corresponds to Eq. (0.2). In this equation a(x) and r(x) are arbitrary
polynomials in x of at most the second and first degree respectively and ). is a
constant.
Equation (0.3) approximates the original differential equation (0.1) up to the
second order of accuracy when h -+ 0 on a nonuniform lattice x = x(s). It is
shown that, for a certain class of nonunifonn lattices, Eq. (0.3) allows the keeping
of a property similar to the fundamental property of the differential equation (0.1)
and the difference equation (0.2): the function
v(s) = y(s + hj2) - y(s - hj2)
x(s + hj2) - x(s - hj2) ,
which is approximately equal to the derivative dyjdx at the point x = x(s),
satisfies an equation of the same type as the function y(s).
For the above class of functions x(s) this property lets us reserve all the basic
points of argument used in Chap. 2 and, by applying elementary mathematical
tools, obtain the basic properties of polynomial solutions of Eq. (0.3) - a dis
crete analog of the Rodrigues formula, the orthogonality property, the recursion
relations, asymptotic representations, etc.
The class of functions x(s) under consideration has the fonn
(0.4)
where Cl, C2, and C3 are arbitrary constants and q is a parameter. This class
also includes linear and quadratic lattices because constants Ct. C2, and C3 may,
3 Particular solutions of Eq. (0.2) with arbitrary complex A are obtained in [NIl] (see also [NI8].
VII
=
in general, depend on q, so that we may choose the constants Ci Ci(q) such
that the expression (0.4) transforms into x(s) = ClSZ + Czs + C3 when q --t 1.
For the functions x(s) of the form (0.4) Eq. (0.3) is called the difference
equation of hypergeometric type. The polynomial solutions of this equation that
have the orthogonality property in the form of a sum over a discrete set of
lattice points are called classical orthogonal polynomials of a discrete variable
(according to terminology accepted in [E7]).
The Racah polynomials and the dual Hahn polynomials for the quadratic
lattice x(s) = s(s + I) that are important in applications are studied in particular
detail. For all nonuniform lattices at q f 1 the systems of polynomials that in the
limit q --t 1 take the form of polynomials orthogonal' on linear and quadratic
lattices are constructed.
In Sect. 3.10 the polynomial solutions of the difference equation of hyperge
ometric type (0.3) are considered for arbitrary complex values of the equation
coefficients. It is shown that under certain conditions these polynomials have the
continuous orthogonality property in the form of an integral in the complex plane
of variable s over a contour C (in particular, the PolIaczek polynomials).
In Sect. 3.11 the explicit expression of polynomial solutions in terms of gen
eralized q-hypergeometric series is obtained from the Rodrigues formula for the
most general case [N22]. These series are introduced by replacing in the gen
eralized hypergeometric series the values (ah = r(a + k)/ r(a) by the values
(alqh = i'q(a + k)/ i'q(a), where i'q(s) = q-(s-1)(s-Z)/4 rq(s) is a generalization
of Euler's gamma function r(s) [11, NI8].
From the expression of polynomials in terms of generalized q-hypergeometric
series that was obtained for the most general case, the formulas for all particular
cases are derived by an appropriate choice of parameters and by taking various
limits. The consideration of these particular cases gives us the classification of
corresponding q-polynomials. All the earlier introduced polynomials are included
in our scheme.
We use generalized q-hypergeometric series instead of basic hypergeometric
series [A27, A29, W8, G7a] because they have more symmetry (for example,
these series do not change after replacing q by 1/q ) and for q --t 1 they transform
=
into generalized hypergeometric series, since limq ..... li'q(S) F(s).
The remaining chapters (4-6), which form the second part of the book, deal
with applications. Chapter 4 discusses applications of classical orthogonal poly
nomials of a discrete variable in computational mathematics and the theory of
difference schemes, in information compression and storage, for the function
approximation in a rectangle and on a sphere, in the theory of probability and
coding theory, in the genetic Moran model, and in some problems of queueing
theory. Here the difference analogs of spherical harmonics orthogonal on a dis
crete set of sphere points are constructed by using the Hahn polynomials and the
q-analogs of the Racah polynomials on a cosinusoidal lattice4•
4 For other important applications see [N2a, VI].
VIII
In Chap. 5 the basic quantities of the theory of representations of the three
dimensional rotation group and the quantum theory of angular momentum -
generalized spherical harmonics, the Clebsch-Gordan coefficients and the 6j
symbols of Wigner - are expressed through the Kravchuk, Hahn, and Racah
polynomials, respectively, which allows the representation of properties of these
quantities in a simple form. Since the Hahn polynomials are difference analogs
of the Jacobi polynomials, the relation between the Clebsch-Gordan coefficients
and the Hahn polynomials easily explains an analogy between these coefficients
and the ~acobic polynomials, noted by 1M. Gel'fand [G13]. Chapter 5 discusses
also close connections of the Hahn polynomials with the group representations
offour-dimensional rotations SO(4) and the Lorenz group SO(3.1), as well as the
Racah polynomials with the representations of group SU(3). It is shown that the
9j-symbols form up to normalization a new system of orthogonal polynomials of
two discrete variables. The main properties of these polynomials are established
by building on the quantum theory of angular momentum.
Chapter 6 considers the method of trees - a simple graphical technique of
solving a multidimensional Laplace equation - proposed by N.Ya. Vilenkin, GJ.
Kuznetsov, and Ya.A. Smorodinsky in [V8, VIO]. Coefficients of transformation
between solutions of the Laplace equation in different systems of spherical co
ordinates (the T-coefficients [KIO]) are expressed through the Racah, Hahn, and
Kravchuk polynomials.
The first four chapters of the book were written by A.F. Nikiforov and V.B.
Uvarov except for Sects. 3.10.3-5, some material from [NI8] being used in the
first part on foundations of the theory; Chaps.5 and 6 were written by S.K.
Suslov.
The book is aimed at a wide range of specialists in theoretical and mathemat
ical physics and computational mathematics. Most of the material is sufficiently
elementary that it is possible to use it as a textbook for undergraduate and grad
uate students of physical and mathematical disciplines, those studying quantum
mechanics, and also those who lecture on mathematics and physics.
Moscow A.F. Nikiforov
April 1991 V.B. Uvarov
S.K. Suslov
IX
Foreword to the Russian Edition
Classical orthogonal polynomials of a discrete variable are an important class of
special functions arising in various problems of mathematics, theoretical physics,
computational mathematics and engineering; this field is now under extensive
development. It should be noted that there is a deep analogy between classical
orthogonal polynomials of continuous and discrete arguments, and the theory
of group representation is one of its basic manifestations. This analogy was
noted by I.M. Gel'fand in the mid-fifties [G 13] in connection with the study
of representations of the rotation group playing an important role in theoretical
physics.
Studies of classical orthogonal polynomials of a discrete variable were initi
ated by P.L. Chebyshev in the middle of the last century and continued by many
prominent scientists. However, there are no books where a theory of these poly
nomials is consistently developed. Up until recently it was not even clear what
polynomials introduced by different authors from various viewpoints belong to
the above class of special functions.
In this book the reader will find for the first time a systematic, compact pre
sentation of both the theory of classical orthogonal polynomials of a discrete
variable and its main applications. The authors have made a significant contribu
tion to this field. They have developed a simple approach to the construction of
the theory of classical orthogonal polynomials of a discrete variable as solutions
of a difference equation of hypergeometric type.
Also of interest is a nonstandard approach to investigating the representations
of the three-dimensional space rotation group by using the theory of classical
orthogonal polynomials of a discrete variable.
This comprehensive book will be useful for both mathematicians and physi
cists.
Moscow M.I.Graev
February 1984 (editor of the Russian edition)
XI
