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Classical orthogonal polynomials of a discrete variable. Transl. from the Russian. (English) Zbl 0743.33001

Springer Series in Computational Physics. Berlin etc.: Springer-Verlag. xvi, 374 p., 26 fig. (1991).
The present book represents the translated and revised and extended version of the authors’ Russian original (1985; Zbl 0576.33001). The classical orthogonal polynomials (C.O.P.) of discrete variables form an important class of special functions which arise in the solution of the hypergeometric type difference equations governing the behavior of certain physical quantities of theoretical physics, computational mathematics and engineering. P. L. Chebyshev appears to be the first mathematician who initiated such type of study which is continued by many authors but is still scattered here and there. In this book the reader will find, for the first time, a systematic and compact presentation of both the theory and application of C.O.P. of a discrete variable.
The first three chapters of this book form the basis of the study. Chapter I gives a concise review of the theory of C.O.P. of continuous arguments. Since a lot of literature is available on this topic so the reader will not find any difficulty due to this scanty survey. In Chapter II, using the Nörlund’s difference quotient operator [L. M. Milne- Thomson, The calculus of finite differences (1981; Zbl 0477.39001), page 23] with some modifications a difference equation of hypergeometric type is derived which is similar to Laplace difference equation [ibid. page 481]. The solutions with their properties have been discussed. The difference equations discussed in Chapter II have been generalized to the case of lattice for \(x(s)\) with a varying mesh in Chapter III. This chapter is not only fundamental in nature but gives a new dimension to the study of C.O.P. The different O.P. are generated by the classification of lattices for \(x(s)\). The most striking feature (appears to be new) of this classification is \(x(s)=e^{2ws}\) which generates \(q\)-polynomials. Thus a new path is opened for further study. The remaining Chapters IV to VI deal with the applications. The difference analogues of spherical harmonics orthogonal on a discrete set and basic quantities of the theory of representation of the three dimensional rotation group etc. are expressed through these polynomials. Lastly the method of trees – a simple graphical technique of solving multidimensional Laplace equation – has been discussed. It is well known that every O.P. satisfies the difference equation \[ a_ n(z)\varphi_{n+2}(z)+b_ n(z)\varphi_{n+1}(z)+c_ n(z)\varphi_ n(z)=0, (A) \] but the converse is obviously not true [H. Bagchi J. Indien Math. Soc., N. Ser. 4, 13-24 (1940; Zbl 0022.33404)]. It is therefore natural to examine this equation so that the solutions may be O.P. It is also well-known [A. K. Rajagopal, Proc. Nat. Inst. Sci. India, Part A 24, 309-313 (1958; Zbl 0085.060)] that the polynomials defined by \[ p_ m(n,x)=[1/k_ mw(x)]D^ m[\omega(x)X^ n] \] satisfy some differential equation of second order. A similar problem can also be considered for \(y_ m(n,x)=[1/k_ mw(n)]\Delta^ m[w(x)p_ n(x)]\). Most of the material (except the last two chapters) is sufficiently simple, hence it can be used as a textbook for undergraduate students.
The reviewer is confident that the present book will give birth to many new possibilities as pointed out above.

MSC:

33-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to special functions
33C50 Orthogonal polynomials and functions in several variables expressible in terms of special functions in one variable
39B99 Functional equations and inequalities
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