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Classical orthogonal polynomials. (English) Zbl 0596.33016

Polynômes orthogonaux et applications, Proc. Laguerre Symp., Bar-le- Duc/France 1984, Lect. Notes Math. 1171, 36-62 (1985).
[For the entire collection see Zbl 0572.00007.]
This very stimulating connection is more than just a collection of miscellaneous mathematical papers. It provides a unique overview, 150 years after the birth of Laguerre, of the accomplishments, implications, and directions for future progress of that area of mathematical analysis known as orthogonal polynomials (although the subject tends in this volume to broaden to include virtually all other kinds of special functions), a subject to which so much of Laguerre’s own work was devoted. My own view is that special functions is the great unifying fabric of mathematics; it impinges on the most abstract of mathematical specialities, as well as the most physical. It is appropriate that these papers involve so many disciplines: number theory, Lie groups, partial differential equations, approximation theory, combinatorics, function theory. Applications include work on solid state physics, tomography, nerve impulse transmission, radar/sonar detection, potential flow.
The importance of orthogonal polynomials, as A. Magnus points out in an introductory note, can be judged from statistics alone. The MATHFILE database contains about 2984 general entries since 1973 on the subject. To be added to these are 1100-1200 entries apiece on the special polynomial families of Chebyshev, Hermite, Jacobi, Laguerre, Legendre. The name of Laguerre appears in the database 1405 times. In fact, as Magnus points out, Laguerre’s work in this field was uncommonly brilliant - not always true of the efforts of those who originate. The ramifications of his work are still being investigated, and there are a number of unsolved problems which will, no doubt, occupy mathematicians for many generations. As a single example: it is still not known (except in special cases) how the weight function for a system of orthogonal polynomials affects the asymptotic behavior of the coefficients in the recurrence relation for the polynomials.
The organizers of the congress, C. Brezinski, A. Draux, A. Magnus, P. Maroni, and A. Ronveaux, are to be congratulated for the clarity and organization they have enforced on the contributions in this mammoth volume. Their concerns have been catholic, and much of the richness of the present collection can be attributed to their wide ranging individual interests. J. Dieudonné contributed the keynote address, Continued fractions and orthogonal polynomials in the work of E. N. Laguerre. The other invited addresses were by W. Hahn (On orthogonal polynomials that satisfy linear functional equations), G. E. Andrews and R. Askey (the present paper) and W. Gautschi (Some new applications of orthogonal polynomials). In a summary, it would be the same if this remarkable volume were buried among the drab proceedings of so many other conferences, for there is something in it to appeal to all mathematicians, and the number of research ideas that it stimulates will be immense. Further, it should be required reading for any mathematician working in that broad area known as analysis.
Reviewer: J.Wimp

MSC:

33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
33-02 Research exposition (monographs, survey articles) pertaining to special functions

Citations:

Zbl 0572.00007