SIAM AG on Orthogonal Polynomials and Special Functions


Extract from OP-SF NET

Topic #10 --------------- OP-SF NET ---------------- November 15, 1996

From: Richard Askey
Subject: Bibliography on Orthogonal Polynomials

Information about the book:

A bibliography on orthogonal polynomials,
Bulletin of the National Research Council, Number 103
National Academy of Sciences, Washington D.C., 1940

During the depression of the 1930s, there were various projects funded by different governments. One of those in the United States was a bibliography on orthogonal polynomials. The nominal authors were listed as Shohat, Hille and Walsh, and they did much of the background work, preparing the outline format, and in Shohat's case knowing much of the early Russian literature. Much of the actual work was done by H.N. Laden as you can read in two sentences which hint at this in the introduction. The book is titled "A Bibliography on Orthogonal Polynomials", it was published by the National Research Council of the National Academy of Sciences, Washington, D.C., in 1940 as Number 103 of the Bulletin of the National Research Council.

It is a 204 page book which starts with a list of 303 periodicals which are referenced. Then there is a seven page outline of information about orthogonal polynomials. This starts with special polynomials (Classical OP) and includes three in two variables as well as the usual ones of Jacobi, Laguerre and Hermite and special cases of them. Each of these is denoted by a letter which will be used later. Then the area of general orthogonal polynomials is broken down into many different types of polynomials; finite interval of real axis, two finite intervals of real axis, more than two, ... Jordan arc or closed curve in plane, etc., and then a section on types of weight functions. That is the first 1 1/3 pages. The rest of the seven pages break up properties of OP into different groups, listed by letters, Greek for such things as general properties, expansions of functions, moment problem, application to mathematical physics, etc. Under each of these heading there are further details. For example, under Moment problem there is: a. Criteria for the character, determined or indeterminate, of the problem. b. Solution, which contains two subheadings:
1. Infinitely many data,
2. Finite many data.

Then the real information is given by an alphabetical list of authors of papers first, and then books and theses. Here is one listing:

Adams, J.C. 1. On the expression of the product of any two Legendre's coefficients by means of a series of Legendre's coefficients. [8]27(1878)63-71.

*[8] is Abstracts of the Papers Communicated to the Royal Society of London: vol. 6. The last refers to the last volume consulted.*

P: alphab4-f, mu

* P means Legendre polynomials, alpha b 4 refers to general properties for alpha, various representations for b and n-th derivative for 4. The f is recurrence relation. mu is evaluation of sums and definite integrals involving OP (especially products of two or more OP).*

In this case the title of the paper tells what is there. In most cases this is not true, and the outline if it were more easily searched could be useful. The attempt to make it searchable in the book is at the end, where there are 7 pages under the title Abbreviated Topical Index. The first is Hermite polynomials. There are 34 lines of which the following is the first:

Adamoff 2; Agronomoff 1; Aitken 1; Aitken and Oppenheim 1; Angelesco 7,

and this is followed by 13,15,17; etc on the next line. I have known of this book for over 40 years and have owned a copy for more than 20. I have successfully used it for finding something to use in research once or twice, and for historical purposes a few more than that.

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