ORTHOGONAL polynomials were first studied in connexion with a certain type of continued fractions, bearing the name of Stieltjes. They are closely related to problems of interpolation and mechanical integration, and recently have come into prominence in connexion with mathematical statistics and quantum mechanics. They are also connected with trigonometric, hypergeometric, Bessel, and elliptic functions, and occasionally occur in the theories of differential and integral equations. The presant book is the first in the English language on the subject, and indeed the first extensive treatment of the subject in any language. It deals both with special orthogonal polynomials, such as those of Jacobi, Laguerre, and Hermite, and also with the general theory. In particular, there is an account of recent investigations of the distributions of the zeros, of asymptotic representations, of expansion problems, and of certain questions of interpolation and mechanical integration. Some of the results not previously published are due to the author himself. The book concludes with a set of sixty problems, a bibliography, and an index.
By. (American Mathematical Society, Colloquium Publications, Vol. 23.) Pp. ix + 401. (New York: American Mathematical Society, 1939.) 6 dollars.