A Method for the Approximation of Functions Defined by Formal Series Expansions in Orthogonal Polynomials* By Jonas T. Holdeman, Jr. Abstract. An algorithm is described for numerically evaluating functions defined by formal (and possibly divergent) series as well as convergent series of orthogonal functions which are, apart from a factor, orthogonal polynomials. When the orthogonal functions are polynomials, the approximations are rational functions. The algorithm is similar in some respects to the method of Padé approximants. A rational approximation involving Tchebychev polynomials due to H. Maehley and described by E. Kogbetliantz [1] is a special case of the algorithm. ■ 1. Introduction. The solutions to many physical problems are obtained as expansions in (infinite) series of orthogonal functions. When the series are con- vergent they can in principle be approximated to any accuracy by truncating the series at the proper point. When the series are weakly convergent or divergent, the procedures for their numerical evaluation become rather ad hoc and a more general approach would be useful. In the following sections we describe an approximation to functions defined as infinite series of orthogonal functions which are (apart from a factor) orthogonal polynomials. The approximation takes the form of the ratio of two functions, the denominator function being a polynomial. When the orthogonal functions are polynomials, the approximating function is rational. The derivation of the algorithm will be formal and no real proofs are given. Indication that the algorithm is at least sometimes valid is provided by the numer- ical examples in Section 6. While the derivations could probably be made rigorous in the case of absolutely convergent series, interesting cases occur with divergent series such as the examples of Section 6. Numerous other practical applications of the algorithm have been made in the past year at Los Alamos Scientific Laboratory. The success of these examples would justify an effort at finding a class of functions representable by divergent series for which the algorithm is applicable. 2. The Approximation. The approximations we shall discuss fall roughly into two problems. The first of these is the approximation problem, that is, given a function (or equivalently its expansion), find an easily calculated approximation to the function. The second problem is the summation of infinite series, that is, given an infinite series (which may not be convergent in the ordinary sense), assign a sum to the series which gives the same value as the ordinary summation method when the series is convergent. Clearly the two problems cannot be sharply separated. In this paper however we will lean more toward the second problem. For simplicity we assume the functions we will encounter are real on the real Received December 14, 1967, revised August 5, 1968. * Work performed under the auspices of the U.S. Atomic Energy Commission. 275 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use