Jacobi polynomial expansions of a generalized hypergeometric function over a semi-infinite ray

作者：Yudell L. Luke, Jet Wimp · 发表于：Mathematics of Computation · 年份：1963 · DOI：10.1090/s0025-5718-1963-0157014-4 · 被引用次数：21 · 研究领域：Mathematical functions and polynomials、Advanced Numerical Analysis Techniques、Numerical methods for differential equations

Introduction.Suppose fix) is continuous and has a piecewise continuous derivative for 0 ^ x/\ ?£ 1.Then fix) may be expanded into a uniformly convergent series of shifted Jacobi polynomials in the formwhere Rnia'ß)ix) = Pn(aß,(2a; -1) and the latter is the usual notation for the Jacobi polynomial [1, Ch. 10].Various techniques are available for the determination of the coefficients a"(X).In this connection, see, for example, the references [2,3,4,5,6,7].Suppose that fix) satisfies the above conditions for 1 i= x/\ ^ °o where | arg X | < 0; a > -1, ß > -1.If fix) has an asymptotic expansion of the form oothen (1.2) may be interpreted as a summability process which converts the generally divergent expansion (1.3) into a convergent expansion.If fix) in (1.3) is of hypergeometric type, then the coefficients fc«(X) may be found formally at least using the procedures [5,6].These yield for 6n(X) an asymptotic series in X which is also of hypergeometric type.The asymptotic representation for 6n(X) in general is not suitable for computation.We are confronted with two problems; one is the interpretation of the asymptotic series for 6re(X), and the other is the computation of 6"(X).In this paper, we show how both problems can be solved for a confluent hypergeometric function.Actually we derive a representation for 6"(X) when fix) is the (r-funetion, which includes the confluent hypergeometric function as a special case.Our computational scheme for 6"(X) is exhibited only when fix) is a ...