科研之友
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摘要
This is the second part of a project which provides asymptotic approximations to the Jacobi polynomials P(n)((alpha,beta)) (x) and to the Racah coefficients P(n)((an+c,bn+d)) (x), as n -> infinity, where a,b,c,d are constants. The approximations to P(n)((alpha,beta)) (x) are generated by the construction of certain fundamental sets of solutions to a hypergeometric differential equation. In a first step we construct approximations to the Jacobi polynomials and the Racah coefficients on a closed interval [z(1), 1] where the solutions are free of zeros. This poses a special challenge since the two endpoints of the interval are a regular-singular point and a turning point of the corresponding differential equation. In the second step we "connect" the approximations of the Jacobi polynomials on [1, infinity) through the singular regular point x = 1 to yield a global approximation on [z(1), infinity). Our global approximation of the Jacobi polynomials on [z(1), infinity) is obtained without the intervention of "special functions."
- 出版日期2010
