Asymptotics for the \({}_ 4F_ 3\) polynomials
About 15 years ago, Wilson showed that the 6-j symbols of angular momentum theory could be transformed into a set of polynomials orthogonal with respect to a measure on a finite set of points. For different choices of the parameters in these orthogonal polynomials the orthogonality is on \([0,\infty)\) with respect to a positive absolutely continuous measure, or such a measure plus a finite number of discrete masses for x negative. The absolutely continuous part of the measure decays like \(\exp [-kx^{1/2}]\). Here the asymptotics of the polynomials is found. Regular oscillations occur, but they are stretched out by replacing n by ln n, unlike the n which occurs for Jacobi polynomials (on a finite interval), or \(n^{1/2}\) for Laguerre polynomials (on \([0,\infty)\) but the weight function decays like exp(-x)). This interesting and important result is first found by giving different representations for these polynomials and then using a nice convexity argument. The asymptotic formula is then used to obtain an equiconvergence theorem with respect to a Dirichlet series. Finally, q- versions of these results are obtained.
scientific article; zbMATH DE number 4204800
scientific article