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Asymptotics for the ${}_{4}{F}_{3}$ polynomials. (English) Zbl 0729.33006
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 $\left[0,\infty \right)$ 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 $exp\left[-k{x}^{1/2}\right]$. 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 $\left[0,\infty \right)$ 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.
##### MSC:
 33C45 Orthogonal polynomials and functions of hypergeometric type 42A10 Trigonometric approximation
##### Keywords:
Wilson polynomials; asymptotics; equiconvergence theorem