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Associated Wilson polynomials. (English) Zbl 0746.33003
The Wilson polynomials appear on top of the Askey table of hypergeometric orthogonal polynomials and thus are, together with the Racah polynomials, the most general system of hypergeometric orthogonal polynomials. They can be written as an hypergeometric ${}_{4}{F}_{3}\left(1\right)$ in which the variable $x$ appears in two of the numerator parameters as the complex conjugate pair $a+i\sqrt{x}$ and $a-i\sqrt{x}$. If one replaces the index $n$ in the coefficients of the three-term recurrence relation for the Wilson polynomials by $n+c$, then one obtains the associated Wilson polynomials. The author obtains all solutions of the recurrence relation for the associated Wilson polynomials as a linear combination of two very well-poised ${}_{7}{F}_{6}$ hypergeometric functions of unit argument. Some applications are given, in particular continued fraction representations, the weight function for the associated Wilson polynomials and a generalization of Dougall’s theorem for the evaluation of a well-poised ${}_{7}{F}_{6}\left(1\right)$.
##### MSC:
 33C45 Orthogonal polynomials and functions of hypergeometric type 33C20 Generalized hypergeometric series, ${}_{p}{F}_{q}$
##### References:
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