A Product Formula and a Non-Negative Poisson Kernel for Racah-Wilson Polynomials

Rahman, Mizan

Canadian journal of mathematics, Tome 32 (1980) no. 6, pp. 1501-1517

Résumé

Physicists have long been using Racah's [7] 6-j symbols as a representation for the addition coefficients of three angular momenta. Racah himself discovered a series representation of the 6-j symbol which can be expressed as a balanced 4 F 3 series of argument 1, that is, a generalized hypergeometric function such that the sum of the 3 denominator parameters exceeds that of the 4 numerator parameters by 1. What Racah does not seem to have realized or, perhaps, cared to investigate, is that his 4 F 3 functions, with variables and parameters suitably identified, form a system of orthogonal polynomials in a discrete variable. The orthogonality of 6-j symbols as an orthogonality of 4 F 3 polynomials was recognized much later by Biedenharn et al. [3] in some special cases. Recently J. Wilson [13, 14] introduced a very general system of orthogonal polynomials expressible as balanced 4 F 3 functions of argument 1 orthogonal with respect to an absolutely continuous measure and/or a discrete weight function. Wilson's polynomials contain Racah's 6-j symbols as a special case. These polynomials might rightfully be credited to Wilson alone, but justice might be better served if we call them Racah-Wilson polynomials.

Détail

DOI : 10.4153/CJM-1980-118-3

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