1987/08/01 by Jet Wimp · 6 citations
Mathematics · Physics and Astronomy · #Mathematical functions and polynomials #Advanced Mathematical Identities #Nonlinear Waves and Solitons
paper · pdf · doi:10.4153/cjm-1987-050-4
In this paper we determine closed-form expressions for the associated Jacobi polynomials, i.e., the polynomials satisfying the recurrence relation for Jacobi polynomials with n replaced by n + c , for arbitrary real c ≧ 0. One expression allows us to give in closed form the [ n — 1/ n ] Padé approximant for what is essentially Gauss' continued fraction, thus completing the theory of explicit representations of main diagonal and off-diagonal Padé approximants to the ratio of two Gaussian hypergeometric functions and their confluent forms, an effort begun in [2] and [19] . (We actually give only the [ n — 1/ n ] Padé element, although other cases are easily constructed, see [19] for details.) We also determine the weight function for the polynomials in certain cases where there are no discrete point masses. Concerning a weight function for these polynomials, so many writers have obtained so many partial results that our formula should be considered an epitome rather than a real discovery, see the discussion in Section 3.