Generating Functions of Jacobi Polynomials

Multiplicative renormalization method (MRM) for deriving generating functions of orthogonal polynomials is introduced by Asai–Kubo– Kuo. They and Namli gave complete lists of MRM-applicable measures for MRM-factors h(x) = ex and (1 − x)−κ. In this paper, MRM-factors h(x) for which the beta distribution B(p, q) over [0, 1] is MRM-applicable are determined. In other words, all generating functions of Boas-Buck type of Jacobi polynomials over [0, 1] are obtained. There are only two types 2F1 „ p + q 2 , p + q ± 1 2 ; p; 4x « up to scaling. For the proofs, a general framework will be given together with an example. 1. Multiplicative Renormalization Method A probability measure μ on R with density fμ(x) is said to be applicable to the multiplicative renormalization method for h(x) (or simply, MRM-applicable), if there exists a suitable analytic function ρ(t) around t = 0 with ρ(0) = 0, r1 = ρ′(0) 6= 0 such that ψ(t, x) = h(ρ(t)x) θ(ρ(t)) with θ(t) = ∫ h(tx) dμ(x) (1.1) is a generating function of the orthogonal polynomials {Pn(x)} in L(μ) with leading coefficient of one. Then there exist Jacobi-Szegö parameters {αn, ωn} satisfying the recursive relation Pn+1(x) = (x− αn)Pn(x)− ωnPn−1(x) (1.2) with ω0 = 1, P−1(x) = 0. Let us suppose that an MRM-factor h(x) is expanded as