Linearization of the Product of Jacobi Polynomials. I

Abstract

Let Pn(α,β) be the Jacobi polynomial of degree n, order (α,β), α,β > – 1, defined by[9, p. 67], and let Rn(α,β)(x) = Pn(α,β)(x)/Pn(αβ)(1). Then for n ≧ m,whereSince Rn(α, β)(l) = 1, it follows that(1)It is known that if (the ultraspherical case) or if α = β + 1, then α = β + 1, then g(k, n, m) ≧ 0.