Inequalities for ultraspherical polynomials and the gamma function

Abstract

Let M n (λ) = (n + λ) 1 − λ max 0⩽θ⩽π( sin θ) λ¦P n (λ)( cos θ)¦, where P n (λ)(x) is the ultraspherical polynomial of degree n and parameter λ. It is shown that M n (λ) < 2 1 − λ Γ(λ) , for 0 < λ < 1 and n = 0, 1, 2… When λ = 0 and when λ = 1 , this inequality becomes an equality. It refines inequality (7.33.5) of G. Szegö's “Orthogonal Polynomials” (4th edition 1975, p. 171), wherein the factor ( n + λ) 1 − λ is replaced by n 1 − λ . The method of proof requires sharpening some inequalities for the ratio Γ(n + λ) Γ(n + 1) , n = 0, 1, 2,… .