Integral representations for Jacobi polynomials and some applications

Abstract

An integral for $\text{[P}{}_{\mathrm{n}}{}^{\mathrm{(\alpha \; +\; \mu ,\beta )}}\text{(x)]}\text{[P}{}_{\mathrm{n}}{}^{\mathrm{(\alpha \; +\; \mu ,\beta )}}\text{(1)]}$ in terms of $\text{[P}{}_{\mathrm{n}}{}^{\mathrm{(\alpha ,\beta )}}\text{(y)]}\text{[P}{}_{\mathrm{n}}{}^{\mathrm{(\alpha ,\beta )}}\text{(1)]}$ with a positive kernel is obtained. For $\text{β = ±}\text{1}\text{2}$ this integral is equivalent to an important integral of Feldheim and Vilenkin connecting ultraspherical polynomials. As an application we show that $\text{P}{}_{\mathrm{n}}{}^{\mathrm{(\alpha ,\alpha )}}\text{(x)}\text{P}{}_{\mathrm{n}}{}^{\mathrm{(\alpha ,\alpha )}}\text{(1)}\text{= ∫}{}_{\mathrm{-1}}{}^1\text{P}{}_{\mathrm{n}}{}^{\mathrm{(\beta ,\beta )}}\text{(y)}\text{P}{}_{\mathrm{n}}{}^{\mathrm{(\beta ,\beta )}}\text{(1)}\text{dμ(y)}$ where $\text{α > β ⩾ −}\text{1}\text{2}\text{, − 1 ⩽ x ⩽ 1,}\text{and}\text{dμ(y)}$ is a positive measure which depends on x but not n. For $\text{β = −}\text{1}\text{2}$ this is a result of Seidel and Szasz. Similar results are obtained for Jacobi polynomials and the positivity of certain sums of ultraspherical and Jacobi polynomials is obtained.