An integral formula for generalized Gegenbauer polynomials and Jacobi polynomials

Abstract

The generalized Gegenbauer polynomials are orthogonal polynomials with respect to the weight function | x| 2 μ (1− x 2) λ−1/2 . An integral formula for these polynomials is proved, which serves as a transformation between h-harmonic polynomials associated with Z 2 invariant weight functions on the plane. The formula also gives a new integral transform for the Jacobi polynomials, which contains several well-known formulae as special cases. The new formulae can be used to prove the positivity of certain sums of the generalized Gegenbauer and Jacobi polynomials.