A connection between ultraspherical and pseudo-ultraspherical polynomials

Abstract

The pseudo-ultraspherical polynomial of degree n is defined by C˜n(λ)(x)=(−i)nCn(λ)(ix) where Cn(λ)(x) is the ultraspherical polynomial. We derive a simple expression linking the polynomials Cn(λ) and C˜n(12−λ−n) and show how to derive various properties of the zeros of C˜n(λ) when λ<1−n from properties of the zeros of Cn(λ) when λ>−12.