A connection between ultraspherical and pseudo-ultraspherical polynomials

The pseudo-ultraspherical polynomial of degree n is defined by (C) over tilde ((lambda))(n)(x) = (-i)n C-n((lambda)) (ix) where C-n((lambda)) (x) is the ultraspherical polynomial. We derive a simple expression linking the polynomials C-n((lambda)) and C-n((1/2-lambda-n)) and show how to derive various properties of the zeros of C-n((lambda)) when A < 1 n from properties of the zeros of C-n((lambda)) when lambda > -1/2. (C) 2016 Elsevier Inc. All rights reserved.