Zeros of pseudo-ultraspherical polynomials

Abstract

The pseudo-ultraspherical polynomial of degree n can be defined by [Formula: see text] where [Formula: see text] is the ultraspherical polynomial. It is known that when λ < -n, the finite set [Formula: see text] is orthogonal on (-∞, ∞) with respect to the weight function (1 + x2)λ-½ and when λ < 1 - n, the polynomial [Formula: see text] has exclusively real and simple zeros. Here, we undertake a deeper study of the zeros of these polynomials including bounds, numbers of real zeros, monotonicity and interlacing properties. Our methods include the Sturm comparison theorem, recurrence relations, and the explicit expression for the polynomials.