Sieved ultraspherical polynomials

Abstract

The continuous q q -ultraspherical polynomials contain a number of important examples as limiting or special cases. One of these arose in Allaway’s Ph.D. thesis. In a previous paper we solved a characterization problem essentially equivalent to Allaway’s and showed that these polynomials arose from the q q -ultraspherical polynomials when q q approached a root of unity. A second class of such polynomials is found, and the recurrence relation and orthogonality relation are found for each of these polynomials. The orthogonality is interesting because the weight function has a finite number of zeros in ( − 1 , 1 ) (-1, 1) . Generating functions and other formulas are also found.