Orthogonal polynomials associated with the Rogers-Ramanujan continued fraction

Abstract

We characterize the symmetric orthogonal polynomials {P n (x)} such that {P n (q n x)} is also orthogonal. This leads to orthogonal polynomials related to the denominator polynomials of the continued fractions of Rogers, Ramanujan, and Carlitz. We establish the orthogonality relation for these polynomials and show that the function Q q n z n /(q; q) n that appear in the aforementioned continued fractions have only real and simple zeros.