Orthogonal polynomials associated with the Rogers-Ramanujan continued fraction

Waleed A. Al-Salam and Mourad Ismail

Vol. 104 (1983), No. 2, 269–283

DOI: 10.2140/pjm.1983.104.269

Abstract

We characterize the symmetric orthogonal polynomials {P_n(x)} such that {P_n(qⁿx)} 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 Σ₀^∞qⁿ² zⁿ∕(q;q)_n that appear in the aforementioned continued fractions have only real and simple zeros.

Mathematical Subject Classification

Primary: 33A65, 33A65

Milestones

Received: 21 April 1981

Published: 1 February 1983