Zeros of generalized Rogers–Ramanujan series: Asymptotic and combinatorial properties

Abstract

In this paper we study the properties of coefficients appearing in the series expansions for zeros of generalized Rogers–Ramanujan series. Our primary purpose is to address several conjectures made by M.E.H. Ismail and C. Zhang. We prove that the coefficients in the series expansion of each zero approach rational multiples of π and π 2 as q → 1 − . We also show that certain polynomials arising in connection with the zeros of Rogers–Ramanujan series generalize the coefficients appearing in the Taylor expansion of the tangent function. These polynomials provide an enumeration for alternating permutations different from that given by the classical q -tangent numbers. We conclude the paper with a method for inverting an elliptic integral associated with the zeros of generalized Rogers–Ramanujan series. Our calculations provide an efficient algorithm for the computation of series expansions for zeros of generalized Rogers–Ramanujan series.