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

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 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 recipe 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.