Tim Huber

Combinatorics of generalized q-Euler numbers

New enumerating functions for the Euler numbers are considered. Several of the relevant generating functions appear in connection to entries in Ramanujans Lost Notebook. The results presented here are, in part, a response to a conjecture made by M.E.H. Ismail and C. Zhang about the symmetry of polynomials in Ramanujans expansion for a generalization of the Rogers-Ramanujan series. Related generating functions appear in the work of H. Prodinger and L.L. Cristea in their study of geometrically distributed random variables. An elementary combinatorial interpretation for each of these enumerating functions is given in terms of a related set of statistics.

DIFFERENTIAL EQUATIONS FOR CUBIC THETA FUNCTIONS

We show that the cubic theta functions satisfy two distinct coupled systems of nonlinear differential equations. The resulting relations are analogous to Ramanujans differential equations for Eisenstein series on the full modular group. We deduce the cubic analogs presented here from trigonometric series identities arising in Ramanujans original paper on Eisenstein series. Several consequences of these differential equations are established, including a short proof of a famous cubic theta function identity derived by J. M. Borwein and P. B. Borwein.

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 @p and @p^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.

On cubic multisections of Eisenstein series

A systematic procedure for generating cubic multisections of Eisenstein series is given. The relevant series are determined from Fourier expansions for Eisenstein series by restricting the congruence class of the summation index modulo three. We prove that the resulting series are rational functions of η(τ) and η(3τ), where η is the Dedekind eta function. A more general treatment of cubic dissection formulas is given by describing the dissection operators in terms of linear transformations. These operators exhibit properties that mirror those of similarly defined quintic operators.