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 Ramanujan's 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 Ramanujan's 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.
Huber, T., & Yee, A. J. (2010). Combinatorics of generalized q-Euler numbers. Journal of Combinatorial Theory, Series A, 117(4), 361–388. https://doi.org/10.1016/j.jcta.2009.07.012
Journal of Combinatorial Theory, Series A