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Combinatorics of generalized \(q\)-Euler numbers. (English) Zbl 1228.05015

Summary: 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 [“Zeros of entire functions and a problem of Ramanujan,” Adv. Math. 209, No.1, 363–380 (2007; Zbl 1107.33018)] 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.

MSC:

05A05 Permutations, words, matrices
11B65 Binomial coefficients; factorials; \(q\)-identities
11B68 Bernoulli and Euler numbers and polynomials
05A15 Exact enumeration problems, generating functions
11P84 Partition identities; identities of Rogers-Ramanujan type
33D15 Basic hypergeometric functions in one variable, \({}_r\phi_s\)
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