Generating functions and generalized Euler numbers

Generating Functions of the Incomplete Generalized Fibonacci and Generalized Lucas Numbers

Arithmetic Properties of Generalized Euler Numbers

The generalized Euler number En|k counts the number of permutations of {1, 2, . . . , n} which have a descent in position m if and only if m is divisible by k. The classical Euler numbers are the special case when k = 2. In this paper, we study divisibility properties of a q-analog of En|k. In particular, we generalize two theorems of Andrews and Gessel [3] about factors of the q-tangent numbers.

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 g...

Generating Functions for q-Apostol Type Frobenius-Euler Numbers and Polynomials

The aim of this paper is to construct generating functions, related to nonnegative real parameters, for q-Eulerian type polynomials and numbers (or q-Apostol type Frobenius–Euler polynomials and numbers). We derive some identities for these polynomials and numbers based on the generating functions and functional equations. We also give multiplication formula for the generalized Apostol type Fro...

Relationships between Generalized Bernoulli Numbers and Polynomials and Generalized Euler Numbers and Polynomials

In this paper, concepts of the generalized Bernoulli and Euler numbers and polynomials are introduced, and some relationships between them are established.