Generalized \(q\)-Euler numbers and polynomials of higher order and some theoretic identities
The purpose of this paper is to give a new approach to the \(q\)-Euler numbers and polynomials of higher order attached to Dirichlet's character \(\chi\). From this, the authors derive some theoretic identities involving generalized \(q\)-Euler numbers and polynomials of higher order. In Section 2, they present new generating functions which are related to \(q\)-Euler numbers and polynomials of higher order attached to \(\chi\). They obtain distribution relations for the \(q\)-Euler polynomials attached to \(\chi\), and find some identities involving these \(q\)-Euler polynomials. Using the Cauchy residue theorem, it is shown that these \(q\)-extensions of the \(q\)-\(l\)-function of order \(r\) attached to \(\chi\) interpolate the \(q\)-Euler polynomials of order \(r\) at negative integers.
generalized Euler numbers and polynomials
scientific article; zbMATH DE number 5823190
scientific article