zbMATH — the first resource for mathematics

Some explicit formulas for certain new classes of Bernoulli, Euler and Genocchi polynomials. (English) Zbl 1353.11031
Summary: In recent years, the subject of Apostol-Bernoulli polynomials, Apostol-Euler polynomials and Apostol-Genocchi polynomials have been studied extensively. Recently, the authors have introduced in [Honam Math. J. 34, No. 3, 311–326 (2012; Zbl 1327.11020); Int. J. Math. Math. Sci. 2012, Article ID 182785, 14 p. (2012; Zbl 1258.11047)] some new generalized classes of Apostol-Bernoulli, Apostol-Euler and Apostol-Genocchi polynomials. In this paper, with the help of a result involving an explicit formula for the generalized potential polynomials obtained by M. Cenkci [Discrete Math. 309, No. 6, 1498–1510 (2009; Zbl 1213.11046)], we develop some explicit formulas related with these new classes of polynomials.

11B68 Bernoulli and Euler numbers and polynomials
33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
11B65 Binomial coefficients; factorials; \(q\)-identities