Generalizations of the Bernoulli and Appell polynomials

Abstract

We first introduce a generalization of the Bernoulli polynomials, and consequently of the Bernoulli numbers, starting from suitable generating functions related to a class of Mittag-Leffler functions. Furthermore, multidimensional extensions of the Bernoulli and Appell polynomials are derived generalizing the relevant generating functions, and using the Hermite-Kampé de Fériet (or Gould-Hopper) polynomials. The main properties of these polynomial sets are shown. In particular, the differential equations can be constructed by means of the factorization method.

A Note on Truncated Exponential-Based Appell Polynomials

“…The examples of Appell sequences are the sequences of growing powers of variable x {x n } ∞ n=0 , the Bernoulli sequences B n (x) [3,15,17], the Euler sequences E n (x) [13,15,17], the Hermite normalized sequences He n (x) [4], and the Laguerre sequences L n (x) [4]. Further, generalizations of above polynomials have been considered [4][5][6].…”

Algebraic properties and Fourier expansions of two-dimensional Apostol–Bernoulli and Apostol–Euler polynomials

“…The series (1) converges for |t + log λ| < 2π (we use the principal branch of the logarithm). The 2D Bernoulli polynomials, corresponding to λ = 1, were defined in [3,4]. They have been studied recently in [9] from the point of view of differential equations and further generalizations are found in [5].…”

Generalized Bernoulli Polynomials and Numbers, Revisited

“…Here, B 2k+2 are Bernoulli numbers, B k = B (1) k (0). They are calculated through another one standard recursion:…”