We consider two aspects of generalized Bernoulli polynomials B n (z). One aspect is connected with deening functions instead of polynomials by making the degree n of the polynomial a complex variable. In the second problem we are concerned with the asymptotic behaviour of B n (z) when the degree n tends to innnity.

In particular, B^\0) = B^\ the Bernoulli number of order k, and BJp = Bn, the ordinary Bernoulli number. Note also that B^ = 0 for n > 0. The polynomials B^\z) and the numbers B^ were first defined and studied by Niels Norlund in the 1920s; later they were the subject of many papers by L. Carlitz and others. For the past twenty-five years not much has been done with them, although recently the ...

Keywords: Hurwitz–Lerch Zeta function Lipschitz–Lerch Zeta function Lerch Zeta function Hurwitz Zeta function Riemann Zeta function Legendre chi function Bernoulli polynomials Bernoulli numbers Discrete Fourier transform a b s t r a c t It is shown that there exists a companion formula to Srivastava's formula for the Lipschitz–Lerch Zeta function [see H.M. Srivastava, Some formulas for the Bern...

In this paper, the author presents several closed forms and determinantal expressions involving Stirling numbers of second kind for higher-order Bernoulli Euler polynomials by applying Faà di Bruno formula some properties Bell polynomials.