Given a formal map F = (F1 . . . , Fn) of the form z + higher order terms, we give tree expansion formulas and associated algorithms for the D-Log of F and the formal flow Ft. The coefficients which appear in these formulas can be viewed as certain generalizations of the Bernoulli numbers and the Bernoulli polynomials. Moreover the coefficient polynomials in the formal flow formula coincide wit...

1. Introduction. In this paper, we generalize the current results on the p-Eisenstein behavior of first and higher order Bernoulli polynomials [4], [6–9], using the machinery of [1]. In so doing, we provide a broader framework for the known results, all of which are either immediate consequences or special cases of our more general results. Because of an explicit formula for the coefficients in...

The aim of this paper is to give generating functions and to prove various properties for some new families of special polynomials and numbers. Several interesting properties of such families and their connections with other polynomials and numbers of the Bernoulli, Euler, Apostol-Bernoulli, ApostolEuler, Genocchi and Fibonacci type are presented. Furthermore, the Fibonacci type polynomials of ...

We analyze the asymptotic behavior of the Apostol-Bernoulli polynomials Bn(x;λ) in detail. The starting point is their Fourier series on [0, 1] which, it is shown, remains valid as an asymptotic expansion over compact subsets of the complex plane. This is used to determine explicit estimates on the constants in the approximation, and also to analyze oscillatory phenomena which arise in certain ...