Closed formulas and determinantal expressions for higher-order Bernoulli and Euler polynomials in terms of Stirling numbers

Explicit Formulas for Bernoulli and Euler Numbers

Explicit and recursive formulas for Bernoulli and Euler numbers are derived from the Faá di Bruno formula for the higher derivatives of a composite function. Along the way we prove a result about composite generating functions which can be systematically used to derive such identities.

Congruences for Bernoulli , Euler , and Stirling Numbers Paul

The values at x=0 are called Bernoulli and Euler numbers of order w; when w=1, the polynomials or numbers are called ordinary. When x=0 or w=1, we often suppress that part of the notation; e.g., B (w) n denotes B n (0), En(x) denotes E (1) n (x), and Bn denotes B (1) n (0). These numbers have been extensively studied and many congruences for them are known. Among the most important results are ...

Some Determinantal Expressions and Recurrence Relations of the Bernoulli Polynomials

In the paper, the authors recall some known determinantal expressions in terms of the Hessenberg determinants for the Bernoulli numbers and polynomials, find alternative determinantal expressions in terms of the Hessenberg determinants for the Bernoulli numbers and polynomials, and present several new recurrence relations for the Bernoulli numbers and polynomials.

Generalized higher order Stirling numbers

Some Formulas for a Family of Numbers Analogous to the Higher-Order Bernoulli Numbers

In this paper the authors establish several formulas and results for the D numbers D (k) 2n and d (k) 2n , which are analogous to the higher-order Bernoulli numbers. Some applications of these families of D numbers are also presented. Corresponding author. 1