this paper, we generalize the current results on the p--Eisenstein

In this paper we establish some new congruences of p-adic integer order Bernoulli numbers. These generalize the Kummer congruences for ordinary Bernoulli numbers. We apply our congruences to prove irreducibility of certain Bernoulli polynomials with order divisible by p and to get new congruences for Stirling numbers. 1. INTRODUCTION The arbitrary order Bernoulli polynomials B (!) n (x) are defined by (cf. [1, 12]) / t e t \Gamma 1 ! ! e xt = 1 X n=0 B (!) n (x) t n n! : (1) In particular, B (!) n (0) = B (!) n is the Bernoulli number of order ! and degree n. If the order ! = 1 the polynomials B (1) n (x) = B n (x) and the numbers B (1) n = B n are called ordinary. More generally, if ! 2 f1; : : : ; ng, the polynomials and numbers are called higher order, or more properly first or higher order. In this paper we consider Bernoulli polynomials and numbers where the order is an arbitrary p-adic integer. In [1] we considered only higher order polynomials and nu...

For any prime p we establish a congruence of Kummer type for the Norlund polynomial B (x) n , and determine the highest power of p which divides the coefficient denominators of the Norlund polynomial. This improves the result of [4] where only an upper bound was proven. We deduce a simple formula for the least common denominator of the coefficients. Applications are made for the Stirling polynomials. Keywords Norlund polynomials, Bernoulli numbers and polynomials, Stirling polynomials, Kummer congruences. 1. Introduction We are pleased to contribute to this volume dedicated to H. W. Gould, whose collection of binomial identities [6] led us into the field of Bernoulli and related polynomials [2]. The Norlund polynomials B (x) n are defined by [10, Chapter 6] X n B (x) n t n n! = / t e t \Gamma 1 ! x : They have many important applications, e.g., B (k) n is the Bernoulli number of order k and degree n, and in particular B (1) n = B n is the ordinary Bernoulli nu...

this paper, we are primarily concerned with factorization questions of the Bernoulli polynomials, both over the rational number field Q and over the field of

The degenerate Stirling numbers and degenerate Eulerian polynomials are intimately connected to the arithmetic of generalized factorials. In this article we show that these numbers and similar sequences may in fact be expressed as p-adic integrals of generalized factorials. As an application of this identification we deduce systems of congruences which are analogues and generalizations of the Kummer congruences for the ordinary Bernoulli numbers. Keywords: Degenerate weighted Stirling numbers; Degenerate Eulerian polynomials; Partial Stirling numbers; Kummer congruences; p-adic integration 1.

. We establish the p--adic singularity pattern of the coefficients of the higher order Bernoulli polynomials, and use this to determine all instances of p--Eisenstein behavior. This approach provides new proofs for and generalizes known irreducibility results. 1. INTRODUCTION This announcement summarizes the author's most significant results on higher order Bernoulli polynomials. Details of the proofs will appear in [2]. In this paper, we will use standard notations rather than the terminology of [1] and [2]. The Bernoulli polynomials B n (x) are defined by [11,12] te xt e t \Gamma 1 = 1 X n=0 B n (x) t n n! ; and the arbitrary order Bernoulli polynomials B (!) n (x) are defined by / t e t \Gamma 1 ! ! e xt = 1 X n=0 B (!) n (x) t n n! : These are called higher order, or more properly first or higher order, if ! = ff 2 f1; : : : ; ng. It is well known that B (!) n (x) is a polynomial with rational coefficients, of degree n in x and !, which is monic in ...

The paper generalizes the traditional single factorial function to integer-valued multiple factorial (j-factorial) forms. The generalized factorial functions are defined recursively as triangles of coefficients corresponding to the polynomial expansions of a subset of degenerate falling factorial functions. The resulting coefficient triangles are similar to the classical sets of Stirling numbers and satisfy many analogous finite-difference and enumerative properties as the well-known combinatorial triangles. The generalized triangles are also considered in terms of their relation to elementary symmetric polynomials and the resulting symmetric polynomial index transformations. The definition of the Stirling convolution polynomial sequence is generalized in order to enumerate the parametrized sets of j-factorial polynomials and to derive extended properties of the j-factorial function expansions. The generalized j-factorial polynomial sequences considered lead to applications expressing key forms of the j-factorial functions in terms of arbitrary partitions of the j-factorial function expansion triangle indices, including several identities related to

On the degrees of irreducible factors of higher order Bernoulli polynomials by Arnold Adelberg (Grinnell, Ia.)