Some Congruences for the Partial Bell Polynomials

Let $B_{n,k}$ and $A_{n}=\sum_{j=1}^{n}B_{n,j}$ with

be, respectively, the $(n,k)^{\rm th}$ partial and the $n^{\rm th}$ complete Bell polynomials with indeterminate arguments $x_1,x_2,\ldots$ . Congruences for $A_{n}$ and $B_{n,k}$ with respect to a prime number have been studied by several authors. In the present paper, we propose some results involving congruences for $B_{n,k}$ when the arguments are integers. We give a relation between Bell polynomials and we apply it to several congruences. The obtained congruences are connected to binomial coefficients.

Received January 10 2009; revised version received April 29 2009. Published in Journal of Integer Sequences, May 12 2009.

Some Congruences for the Partial Bell Polynomials

Miloud Mihoubi Faculty of Mathematics P. O. Box 32 16111 El-Alia, Bab-Ezzouar, Algiers Algeria

Miloud Mihoubi
Faculty of Mathematics
P. O. Box 32
16111 El-Alia, Bab-Ezzouar, Algiers
Algeria