Identities behind some congruences for r-Bell and derangement polynomials
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RESEARCH
Identities behind some congruences for r-Bell and derangement polynomials Grzegorz Serafin * Correspondence:
grzegorz.serafi[email protected] Faculty of Pure and Applied Mathematics, Wrocław University of Science and Technology, Ul. Wybrze˙ze Wyspianskiego ´ 27, Wrocław, Poland The author was supported by the National Science Centre Grant No. 2015/18/E/ST1/00239.
Abstract We derive new congruences bounding r-Bell and derangement polynomials, which generalize the existing ones, while the presented approach is significantly simpler and, at the same time, more informative. Namely, we provide precise identities that imply the congruences and explain somehow their nature. Keywords: r-Bell polynomials, Derangement polynomials, Congruences, Identities Mathematics Subject Classification: 11B73, 11A07, 11C08, 05A15
1 Introduction The Bell numbers Bn represent number of partitions of a given set of n elements and are one of the most classical objects in combinatorics. The r-Bell numbers Bn,r extend this definition and count partitions of a set of n + r elements such that r chosen elements are separated [13]. They appear also naturally when dealing with congruences of classical Bell numbers, which will be shown in the sequel. If we restrict the partitions to consist of exactly k + r sets, their number is given by the r-Stirling numbers of the second kind nk r (see [2–4]), which lets us write Bn,r = nk=0 nk r . The framework of this paper is slightly more general. Precisely, we focus on the r-Bell polynomials given by Bn,r (x) =
n n k=0
k
xk ,
x ∈ R.
r
Clearly, we have Bn,r (1) = Bn,r . Furthermore, for r = 0 we obtain the Bell polynomials, known also as Touchard or exponential polynomials. Some of the most intensively studied features of these polynomials are their divisibility properties (see, among others, [7,8,10, 12,15,19–22]). The goal of the paper is to find precise identities that directly lead to some known as well as some new congruences. This method is not only more constructive and informative, but also, at least in this case, turns out to be significantly shorter. The history of research on congruences for Bell number is very long and goes back to 1933 and the famous Touchard congruence [21] Bn+p ≡ Bn + Bn+1
123
(mod p),
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