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Curious congruences related to the Bell polynomials. (English) Zbl 1415.11042

This paper studies some famous classes of polynomials that can be defined via classical combinatorial objects. The main results of the paper are congruences that these polynomials satisfy modulo a prime \(p\). For example, define the \(r\)-derangement polynomials \(\mathcal{D}_{n,r}(x)\) by \(\mathcal{D}_{n,r}(x)=\frac{1}{r!}\sum_{j=0}^n{\binom{n}{j}}(j+r)!(x-1)^{n-j}\). The authors show that \[ \mathcal{D}_{mp+q,r-1}(1-x)\equiv(-x)^{mp}\mathcal{D}_{q,r-1}(1-x)\pmod p. \] As another example, consider the \(n^{\text{th}}\) Lah polynomial \(\mathcal{L}_n(x)=\sum_{k=0}^nL(n,k)(x)_k\), where \(L(n,k)\) is the number of partitions of the set \([n]\) into \(k\) ordered lists and \[ (\alpha)_r=\alpha(\alpha-1)\cdots(\alpha-r+1). \] The authors show that \[ (-x)^{m+n-2}\sum_{k=0}^{p-1}(-m)_{p-1-k}\mathcal{L}_{n+k}(x)\equiv x^{p-1}\sum_{j=0}^n(-1)^jL(n,j)\mathcal{D}_{j+m+n-2}(1-x)\pmod p. \] The paper extends further, giving congruences involving the so-called \((r_1,\ldots,r_q)\)-Bell polynomials.

MSC:

11B73 Bell and Stirling numbers
05A18 Partitions of sets
11A07 Congruences; primitive roots; residue systems
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