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.
Reviewer: Colin Defant (Dade City)
MSC:
11B73 | Bell and Stirling numbers |
05A18 | Partitions of sets |
11A07 | Congruences; primitive roots; residue systems |
Keywords:
generalized Bell umbra; Bell polynomials; derangement polynomials; lah polynomials; congruencesReferences:
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