A new extension of the Sun-Zagier result involving Bell numbers and derangement numbers

2020/06/24 by Zhi‐Wei Sun, Sun, Zhi-Wei
Engineering · Mathematics · #05A15 #05A18 #11A07 #11B73 #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #Combinatorics (math.CO) #FOS: Mathematics #graph theory and CDMA systems

paper · pdf · doi:10.48550/arxiv.2006.16089

openalex publication_date 2020/06/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let p be any prime and let a and n be positive integers with p\nmid n. We show that ∑_k=1^pa-1(B_k)/((-n)^k)≡ a(-1)^n-1D_n-1\pmod p, where B₀,B₁,… are the Bell numbers and D₀,D₁,… are the derangement numbers. This extends a result of Sun and Zagier published in 2011. Furthermore, we prove that (-x)ⁿ∑_k=1^pa-1(B_k(x))/((-n)^k)≡ -∑_r=1^ax^pr∑_k=0^n-1((n-1)!)/(k!)(-x)^k\pmodp\mathbb Z_p[x], where B_k(x)=∑_l=0^kS(k,l)x^l is the Bell polynomial of degree k with S(k,l) (0≤ l≤ k) the Stirling numbers of the second kind, and \mathbb Z_p is the ring of all p-adic integers.

Related

• Partial Deranged Bell Numbers and Their Combinatorial Properties
• Congruences on the Bell polynomials and the derangement polynomials
• Refinements of the Bell and Stirling numbers
• On a curious property of Bell numbers
• Arithmetic Properties of the Sequence of Derangements and its Generalizations