Identities behind some congruences for r-Bell and derangement polynomials

@inproceedings{Serafin2020IdentitiesBS,
  title={Identities behind some congruences for r-Bell and derangement polynomials},
  author={Grzegorz Serafin},
  year={2020}
}
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. 
View via Publisher
link.springer.com
2 Citations
Background Citations
2

2 Citations

Citation Type

Citation Type

Backward Touchard congruence
  • G. Serafin
  • Mathematics
    Bulletin of the Belgian Mathematical Society - Simon Stevin
  • 2022
The celebrated Touchard congruence states that Bn+p ≡ Bn + Bn+1 modulo p, where p is a prime number and Bn denotes the Bell number. In this paper we study divisibility properties of Bn−p and their…
Derangements and the $p$-adic incomplete gamma function.
We introduce a new $p$-adic analogue of the incomplete gamma function. We also introduce a closely related family of combinatorial sequences counting derangements and arrangements in certain wreath…

References

SHOWING 1-10 OF 20 REFERENCES
Congruences on the Bell polynomials and the derangement polynomials
Some congruences concerning the Bell numbers
In this Note we give elementary proofs { based on umbral calculus { of the most fundamental congruences satised by the Bell numbers and polynomials. In particular, we establish the conguences of…
Divisibility properties of the r-Bell numbers and polynomials
ON A CURIOUS PROPERTY OF BELL NUMBERS
Abstract In this paper we derive congruences expressing Bell numbers and derangement numbers in terms of each other modulo any prime.
The Dual of Spivey's Bell Number Formula
We give the dual of Spivey’s recent formula for Bell numbers. The dual involves factorials and Stirling numbers of the first kind. We point out that Spivey’s formula immediately yields the famous…
On a Generalized Recurrence for Bell Numbers
A novel recurrence relation for the Bell numbers was recently derived by Spivey, using a beautiful combinatorial argument. An algebraic derivation is proposed that allows straightforward…
The r-Bell numbers
The notion of generalized Bell numbers was appeared in several works but there is no a systematic treatise on this topic. In this paper we fill this gap. We discuss the most important combinatorial,…
The r-Stirling numbers
A Generalized Recurrence for Bell Numbers
We show that the two most well-known expressions for Bell numbers, n = P n=0 � n � and n+1 = P n=0 n k � k, are both special cases of a third expression for the Bell numbers, and we give a…
New modular properties of bell numbers
...
1
2
...