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. 
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

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