Congruences on the Bell polynomials and the derangement polynomials

Abstract

In this note, by the umbra calculus method, the Sun and Zagier's congruences involving the Bell numbers and the derangement numbers are generalized to the polynomial cases. Some special congruences are also provided.

Identities behind some congruences for r-Bell and derangement polynomials

“…Note that taking r = 0, a = 1 or n = 0, a = 1 we recover the main result of [20] and Theorem 6 from [15], respectively. On the other hand, for n = r = 0, we obtain the main result of [18].…”

“…Furthermore, for r = 0 we obtain the Bell polynomials, known also as Touchard or exponential polynomials. Some of the most intensively studied features of these polynomials are their divisibility properties (see, among others, [7,8,10,12,15,[19][20][21][22]). The goal of the paper is to find precise identities that directly lead to some known as well as some new congruences.…”

“…Later on, this result has been generalized in several directions. In [20] and [15] the authors considered B n+k and B k,r , respectively, instead of B k on the left-hand side of the congruence, while in [18] the sum in (1) is considered up to p a − 1, where a is any positive integer. In this paper, we unify all those results by providing the following equivalence (see Theorem 2)…”

Identities and congruences involving the geometric polynomials

“…This work is motivated by application of the umbral calculus method to determine identities and congruences involving Bell numbers and polynomials in the works of Gessel [13], Sun et al [27], Mező et al [19] and Benyattou et al [1]. In this paper, we will talk about identities and congruences involving the .r; s/-geometric polynomials based on the geometric umbra defined by w n x WD w n .x/: For more information about umbral calculus, see [5,13,[22][23][24][25].…”

Real-rooted polynomials via generalized Bell umbra

“…, B n,r (x) = e −x j≥0 (j + r) n x j j! and B n x = B n (x) be the generalized Bell umbra introduced by Sun et al [19]. It is known that, for any polynomial f and any integer n ≥ 0, the generalized Bell umbra satisfies [19]…”