Some formulas for Bell numbers

  title={Some formulas for Bell numbers},
  author={Takao Komatsu and Claudio Pita-Ruiz},
We give elementary proofs of three formulas involving Bell numbers, including a generalization of the Gould-Quaintance formula and a generalization of Spivey’s formula. We find variants for two of our formulas which involve some well-known sequences, among them the Fibonacci, Bernoulli and Euler numbers. 
3 Citations
Some combinatorial identities for the r-Dowling polynomials
  • M. Shattuck
  • Mathematics
    Notes on Number Theory and Discrete Mathematics
  • 2019
Recently, three new Bell number formulas were proven using algebraic methods, one of which extended an earlier identity of Gould–Quaintance and another a previous identity of Spivey. Here, making use
Shifting powers in Spivey’s Bell number formula
Abstract In this paper, we consider extensions of Spivey’s Bell number formula wherein the argument of the polynomial factor is translated by an arbitrary amount. This idea is applied more generally


Extensions of Spivey's Bell Number Formula
  • A. Xu
  • Mathematics
    Electron. J. Comb.
  • 2012
An extension of Spivey's Bell number formula and its associated Bell polynomial extension is established by using Hsu-Shiue's generalized Stirling numbers and Gould-Quaintance's new Bell number formulas are extended.
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
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
Generalizations of Bell number formulas of Spivey and Mezo
We provide q -generalizations of Spivey’s Bell number formula in various settings by considering statistics on different combinatorial structures. This leads to new identities involving q -Stirling
Implications of Spivey's Bell Number Formula
Recently, Spivey discovered a novel formula for B(n + m), where B(n + m) is the (n + m) th Bell number. His proof was combinatorial in nature. This paper provides a generating function proof of
Carlitz-Type and Other Bernoulli Identities
By using an explicit formula for Bernoulli polynomials we obtained in a recent work (in which Bn (x) is written as a linear combination of the polynomials (x − r) n , r = 1,...,K +1, where K ≥ n), it
On a New Family of Generalized Stirling and Bell Numbers
A new family of generalized Stirling and Bell numbers is introduced by considering powers $(VU)^n$ of the noncommuting variables $U,V$ satisfying $UV=VU+hV^s$. The case $s=0$ (and $h=1$) corresponds
The Generalized Stirling and Bell Numbers Revisited
The generalized Stirling numbers Ss;h(n,k) introduced recently by the authors are shown to be a special case of the three parameter family of generalized Stirling numbers S(n,k;�,�,r) considered by
Enumerative combinatorics
This review of 3 Enumerative Combinatorics, by Charalambos A.good, does not support this; the label ‘Example’ is given in a rather small font followed by a ‘PROOF,’ and the body of an example is nonitalic, utterly unlike other statements accompanied by demonstrations.