Some formulas for Bell numbers
@article{Komatsu2018SomeFF, title={Some formulas for Bell numbers}, author={Takao Komatsu and Claudio Pita-Ruiz}, journal={Filomat}, year={2018}, volume={32}, pages={3881-3889} }
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
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References
SHOWING 1-10 OF 14 REFERENCES
Extensions of Spivey's Bell Number Formula
- MathematicsElectron. 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
- Mathematics
- 2008
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
- Mathematics
- 2012
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
- Mathematics
- 2014
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
- Mathematics
- 2008
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
- Mathematics
- 2016
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
- MathematicsElectron. J. Comb.
- 2011
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
- Mathematics
- 2012
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
- MathematicsSIGA
- 2008
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.
A generalized recurrence for Bell polynomials: An alternate approach to Spivey and Gould-Quaintance formulas
- MathematicsEur. J. Comb.
- 2009