A generalized recurrence formula for Stirling numbers and related sequences

Mark Shattuck
Notes on Number Theory and Discrete Mathematics, ISSN 1310-5132
Volume 21, 2015, Number 4, Pages 74—80
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Authors and affiliations

Mark Shattuck
Department of Mathematics, University of Tennessee
37996 Knoxville, TN, USA

Abstract

In this note, we provide a combinatorial proof of a generalized recurrence formula satisfied by the Stirling numbers of the second kind. We obtain two extensions of this formula, one in terms of r-Whitney numbers and another in terms of q-Stirling numbers of Carlitz. Modifying our proof yields analogous formulas satisfied by the r-Stirling numbers of the first kind and by the r-Lah numbers.

Keywords

  • Stirling numbers
  • r-Whitney numbers
  • q-Stirling numbers

AMS Classification

  • 05A19
  • 05A18

References

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  3. Carlitz, L. (1948) q-Bernoulli numbers and polynomials, Duke Math. J., 15, 987–1000.
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  8. Nyul, G. & G. R´acz (2014) The r-Lah numbers, Discrete Math., http://dx.doi.org/10.1016/j.disc.2014.03.029 (in press).
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Cite this paper

APA

Shattuck, M. (2015). A generalized recurrence formula for Stirling numbers and related sequences. Notes on Number Theory and Discrete Mathematics, 21(4), 74-80.

Chicago

Shattuck, Mark. “A Generalized Recurrence Formula for Stirling Numbers and Related Sequences.” Notes on Number Theory and Discrete Mathematics 21, no. 4 (2015): 74-80.

MLA

Shattuck, Mark. “A Generalized Recurrence Formula for Stirling Numbers and Related Sequences.” Notes on Number Theory and Discrete Mathematics 21.4 (2015): 74-80. Print.

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