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


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.


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

AMS Classification

  • 05A19
  • 05A18


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Cite this paper

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

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