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Inequalities pp 109-112 | Cite as

Concavity Properties and a Generating Function for Stirling Numbers

  • Elliott H. Lieb

Abstract

The Stirling numbers of the first kind, S N k, and of the second kind, σN k, are shown to be strongly logarithmically concave as functions of k for fixed TV. This result is stronger than the unimodality conjecture which was heretofore proved only for σN k (Harper). We also introduce a generating function for the σN k which is different from the conventional one but which has a relatively simple closed form expression.

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References

  1. 1.
    M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions (Nat. Bur. Standards Appl. Math. Ser. 55), U. S. Gov’t. Printing Office, Washington, D. C., 1964, p. 824.zbMATHGoogle Scholar
  2. 2.
    L. H. Harper, Stirling Behavior is Asymptotically Normal, Ann. Math. Statist. 38 (1967), 410–414.MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2002

Authors and Affiliations

  • Elliott H. Lieb
    • 1
  1. 1.Physics DepartmentNortheastern UniversityBostonUSA

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