Concavity properties and a generating function for stirling numbers

Elliott H. Lieb

doi:10.1016/S0021-9800(68)80057-2

Concavity properties and a generating function for stirling numbers

Elliott H. Lieb

Mathematics

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45 Scopus citations

Abstract

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

Original language	English (US)
Pages (from-to)	203-206
Number of pages	4
Journal	Journal of Combinatorial Theory
Volume	5
Issue number	2
DOIs	https://doi.org/10.1016/S0021-9800(68)80057-2
State	Published - Sep 1968

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title = "Concavity properties and a generating function for stirling numbers",

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

author = "Lieb, {Elliott H.}",

note = "Funding Information: * Work supported by National Science Foundation Grant GP-6851. t Present address: Mathematics Department, M.I.T., Cambridge, Massachusetts, 02139.",

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N1 - Funding Information: * Work supported by National Science Foundation Grant GP-6851. t Present address: Mathematics Department, M.I.T., Cambridge, Massachusetts, 02139.

PY - 1968/9

Y1 - 1968/9

N2 - The Stirling numbers of the first kind, SNk, and of the second kind, δNk, are shown to be strongly logarithmically concave as functions of k for fixed N. This result is stronger than the unimodality conjecture which was heretofore proved only for δNk (Harper). We also introduce a generating function for the δNk which is different from the conventional one but which has a relatively simple closed form expression.

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

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Concavity properties and a generating function for stirling numbers

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