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2014•Studies in Applied MathematicsRequires access

Asymptotics of Stirling and Chebyshev‐Stirling Numbers of the Second Kind

Wolfgang Gawronski, Lance L. Littlejohn, Thorsten Neuschel

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Abstract

For the classical Stirling numbers of the second kind, asymptotic formulae are derived in terms of a local central limit theorem. The underlying probabilistic approach also applies to the Chebyshev–Stirling numbers, a special case of the Jacobi–Stirling numbers. Essential features are uniformity properties and the fact that the leading terms of the asymptotics are given explicitly and they contain elementary expressions only. Thereby supplements of the asymptotic analysis of these numbers are established.

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What this paper is about

For the classical Stirling numbers of the second kind, asymptotic formulae are derived in terms of a local central limit theorem. The underlying probabilistic approach also applies to the Chebyshev–Stirling numbers, a special case of the Jacobi–Stirling numbers. Essential features are uniformity properties and the fact that the leading terms of the asymptotics are given explicitly and they contain elementary expressions only. Thereby supplements of the asymptotic analysis of these numbers are established.

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Available abstract

For the classical Stirling numbers of the second kind, asymptotic formulae are derived in terms of a local central limit theorem. The underlying probabilistic approach also applies to the Chebyshev–Stirling numbers, a special case of the Jacobi–Stirling numbers. Essential features are uniformity properties and the fact that the leading terms of the asymptotics are given explicitly and they contain elementary expressions only. Thereby supplements of the asymptotic analysis of these numbers are established.

Key concepts: Stirling numbers of the first kind, Stirling number, Stirling numbers of the second kind, Mathematics, Stirling engine, Bell polynomials, Chebyshev filter, Limit (mathematics)

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