An asymptotic series with Nörlund-number coefficients for higher-order Fubini numbers with a parameter

S. Spektor
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 32, 2026, Number 3, Pages 642–652
DOI: 10.7546/nntdm.2026.32.3.642-652
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S. Spektor
Quantitative Science Department, Canisius University
2001 Main Street, Buffalo, NY 14208–1098, USA

Abstract

For real parameters r, \alpha > 0, let F_n^{(r)}(\alpha) denote the sequence built from Stirling numbers of the second kind by weighting with rising factorials of r and powers of \alpha. Three familiar specializations sit inside this family: the Fubini numbers (r=\alpha=1), the higher-order Fubini numbers (\alpha=1, r a positive integer), and—after rescaling \alpha in terms of p—the integer moments of a negative-binomial random variable. Our main theorem is an asymptotic expansion for F_n^{(r)}(\alpha) in which the successive coefficients turn out to be the Nörlund (higher-order Bernoulli) numbers. A finite-termination phenomenon occurs at positive integer r: the algebraic singular expansion at the dominant singularity has exactly r nonzero terms, and the resulting r-term approximation agrees with F_n^{(r)}(\alpha) to within an exponentially small relative error. We also prove a linear recurrence that holds for all real r,\alpha > 0; it powers the numerical experiments at the end of the paper.

Keywords

  • Higher-order Fubini numbers
  • Nörlund numbers
  • Stirling numbers of the second kind
  • Asymptotic expansion
  • Negative binomial distribution
  • Singularity analysis
  • Higher-order geometric polynomials.

2020 Mathematics Subject Classification

  • 11B73
  • 11B68
  • 05A16
  • 60E05

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Manuscript history

  • Received: 11 May 2026
  • Revised: 5 November 2026
  • Accepted: 10 November 2026
  • Online First: 11 November 2026

Copyright information

Ⓒ 2026 by the Author.
This is an Open Access paper distributed under the terms and conditions of the Creative Commons Attribution 4.0 International License (CC BY 4.0).

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Cite this paper

Spektor, S. (2026). An asymptotic series with Nörlund-number coefficients for higher-order Fubini numbers with a parameter. Notes on Number Theory and Discrete Mathematics, 32(3), 642-652, DOI: 10.7546/nntdm.2026.32.3.642-652.

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