Some identities and an explicit formula for Bernoulli and Stirling numbers

2012/02/09 by Feng Qi, Bai‐Ni Guo · 1 citation
Mathematics · #Advanced Mathematical Identities #Bell polynomials #Bernoulli number #Bernoulli's principle #Binomial coefficient #Combinatorics #Harmonic number #Mathematical Inequalities and Applications #Mathematical functions and polynomials #Mathematics #Pure mathematics #Stirling engine #Stirling number #Stirling numbers of the first kind #Stirling numbers of the second kind #math.CA #math.CO #math.NT #msc:11B73 #msc:26A24 #msc:33B10 #msc:34A30

paper · pdf · doi:10.1016/j.cam.2013.06.020

published as Bai-Ni Guo and Feng Qi, Some identities and an explicit formula for Bernoulli and Stirling numbers, Journal of Computational and Applied Mathematics 255 (2014), 568--579 · 9 pages

arxiv created 2012/02/09 · openalex publication_date 2013/06/25 · arxiv updated 2014/03/07 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05

Abstract

In the paper, the author establishes some identities which show that the functions \frac1(1-e^{± t})^k and the derivatives (\frac1e^{± t}-1)⁽ⁱ⁾ can be expressed each other by linear combinations with coefficients involving the combinatorial numbers and the Stirling numbers of the second kind, where t≠0 and i,k∈ℕ.

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• Diagonal recurrence relations, inequalities, and monotonicity related to the Stirling numbers of the second kind