Sums of Products of Bernoulli numbers of the second kind

2007/09/19 by Ming Wu, Hao Pan, Wu, Ming +1
Mathematics · #05A19 #11B68 #Advanced Mathematical Identities #Analytic Number Theory Research #Combinatorics (math.CO) #FOS: Mathematics #Mathematical functions and polynomials #Number Theory (math.NT) #math.CO #math.NT #msc:05A19 #msc:11B68

paper · pdf · doi:10.48550/arxiv.0709.2947

Accepted by the Fibonacci Quarterly

arxiv created 2007/09/19 · openalex publication_date 2007/09/19 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

The Bernoulli numbers b₀,b₁,b₂,.... of the second kind are defined by ∑_n=0^∞ b_ntⁿ=(t)/(log(1+t)). In this paper, we give an explicit formula for the sum ∑_{j1+j2+...+jN=n, j1,j2,...,jN>=0}b_j1b_j2...b_jN. We also establish a q-analogue for ∑_k=0ⁿ b_kb_n-k=-(n-1)b_n-(n-2)b_n-1.

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