A New Identity Linking Bernoulli Numbers, Stirling Numbers of the First Kind, and Bessel Numbers of the First Kind

2025/05/28 by Benmoussa, Abdelhay
#05A18 #11B68 #11B73 #33C10 #FOS: Mathematics #General Mathematics (math.GM)

paper · doi:10.48550/arxiv.2505.22819

Abstract

We establish a new identity linking Bernoulli, Stirling (first kind), and Bessel (first kind) numbers: ∑_k=0ⁿ 2 ^n-k s(n,k) B_k = ∑_k=0ⁿ b(n,k) ((-1)^k k!)/(k+1). This parallels the classical Stirling--Bernoulli relation B_n = ∑_k=0ⁿ S(n,k) ((-1)^k k!)/(k+1), replacing S(n,k) with s(n,k) and b(n,k), and thus revealing a new structural connection among these families of numbers.

Citations

• Combinatorial proofs of inverse relations and log-concavity for Bessel numbers

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