The Relationship Between Euler Numbers and Bernoulli Numbers with Ordered Partitions

2025/12/04 by Pirayvatloo, Kamyar Sepehri, Azar, Kazem Haghnejad
#FOS: Mathematics #General Mathematics (math.GM)

paper · doi:10.48550/arxiv.2512.06028

Abstract

In this paper, for every n ∈ ℕ, the following relationships between the functions K_b(n) and K_e(n) and the Bernoulli and Euler numbers are proved: B_2n = - \frac(2n)!2²ⁿ-2 K_b(n), E_2n = (2n)! K_e(n). The functions K_b and K_e are defined recursively by K_b(0) = K_e(0) = 1, K_b(n) = - ∑_n'=0 ^n-1 \fracK_b(n')( 2(n-n') + 1 )!, n ≥ 1, K_e(n) = - ∑_n'=0 ^n-1 \fracK_e(n')( 2(n-n') )!, n ≥ 1. Furthermore, we present combinatorial interpretations of these functions in terms of ordered partitions of n: K_b(n) = ∑_{λ\vDash n} \frac(-1)^ℓ(λ) ∏_i=1^ℓ(λ) (2b_i + 1)!, n ≥ 1, K_e(n) = ∑_{λ\vDash n} \frac(-1)^ℓ(λ) ∏_i=1^ℓ(λ) (2b_i)!, n ≥ 1, where λ= (b₁,b₂,…,b_k) \vDash n and ℓ(λ)=k.

Citations

• Bounds for the logarithm of the Euler gamma function and its derivatives
• Finite, closed-form expressions for the partition function and for Euler, Bernoulli, and Stirling numbers
• Some series and integrals involving the Riemann zeta function, binomial coefficients and the harmonic numbers. Volume I
• On Bernoulli Numbers and Its Properties

Related

• Some results associated with Bernoulli and Euler numbers with applications
• Generalized Euler numbers and ordered set partitions
• Bernoulli and Euler Partitions
• New partition function recurrences
• On Euler's Theorem