Dual Bases for Analytic Bernoulli Functions

2025/09/25 by Ken Nagai, Nagai, Ken · 1 citation
Computer Science · Mathematics · #Advanced Optimization Algorithms Research #Algebra over a field #Analytic function #Bernoulli polynomials #Bernoulli's principle #Dual (grammatical number) #Extension (predicate logic) #FOS: Mathematics #Fourier analysis #General Mathematics (math.GM) #Orthogonality #Polynomial and algebraic computation

paper · pdf · doi:10.48550/arxiv.2510.00025

published in arXiv (Cornell University) (Cornell University)

openalex publication_date 2025/09/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05

Abstract

We present a dual-basis framework for analytic Bernoulli functions. On the Hurwitz side, even zeta values arise, while on the Clausen side, odd zeta values appear. Both bases are generated by the same Heisenberg--Weyl ladder and are linked by the Poisson--Lerch transform, which plays the role of a Fourier bridge. The resulting orthogonality relations isolate ζ(2m) and β(2m+1) in strictly separated channels. Low-degree examples confirm the rational evaluations, and appendices connect the picture with selector kernels, Poisson summation, and oscillator analogies.

Citations

• Analytic Bernoulli Functions: Correspondence with Hermite Polynomials
• Trigonometric Selector Kernels, Duality, and Odd Zeta Values

Cited by

• Cyclic Vanishing Identities of Sun-Pan Type: Analytic and Modular Perspectives

Related

• Analytic Bernoulli Functions: Correspondence with Hermite Polynomials
• An introduction to the Bernoulli function
• Bernoulli identities, zeta relations, determinant expressions, Mellin transforms, and representation of the Hurwitz numbers
• Trigonometric Selector Kernels, Duality, and Odd Zeta Values
• The generalized Bernoulli numbers and its relation with the Riemann zeta function at odd-integer arguments