Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Dual Bases for Analytic Bernoulli Functions

Ken Nagai

TL;DR

This work develops a dual-basis framework for analytic Bernoulli functions, revealing a Hurwitz-based basis encoding even zeta values $\zeta(2m)$ and a Clausen-based basis encoding odd Dirichlet beta values $\beta(2m{+}1)$, with both arising from a common Heisenberg–Weyl ladder. The two bases form a dual system connected by the Poisson–Lerch transform, yielding explicit orthogonality relations that strictly separate the two zeta channels. Cross-branch pairings vanish, and a rotated-weight interpolation shows a continuous path between the beta- and zeta-branches. Low-degree checks corroborate the rational evaluations, and appendices link the framework to selector kernels, Poisson summation, and oscillator analogies, unifying several perspectives on analytic Bernoulli functions.

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.

Dual Bases for Analytic Bernoulli Functions

TL;DR

This work develops a dual-basis framework for analytic Bernoulli functions, revealing a Hurwitz-based basis encoding even zeta values and a Clausen-based basis encoding odd Dirichlet beta values , with both arising from a common Heisenberg–Weyl ladder. The two bases form a dual system connected by the Poisson–Lerch transform, yielding explicit orthogonality relations that strictly separate the two zeta channels. Cross-branch pairings vanish, and a rotated-weight interpolation shows a continuous path between the beta- and zeta-branches. Low-degree checks corroborate the rational evaluations, and appendices link the framework to selector kernels, Poisson summation, and oscillator analogies, unifying several perspectives on analytic Bernoulli functions.

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 and in strictly separated channels. Low-degree examples confirm the rational evaluations, and appendices connect the picture with selector kernels, Poisson summation, and oscillator analogies.

Paper Structure

This paper contains 9 sections, 5 theorems, 41 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Examples: low-degree checks
  4. Conclusion
  5. Appendix A: Ladder verification
  6. Appendix B: Kernels and verification tables
  7. Appendix C: Poisson--Lerch derivations
  8. Appendix D: Oscillator analogies
  9. Appendix E: Numerical verification tables

Key Result

Proposition 2.1

Let $F^*$ denote the umbral operator of the analytic Bernoulli system. Then the Bernoulli family $\{B_n(x)\}_{n\ge0}$ and the Hermite family $\{H_n(x)\}_{n\ge0}$ are both governed by the same Heisenberg--Weyl ladder relation In particular, This realizes the Bernoulli--Hermite correspondence as an umbral analogue of the one-dimensional oscillator ladder.

Theorems & Definitions (9)

  • Proposition 2.1: Common Ladder
  • Proposition 2.2: Dual bases: Hurwitz vs. Clausen
  • Theorem 2.3: Dual pairing: alternating branch
  • Theorem 2.4: Dual pairing: symmetric branch
  • Remark 2.5: Selection rules
  • Proposition 2.6: Poisson--Lerch Bridge
  • Remark 3.1: Selection rules revisited
  • Remark 3.2: Rotated weights and continuous interpolation
  • Remark 3.3: Analytic provenance