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Poly-Bernoulli numbers from shifted log-sine integrals

Toshiki Matsusaka

Abstract

In 1999, Arakawa and Kaneko introduced a zeta function whose special values at negative integers yield the poly-Bernoulli numbers and investigated its relation to multiple zeta values. Since the poly-Bernoulli numbers appear in this function essentially by design, it is natural to ask whether they arise as special values of more intrinsic zeta-type objects. In this article, we show that a shifted log-sine integral provides such an example. Its analytically continued values at negative integers are given by anti-diagonal sums of poly-Bernoulli numbers with negative index.

Poly-Bernoulli numbers from shifted log-sine integrals

Abstract

In 1999, Arakawa and Kaneko introduced a zeta function whose special values at negative integers yield the poly-Bernoulli numbers and investigated its relation to multiple zeta values. Since the poly-Bernoulli numbers appear in this function essentially by design, it is natural to ask whether they arise as special values of more intrinsic zeta-type objects. In this article, we show that a shifted log-sine integral provides such an example. Its analytically continued values at negative integers are given by anti-diagonal sums of poly-Bernoulli numbers with negative index.

Paper Structure

This paper contains 3 sections, 6 theorems, 25 equations.

Table of Contents

  1. Introduction
  2. Central binomial series
  3. Lehmer's polynomials

Key Result

Theorem 1.2

For $0 < \sigma < \pi$, the shifted log-sine integral $\mathrm{SLs}(s; \sigma)$ admits the analytic continuation to $s \in \mathbb{C}$. For any integer $n \ge -1$, we have where we put $x = \sin^2(\sigma/2)$. In particular, for $n \ge 0$, we have

Theorems & Definitions (10)

  • Definition 1.1
  • Theorem 1.2
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Theorem 2.3
  • Proposition 3.1
  • Proposition 3.2
  • proof