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Iterated beta integrals

Minoru Hirose, Nobuo Sato

Abstract

We introduce iterated beta integrals, a new class of iterated integrals on the universal abelian covering of the punctured projective line that unifies hyperlogarithms and classical beta integrals while preserving their fundamental properties. We establish various analytic properties of these integrals with respect to both the exponent parameters and the main variables. Their key feature is invariance under simultaneous translation of the exponent parameters, which generates relations between integrals over possibly different coverings. This mechanism recovers notable identities for multiple zeta values and variants -- including Zagier's 2-3-2 formula, Murakami's $t$-value analogue, Charlton's $t$-value analogue, Zhao's $2$-$1$ formula, and Ohno's relation -- and also yields new relations, such as a proof of a Galois descent phenomenon for multiple omega values.

Iterated beta integrals

Abstract

We introduce iterated beta integrals, a new class of iterated integrals on the universal abelian covering of the punctured projective line that unifies hyperlogarithms and classical beta integrals while preserving their fundamental properties. We establish various analytic properties of these integrals with respect to both the exponent parameters and the main variables. Their key feature is invariance under simultaneous translation of the exponent parameters, which generates relations between integrals over possibly different coverings. This mechanism recovers notable identities for multiple zeta values and variants -- including Zagier's 2-3-2 formula, Murakami's -value analogue, Charlton's -value analogue, Zhao's - formula, and Ohno's relation -- and also yields new relations, such as a proof of a Galois descent phenomenon for multiple omega values.

Paper Structure

This paper contains 33 sections, 48 theorems, 409 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Zhao's 2-1 formula vs Zagier's 2-3-2 formula
  3. Reformulation of Zhao and Zagier into integral equalities
  4. Interpretation of Zhao and Zagier by the translation invariance of iterated beta integrals
  5. Other applications of the translation invariance
  6. Structure of the paper
  7. A note on complex powers
  8. Definitions and properties of iterated beta integrals
  9. Definition of iterated beta integrals
  10. Domain of convergence and analytic continuation with respect to exponent parameters
  11. Special values and connection to hyperlogarithms
  12. Contiguous-type relations
  13. Differential equations
  14. Translation invariance
  15. Series expansions
  16. ...and 18 more sections

Key Result

Theorem 1

For $\Bbbk\in\mathbb{I}$, we have Particularly, when $\Bbbk=(1,\{2\}^{l_{1}},\ldots,1,\{2\}^{l_{d}})$, this gives the 2-1 formula

Figures (3)

  • Figure 9.1: The paths $\alpha,\beta_{\mathrm{up}},\beta_{\mathrm{down}}$
  • Figure 9.2: The Pochhammer contour $P$
  • Figure 9.3: The paths $C_{0},C_{1},\alpha$

Theorems & Definitions (91)

  • Theorem 1: Zhao's formula Zhao_21
  • Theorem 2: Zagier's 2-3-2 formula Zagier_232
  • Theorem 3: Reformulated version of Zhao's formula
  • Proposition 4
  • proof : Sketch of proof
  • Theorem 5: Theorem \ref{['thm:Case_2a-1']} later
  • Remark 6
  • Theorem 7
  • Theorem 8: Translation invariance (Theorem \ref{['thm: translation invariance']})
  • Proposition 9
  • ...and 81 more