Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Families of self-inverse functions and dilogarithm identities

Lauri Alha

TL;DR

This paper develops a unified framework of gemini functions, a family of self-inverse, symmetric functions whose integrals naturally decompose into two dilogarithm terms. By introducing a generalized gemini form with a shape factor $a$ and a scale factor $b$, the author provides a robust toolkit—centered on a five-term gemini identity and fixed-point reductions—to generate and prove two-term dilogarithm identities, ladders, and related evaluations. The work yields numerous closed-form Li$_2$ identities in the real and complex domains, including connections to golden and plastic constants, Pisot numbers, Legendre’s chi-function, and various ladders; it also offers geometric interpretations via area decompositions and median concepts, linking dilogarithm values to integration limits. Collectively, the results extend the landscape of dilogarithm identities, offer systematic methods for discovering new relations, and illuminate deep links between number-theoretic constants and polylogarithms with potential implications for higher polylogarithms and related fields. The framework’s versatility suggests broad applicability to conformal field theory, hyperbolic geometry, and number theory, enabling streamlined derivations of both known and novel Li$_2$ relations.

Abstract

We introduce a self-inverse function via an integral equivalent to a two-term combination of dilogarithms. We refer to this function as a fundamental form, since there is a family of extensions of this function that satisfy similar self-inverse and symmetric properties. We also construct a family of functions generalizing the fundamental form via two auxiliary parameters, which we refer to as shape and scale factors. Through new integration techniques, we introduce and prove a variety of dilogarithm identities and evaluations for dilogarithm ladders and for two-term dilogarithm combinations. The functions$ \gemini_{a}^{b}(x)$ we introduce are referred to as gemini functions and may be seen as providing a broad framework in the derivation of and application of dilogarithm identities.

Families of self-inverse functions and dilogarithm identities

TL;DR

This paper develops a unified framework of gemini functions, a family of self-inverse, symmetric functions whose integrals naturally decompose into two dilogarithm terms. By introducing a generalized gemini form with a shape factor and a scale factor , the author provides a robust toolkit—centered on a five-term gemini identity and fixed-point reductions—to generate and prove two-term dilogarithm identities, ladders, and related evaluations. The work yields numerous closed-form Li identities in the real and complex domains, including connections to golden and plastic constants, Pisot numbers, Legendre’s chi-function, and various ladders; it also offers geometric interpretations via area decompositions and median concepts, linking dilogarithm values to integration limits. Collectively, the results extend the landscape of dilogarithm identities, offer systematic methods for discovering new relations, and illuminate deep links between number-theoretic constants and polylogarithms with potential implications for higher polylogarithms and related fields. The framework’s versatility suggests broad applicability to conformal field theory, hyperbolic geometry, and number theory, enabling streamlined derivations of both known and novel Li relations.

Abstract

We introduce a self-inverse function via an integral equivalent to a two-term combination of dilogarithms. We refer to this function as a fundamental form, since there is a family of extensions of this function that satisfy similar self-inverse and symmetric properties. We also construct a family of functions generalizing the fundamental form via two auxiliary parameters, which we refer to as shape and scale factors. Through new integration techniques, we introduce and prove a variety of dilogarithm identities and evaluations for dilogarithm ladders and for two-term dilogarithm combinations. The functions we introduce are referred to as gemini functions and may be seen as providing a broad framework in the derivation of and application of dilogarithm identities.

Paper Structure

This paper contains 53 sections, 28 theorems, 326 equations, 14 figures.

Table of Contents

  1. Introduction
  2. Motivating results
  3. Generalized gemini functions
  4. Derivation of a five-term gemini identity
  5. On the derivation of a three-term gemini identity
  6. A degenerate form of a gemini function
  7. The inversion identity, Landen's identity, and the duplication formula
  8. Further three-term relations
  9. A framework for generating two-term dilogarithm identities
  10. Application examples of gemini identities in the real domain
  11. On the evaluation of $\operatorname{Li_{2}}(\frac{1}{\phi^{2}})$
  12. On Legendre's chi-function evaluated at $\frac{1}{\phi^{3}}$
  13. On the Legendre chi-function evaluated at $\sqrt{2} - 1$
  14. On a relation between $\operatorname{Li_{2}}(\sqrt{2}-1)$ and $\operatorname{Li_{2}}(\frac{2-\sqrt{2}}{4})$
  15. A new two-term dilogarithm identity involving $\phi$
  16. ...and 38 more sections

Key Result

Theorem 1

The relation holds for complex $a$ and $x \not\in \{ 0, 1 \}$ such that $x + a \neq 0$ and such that the arguments of the dilogarithmic expressions given above are such that the associated power series converge absolutely.

Figures (14)

  • Figure 1: A self-inverse function derived using \ref{['splitimply']}.
  • Figure 2: These given graphs illustrate gemini functions for various values of $b$ and with $a = 1$.
  • Figure 3: This plot illustrates the curve of a $\gemini_{-\frac{1}{2}}(x)$-function and required area components needed to derive dilogarithm identities introduced in this paper.
  • Figure 4: The graph of a degenerate gemini function and the schematics of area sections needed to derive the reflection formula are illustrated in this figure. The fixed point of a degenerate form is given by $x_{0}=\ln(1+\sqrt{1+0})=\ln(2)$. The area of a middle square is such that $A_{0}=\ln^{2}(2)$. The corresponding two apex areas are equal, which are given by $A_{a}=\frac{1}{2}[A_{tot}-A_{0}]=\frac{1}{2}[\frac{\pi^{2}}{6}-\ln^{2}(2)]=\frac{\pi^{2}}{12}-\frac{1}{2}\ln^{2}(2)= \operatorname{Li}_{2}\left(\frac{1}{2}\right)$.
  • Figure 5: The graphs on the left side illustrates the arguments of all three terms of the first identity as a function of the shape factor $a$. The graphs on the right side illustrates the arguments of all three terms of the second identity as a function of the shape factor $a$.
  • ...and 9 more figures

Theorems & Definitions (47)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Theorem 6
  • Theorem 7
  • Theorem 8
  • Theorem 9
  • Remark 1
  • ...and 37 more