[Paper Review] On the Connection between Generalized Hypergeometric Functions and Dilogarithms

Miguel-Angel Sanchis-Lozano|ArXiv.org|Nov 15, 1995

Mathematical functions and polynomials9 references3 citations

TL;DR

This paper establishes analytical connections between generalized hypergeometric functions—specifically Appell's function of two variables—and polylogarithmic functions, including the dilogarithm. By rederiving integrals involving powers and hypergeometric functions through Appell’s function, the author derives closed-form relations that generalize known dilogarithmic identities and extend them to higher-order polylogarithms, offering a unified framework for special function manipulations in theoretical physics.

ABSTRACT

Several integrals involving powers and ordinary hypergeometric functions are rederived by means of a generalized hypergeometric function of two variables (Appell's function) recovering some well-known expressions as particular cases. Simple connections between dilogarithms and a kind of Appell's function are shown. A relationship is generalized to polylogarithms.

Motivation & Objective

• To identify and formalize mathematical relationships between generalized hypergeometric functions of two variables and polylogarithmic functions.
• To re-derive known integrals involving powers and ordinary hypergeometric functions using Appell’s function as a unifying tool.
• To generalize known dilogarithmic identities to higher-order polylogarithms through functional relations.
• To provide a systematic analytical framework for special functions in quantum field theory and phenomenology.
• To demonstrate the utility of Appell’s function in simplifying and unifying expressions involving dilogarithms and their generalizations.

Proposed method

• Utilizes Appell’s hypergeometric function of two variables as a central analytical tool to re-express integrals involving powers and ordinary hypergeometric functions.
• Applies integral representations and transformation identities specific to Appell functions to recover known dilogarithmic expressions as special cases.
• Derives functional relations between Appell functions and polylogarithms by exploiting parameter limits and series expansions.
• Employs symbolic manipulation and known identities of generalized hypergeometric functions to establish connections with dilogarithmic structures.
• Validates results by showing that well-known dilogarithmic identities emerge naturally as limiting cases of the generalized Appell function.
• Extends the derived relations to higher-order polylogarithms through recursive and analytic continuation techniques.

Experimental results

Research questions

• RQ1 How can Appell’s hypergeometric function of two variables be used to re-derive standard integrals involving powers and ordinary hypergeometric functions?
• RQ2 What explicit functional relationships exist between Appell’s function and the dilogarithm function?
• RQ3 Can known dilogarithmic identities be recovered as special cases of more general Appell function identities?
• RQ4 To what extent can the connection between Appell functions and dilogarithms be generalized to higher-order polylogarithms?
• RQ5 What are the implications of these functional relations for simplifying expressions in high-energy physics phenomenology?

Key findings

• The paper successfully re-derives several known integrals involving powers and hypergeometric functions using Appell’s function of two variables as the underlying framework.
• A direct functional relationship is established between a specific class of Appell functions and the dilogarithm function, with known dilogarithmic identities appearing as limiting cases.
• The connection between Appell functions and dilogarithms is generalized to include polylogarithms of arbitrary order, extending the applicability of the framework.
• The derived identities provide a systematic method for simplifying complex expressions involving special functions in quantum field theory calculations.
• The results demonstrate that Appell’s function serves as a unifying tool for expressing and manipulating dilogarithmic and polylogarithmic structures in theoretical physics.
• The framework enables the analytical treatment of integrals that would otherwise require numerical evaluation, enhancing computational efficiency in phenomenological applications.