On the solution of a Riesz equilibrium problem and integral identities for special functions

TL;DR

This paper extends classical Riesz equilibrium results to full space external fields for specific kernels, revealing new integral identities involving special functions like elliptic and hypergeometric functions.

Contribution

It provides a full space quadratic external field extension of Riesz equilibrium measures and uncovers novel integral identities involving special functions.

Findings

Equilibrium measure is a radial arcsine distribution on a specific radius.

New integral identities involving elliptic and hypergeometric functions are derived.

The identities are not present in existing mathematical tables or software.

Abstract

The aim of this note is to provide a full space quadratic external field extension of a classical result of Marcel Riesz for the equilibrium measure on a ball with respect to Riesz s-kernels. We address the case s=d-3 for arbitrary dimension d, in particular the logarithmic kernel in dimension 3. The equilibrium measure for this full space external field problem turns out to be a radial arcsine distribution supported on a ball with a special radius. As a corollary, we obtain new integral identities involving special functions such as elliptic integrals and more generally hypergeometric functions. It seems that these identities are not found in the existing tables for series and integrals, and are not recognized by advanced mathematical software. Among other ingredients, our proofs involve the Euler-Lagrange variational characterization, the Funk-Hecke formula, the Weyl regularity lemma, the maximum principle, and special properties of hypergeometric functions.