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

The aim of this note is to provide a quadratic external field extension of a classical result of Marcel Riesz for the equilibrium measure on a ball with respect to Riesz $s$-kernels, including the logarithmic kernel, in arbitrary dimensions. The equilibrium measure is a radial arcsine distribution. As a corollary, we obtain new integral identities involving special functions such as elliptic integrals and more generally hypergeometric functions. 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, and the Weyl lemma for the regularity of elliptic equations.