On the solution of a Riesz equilibrium problem and integral identities for special functions
Chafai, Djalil; Saff, Edward B.; Womersley, Robert S. (2021), On the solution of a Riesz equilibrium problem and integral identities for special functions. https://basepub.dauphine.psl.eu/handle/123456789/22354
TypeDocument de travail / Working paper
External document linkhttps://hal.archives-ouvertes.fr/hal-03312868
Series titleCahier de recherche du CEREMADE
MetadataShow full item record
CEntre de REcherches en MAthématiques de la DEcision [CEREMADE]
Saff, Edward B.
Center for Constructive Approximation [Vanderbilt]
Womersley, Robert S.
School of Mathematics and Statistics [Sydney] [UNSW]
Abstract (EN)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.
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