Coupling of the Legendre polynomials with kernels $|x-y|^α$ and $\ln|x-y|$

2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/68748Double integrals that represent matrix elements of the power and logarithmic potentials in the Legendre polynoiomial basis on [-1,1] are found in a closed form. Several proofs are given, which involve different special functions and identities. A connection of the new formulae and Whipple's hypergeometric summation formula is shown.23 pp. LaTeX, Preprint of the Keldysh Institute for Applied MathematicsClassical Analysis and ODEs33C20; 45H05Coupling of the Legendre polynomials with kernels $|x-y|^α$ and $\ln|x-y|$text