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

S. Sadov

paper

Coupling of the Legendre polynomials with kernels $∣ x - y ∣^{α}$ and $ln ∣ x - y ∣$

arXiv:math/0310063

Abstract

Double 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 Mathematics

Coupling of the Legendre polynomials with kernels ∣x−y∣α and ln∣x−y∣

Abstract

Coupling of the Legendre polynomials with kernels $∣ x - y ∣^{α}$ and $ln ∣ x - y ∣$