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

S. Sadov
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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.

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S. Sadov. 2003-10-06. Coupling of the Legendre polynomials with kernels $|x-y|^α$ and $\ln|x-y|$. https://arxiv.org/abs/math/0310063

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