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arXiv · 1603.03547

Two Definite Integrals Involving Products of Four Legendre Functions

Yajun Zhou
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Abstract

The definite integrals $ \int_{-1}^1x[P_ν(x)]^4\,\mathrm{d} x$ and $ \int_{0}^1x[P_ν(x)]^2\{[P_ν(x)]^2-[P_ν(-x)]^2\}\,\mathrm{d} x$ are evaluated in closed form, where $ P_ν$ stands for the Legendre function of degree $ ν\in\mathbb C$. Special cases of these integral formulae have appeared in arithmetic studies of automorphic Green's functions and Epstein zeta functions.

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Yajun Zhou. 2017-04-29. Two Definite Integrals Involving Products of Four Legendre Functions. https://doi.org/10.1007/s11139-017-9916-3

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