Abstract
The definite integrals \( \int _{-1}^1x[P_\nu (x)]^4\mathrm{d}x\) and \( \int _{0}^1x[P_\nu (x)]^2\{[P_\nu (x)]^2-[P_\nu (-x)]^2\}\mathrm{d}x\) are evaluated in closed form, where \( P_\nu \) stands for the Legendre function of degree \( \nu \in \mathbb C\). Special cases of these integral formulae have appeared in arithmetic studies of automorphic Green’s functions and Epstein zeta functions.
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. Translated from the Russian by N. M. Queen
. Translated from the Russian by G. G. GouldAcknowledgements
The author is grateful to an anonymous referee for thoughtful comments on polishing the presentation of this paper.
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Zhou, Y. Two definite integrals involving products of four Legendre functions. Ramanujan J 45, 299–317 (2018). https://doi.org/10.1007/s11139-017-9916-3
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DOI: https://doi.org/10.1007/s11139-017-9916-3