We prove the Domb Apéry-limit, listed without proof in the literature since 2008, by deriving an Atkin–Lehner transformation law for the Eichler integral of an Eisenstein form on the genus-zero modular curve of level six, then extracting the connection constant from the derivative of the period polynomial at the CM fixed point. As a corollary we establish a continued fraction identity for the Apéry constant conjectured by the Ramanujan Machine project and open since 2019.
Alex Shvets (Fri,) studied this question.
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