Interpolated Apéry numbers, quasiperiods of modular forms, and motivic gamma functions

Abstract

We extend the definitions of the sequences used by Apéry in his proof of the irrationality of  ζ ( 3 ) \zeta (3) to non-integral values of the index and relate the value with index  − 1 / 2 -1/2 to the central value of the L L -series of the unique normalized cusp form of weight 4 on  Γ 0 ( 8 ) \Gamma _0(8) . We also discuss the notion of quasiperiods of modular forms and relate the Apéry numbers of other half-integral indices to these. We further explain the conjectural relationship of the Taylor expansion around 0 of a different interpolation of the Apéry numbers to a generalized version of the Gamma Conjecture, and discuss interpretations of the various results with families of Calabi-Yau manifolds, mirror symmetry, and motivic gamma functions.