Hodge structure and motivic gamma function associated to the Apéry family

(Joint work with M. Vlasenko) Inspired by recent work of V. Golyshev and D. Zagier, we associate to the  Apéry family (a rank 3 variation of Hodge structure on an open set in the projective line) a "motivic Gamma function" which is a Mellin transform of a suitable Picard Fuchs solution. The Taylor series at $s=0$ of this  Mellin transform has coefficients which numerical calculations suggest are (multiple) zeta values. We show how this data can be interpreted as a variation of mixed Hodge structure.