Motivic Gamma functions

I will describe some aspects of work by V. Golyshev and collaborators on motivic gamma functions. When a variety $X$ is “spread out”, i.e. one is given $f: X -> P^1$, then periods on $X$ can be interpreted as periods of the Gauss-Manin connection on$ P^1$. The Mellin transforms of these periods, which Golyshev calls motivic Gamma functions, satisfy the same recursion as the Picard Fuchs equations. In special cases, their derivatives at $s=0$ yield Mahler measure. Inhomogeneous solutions of the recursion are linked to the Apery program and to limits of Beilinson regulators.