  We study the hypergeometric functions associated to five one-parameter
deformations of Delsarte K3 quartic hypersurfaces in projective space. We
compute all of their Picard--Fuchs differential equations; we count points
using Gauss sums and rewrite this in terms of finite field hypergeometric sums;
then we match up each differential equation to a factor of the zeta function,
and we write this in terms of global L-functions. This computation gives a
complete, explicit description of the motives for these pencils in terms of
hypergeometric motives.
