  We study analytic properties of the pair consisting of a rather general form
of zeta-function with an Euler product and a periodic Hurwitz zeta-function
with a transcendental parameter. We first survey briefly previous results, and
then investigate the mixed joint discrete value distribution and prove the
mixed joint discrete universality theorem for these functions, in the case when
common differences of relevant arithmetic progressions are not necessarily the
same. Also some generalizations are given. For this purpose, certain arithmetic
conditions on the common differences are necessary.
