  In their paper on the gamma conjecture in mirror symmetry, Golyshev and
Zagier introduce what we refer to as Frobenius constants associated to an
ordinary linear differential operator L with a reflection type singularity.
These numbers describe the variation around the reflection point of Frobenius
solutions to L defined near other singular points. Golyshev and Zagier show
that in certain geometric cases Frobenius constants are periods, and they raise
the question quite generally how to describe these numbers motivically. In this
paper we give a relation between Frobenius constants and Taylor coefficients of
generalized gamma functions, from which it follows that Frobenius constants of
Picard--Fuchs differential operators are periods. We also study the relation
between these constants and periods of limiting Hodge structures.
  This is a major revision of the previous version of the manuscript. The
notion of Frobenius constants and our main result are extended to the general
case of regular singularities with any sets of local exponents. In addition,
the generating function of Frobenius constants is given explicitly for all
hypergeometric connections.
