  Infinite order differential equations have come to play an increasingly
significant role in theoretical physics. Field theories with infinitely many
derivatives are ubiquitous in string field theory and have attracted interest
recently also from cosmologists. Crucial to any application is a firm
understanding of the mathematical structure of infinite order partial
differential equations. In our previous work we developed a formalism to study
the initial value problem for linear infinite order equations with constant
coefficients. Our approach relied on the use of a contour integral
representation for the functions under consideration. In many applications,
including the study of cosmological perturbations in nonlocal inflation, one
must solve linearized partial differential equations about some time-dependent
background. This typically leads to variable coefficient equations, in which
case the contour integral methods employed previously become inappropriate. In
this paper we develop the theory of a particular class of linear infinite order
partial differential equations with variable coefficients. Our formalism is
particularly well suited to the types of equations that arise in nonlocal
cosmological perturbation theory. As an example to illustrate our formalism we
compute the leading corrections to the scalar field perturbations in p-adic
inflation and show explicitly that these are small on large scales.
