A new type of factorial series expansions and applications

Abstract

We construct a new type of convergent asymptotic representations, dyadic factorial expansions. Their convergence is geometric and the region of convergence can include Stokes rays, and often extends down to 0^+. For special functions such as Bessel, Airy, Ei, Erfc, Gamma and others, this region is C without an arbitrarily chosen ray effectively providing uniform convergent asymptotic expansions for special functions. We prove that relatively general functions, Ecalle resurgent ones possess convergent dyadic factorial expansions. We show that dyadic expansions are numerically efficient representations. The expansions translate into representations of the resolvent of self-adjoint operators in series in terms of the associated unitary evolution operator evaluated at some prescribed points (alternatively, in terms of the generated semigroup for positive operators).