A Feynman integral and its recurrences and associators

Abstract

We determine closed and compact expressions for the ϵ-expansion of certain Gaussian hypergeometric functions expanded around half-integer values by explicitly solving for their recurrence relations. This ϵ-expansion is identified with the normalized solution of the underlying Fuchs system of four regular singular points. We compute its regularized zeta series (giving rise to two independent associators) whose ratio gives the ϵ-expansion at a specific value. Furthermore, we use the well known one-loop massive bubble integral as an example to demonstrate how to obtain all-order ϵ-expansions for Feynman integrals and how to construct representations for Feynman integrals in terms of generalized hypergeometric functions. We use the method of differential equations in combination with the recently established general solution for recurrence relations with non-commutative coefficients.