From Invariance Under Binomial Thinning to Unification of the Cauchy and the Gołąb–Schinzel-Type Equations

Abstract

We point out to a connection between a problem of invariance of power series families of probability distributions under binomial thinning and functional equations which generalize both the Cauchy and an additive form of the Gołąb–Schinzel equation. We solve these equations in several settings with no or mild regularity assumptions imposed on unknown functions.

Non-periodic solutions of the Goła̧b–Schinzel type functional equation

“…and some its conditional versions have been investigated under various regularity assumptions in [9][10][11] and [14,15]. It is remarkable that solutions of (6) and their applications to invariance under binomial thinning have been recently considered in [3].…”