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

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