The Translation Equation in the Ring of Formal Power Series Over ℂ and Formal Functional Equations

In this survey we describe the construction of one-parameter subgroups (iteration groups) of Γ, the group of all (with respect to substitution) invertible power series in one indeterminate x over ℂ $$\mathbb{C}$$ . In other words, we describe all solutions of the translation equation in ℂ [ [ x ] ] $$\mathbb{C}[\![\,x\,]\!]$$ , the ring of formal power series in x with complex coefficients. For doing this the method of formal functional equations will be applied. The coefficient functions of solutions of the translation equation are polynomials in additive and generalized exponential functions. Replacing these functions by indeterminates we obtain formal functional equations. Applying formal differentiation operators to these formal translation equations we obtain three types of formal differential equations. They can be solved in order to get explicit representations of the coefficient functions. For solving the formal differential equations we apply Briot–Bouquet differential equations in a systematic way.