Formal functional equations and generalized Lie-Gröbner series

Formal functional equations and generalized Lie-Gröbner series

Journal of Difference Equations and Applications 24 Issue 5, 1-28 (2017). DOI 10.1080/10236198.2017.1328507

Abstract. Studying the translation equation F(s+t,x)=F(s,F(t,x)), s,t∈ ℂ, for F_t(x)=F(t,x)=∑_{ν≥ 1}c_ν(t)x^ν, t∈ ℂ, or the associated system of cocycle equations in rings of formal power series it is well known that the coefficient functions of their solutions 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 equations we obtain different types of formal differential equations. They can be solved in order to get explicit representations of the coefficient functions. In the present paper we consider iteration groups of type II, i.e. solutions of the translation equation of the form F(t,x)=x+∑_{n≥ k}c_n(t)xⁿ, t∈ ℂ, where k≥ 2 and c_k: ℂ→ ℂ is an additive function different from 0. They correspond to formal iteration groups G(y,x)∈ (ℂ[y])[[x]] of type II, which turn out to be the Lie-Gröbner series LG_y(x)=∑_{n≥ 0}(1)/(n!)Dⁿ(x)yⁿ. Here the Lie-Gröbner operator D is defined by D(f(x))=f'(x)H(x) for f∈ C[[x]] where H is the formal generator of G. Using this particular form of the formal iteration group G we are able to find short proofs and elegant representations of the solutions of the cocycle equations. In connection with the second cocycle equation we study the generalized Lie-Gröbner operator D(f)= (∑_j=1^k-1-κ_jx^j)f(x)+f'(x)H(x), f∈ C[[x]], where κ₁,…,κ_k-1∈ ℂ are given. It yields the corresponding generalized Lie-Gröbner series LG_y(x)=∑_{n≥ 0}(1)/(n!)Dⁿ(x)yⁿ which appears in the presentation of the solution of the second cocycle equation.

GDPR

Formal functional equations and generalized Lie-Gröbner series