The formal translation equation for iteration groups of type II (Q623383)

The article deals with the functional translation equation \[ F(s + t,x) = F(s,F(t,x)), \qquad s,t \in {\mathbb C},\tag{1} \] in the ring \({\mathbb C}[[x]]\) of formal power series over \({\mathbb C}\). More precisely, the authors study solutions to (1) of form \[ F(s,x) = x + \sum_{n \geq k} c_n(s)x^n,\tag{2} \] where \(k \geq 2\), \(c_k\) is a nonzero additive function, and other coefficient functions \(c_n(s)\), \(n > k\), are polynomials in \(c_k(s)\). The problem is reduced to the finding of a function \(G(y,z) \in {\mathbb C}([y,z])[[x]]\), \(G(0,x) = x\), satisfying the equation \[ G(y + z) = G(y,G(z,x)) \] and the Aczél-Jabotinsky type differential equation \[ H(x)\frac{\partial}{\partial y}\;G(y,x) = H(G(y,x)), \qquad H(x) = \frac{\partial}{\partial y}\;G(y,x)\bigg|_{y=0}. \] Rewriting \(G(y,x)\) in the form \(\sum_{n\geq 0} \varphi_n(x)y^n\) the authors obtain explicit formulas for \(\varphi_n\) in terms of derivatives \(H^{(j)}(x)\) of the infinitesimal generator \(H(x)\); further, they also obtain the representation of \(G(y,x)\) as a Lie-Gröbner series. Using the canonical form of \(H\) as \(x^k + hx^{2k-1}\) the authors find the expression \(G(y,x) = \sum_{r \geq 0} G_r(y,x)h^r\) yet.