On the
formal second cocycle equation for iteration groups of type
II |

Journal of Difference Equations and Applications **21**
Issue 7, (2015). DOI 10.1080/10236198.2015.1040783

**Abstract.** Let x be an indeterminate over ℂ. We
investigate solutions

β

β(s,x)=∑ _{n≥ 0}β_{n}(s)x^{n},^{.}_{.}

in

β(s+t,x)= β(s,x)α(t,F (s,x)) +β(t,F (s,x)), s,t∈ ℂ (Co2) ^{.}_{.}

of the form F(s,x)≡ x+c

F(s+t,x)=F(s,F(t,x)), s,t∈ ℂ, (T) ^{.}_{.}

Using the method of "formal functional equations" applied already for the study of F and α in previous manuscripts we obtain a formal version of the second cocycle equation in the ring (ℂ[S,U,σ])[[x]]. We solve this equation in a completely algebraic way, by deriving formal differential equations. This way it is possible to get the structure of the coefficients in great detail which are now polynomials. We prove the universal character of these polynomials depending on certain parameters, the coefficients of the generator K of a formal first cocycle, and the coefficients of three generators N

α(s+t,x)= α(s,x)α(t,F (s,x)), s,t∈ ℂ. (Co1) ^{.}_{.}

harald.fripertinger "at" uni-graz.at, May 6, 2024

GDPR | On the
formal second cocycle equation for iteration groups of type
II |