Second Order Transfer Equations; and Generalizations to Arbitrary Orders

The author provides a solution to the equation $ y(s+2) = $\mathcal{T}$^2 y = F(s,y,$\mathcal{T}$ y) = F(s,y(s),y(s+1))$;where$y$isholomorphic;and$F$ is a holomorphic function with specific decay conditions. This result is provided using infinite compositions, and a limiting process. The technique is generalized to arbitrary $k$’th order transfer equations: $u(s+k) = F(s,u(s),u(s+1),…,u(s+k-1))$\mathrm{.}Thetechniqueisderivedbyutilizingsolutions$w(s+1) = $\mathcal{T}$ w = F(s,w)$andsequentiallyapproximating$u$,or$y$, with a sequence of said $w$.