On a linear functional equation for formal power series.

On a linear functional equation for formal power series.

Abstract: Let ρ be a primitive j₀ -th complex root of 1, C[[x]] the ring of formal power series in x over ℂ, and let a(x), b(x) in C[[x]]. We study the two equations

φ(ρx)=a(x)φ(x)+b(x)	(L)

and

φ(ρx)=a(x)φ(x)	(Lh)

for φ in C[[x]], which occurred in connection with an interesting and important special case when dealing with the problem of a covariant embedding of (L) with respect to an iteration group. (See H. Fripertinger and L. Reich. On covariant embeddings of a linear functional equation with respect to an analytic iteration group.) We describe necessary and sufficient conditions for finding nontrivial solutions of (L_h) and for finding solutions of (L) in the form of "cyclic" functional equations for a and b. Then we describe the set of all solutions of these functional equations and present different representations of their general solutions.

GDPR

On a linear functional equation for formal power series.