On a convolution of linear recurring sequences over finite fields. II (Q1322066)

[For part I, cf. J. Algebra 149, 179-182 (1992; Zbl 0761.11042).] Let \(F\) be a finite field and \(S_ F\) the \(F\)-algebra of all semi- infinite sequences over \(F\). For a nonconstant polynomial \(f\) over \(F\) let \(S_ F (f(x))\) denote the set of all homogeneous linear recurring sequences in \(F\) with characteristic polynomial \(f\). For \({\mathbf s}, {\mathbf t}\in S_ F\) denote the convolution \(u_ n= \sum_{i=0}^ n s_ i t_{n-i}\) by \({\mathbf u}= {\mathbf s}*{\mathbf t}\). Denote by \(S_ F (f(x))* S_ F(g(x))\) the subspace of \(S_ F\) spanned by all such convolutions with \({\mathbf s}\in S_ F (f(x))\) and \({\mathbf t}\in S_ F(g(x))\). It is shown here that \(S_ F (xf(x))* S_ F(g(x))= S_ F(f(x) g(x))\) and also, by counterexample, that \(S_ F (f(x))* S_ F (g(x))\neq S_ F (f(x) g(x))\).