Conditional recurrent sequences (Q1190351)

For a recurrent sequence \(u_ 0,u_ 1,u_ 2,\dots\) let \(E\) be the shift operator \((Eu_ n=u_{n+1})\) and define this sequence by equations \(f(E)u_{3n}=0\), \(g(E)u_{3n+1}=0\) and \(h(E)u_{3n+2}=0\), where \(f\), \(g\), \(h\) are polynomials in \(E\) with coefficients from the field of rational functions in \(n\) over a ground field. The present paper discusses the problem of defining the sequence by one polynomial \(w(E)\) such that \(w(E)u_ n=0\) (\(n=0,1,2,\dots\)) and to express \(u_{3n}\), \(u_{3n+1}\) and \(u_{3n+2}\) in terms of an auxiliary sequence, which satisfies a recurrence relation of lowest degree.