Products of linear recurring sequences

Abstract Let L ( f ) denote the space of all sequences of elements of a field F generated by the linear recurrence corresponding to the polynomial f over F . It is well known that if f 1 , f 2 ,…, f m are polynomials over F none of which has multiple roots, then L ( f 1 ) L ( f 2 ) … L ( f m ) = L ( h ), where h is a polynomial whose roots are the distinct products a 1 a 2 … a m , a i a root of f i . Here we find such an h in the general case.