Diophantine equations for polynomial recursive sequences

We study the Diophantine equation of type [Formula: see text], where [Formula: see text] and [Formula: see text] are polynomial power sums defined over a number field [Formula: see text]. By applying the finiteness criterion of Bilu and Tichy, we show under appropriate assumptions that equation [Formula: see text] has infinitely many solutions with bounded [Formula: see text]-denominator. We also study decomposable polynomials in third and second order linear recurrence sequences. In particular, we show that if [Formula: see text] for a simple third order linear recurrence sequence [Formula: see text] of complex polynomials, then [Formula: see text] is bounded. Furthermore, we show that if [Formula: see text] for a binary recurrence sequence [Formula: see text] then [Formula: see text] is bounded.