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.