We study the Diophantine equation of type $U_n(x)=V_m(y)$, where $(U_n)_{ n\geq 0}$ and $(V_m)_{ m\geq 0}$ are polynomial power sums defined over a number field $K$. By applying the finiteness criterion of Bilu and Tichy, we show under appropriate assumptions that equation $U_n(x)=V_m(y)$ has infinitely many solutions with bounded $\mathcal{ O}_S$-denominator. We also study decomposable polynomials in third and second order linear recurrence sequences. In particular, we show that if $W_n(x)=g(h(x))$ for a simple third order linear recurrence sequence $(W_n(x))_{ n\geq 0}$ of complex polynomials, then deg $g$ is bounded. Furthermore, we show that if $(u_{ n_1}+u_{ n_2})(x)=g(h(x))$ for a binary recurrence sequence $(u_n(x))_{ n\geq 0}$ then deg $g$ is bounded.
Categories: math.NT