On polynomial values of the sum and the product of the terms of linear recurrences (Q2714420)

Let \(x\geq 2\), \(x_1,x_2,\dots,x_m\) be positive integer variables. The author discusses the solubility of Diophantine equations of the form NEWLINE\[NEWLINE \sum^m_{i=1} G^{(i)}_{x_i}=F(x)\;\text{and} \prod^m_{i=1} G^{(i)}_{x_i}=F(x), NEWLINE\]NEWLINE where \(G^{(i)}=\{G^{(i)}_{x_i}\}^\infty_x 0\), \(i=1,2,\dots,m\), is a system a linear recurrences of order \(k_i\) defined by NEWLINE\[NEWLINE G^{(i)}_x=A^{(i)}_1 G^{(i)}_{x-1}+A^{(i)}_2G^{(i)}_{x-2}+\dots + A^{(i)}_{k_i}G^{(i)}_{x-k_i},\;2\leq k_i\leq x, NEWLINE\]NEWLINE where \(F(x)=dx^q+d_px^p+d_{p-1}x^{p-1}+\dots+d_0\in \mathbb Z [x]\) has degree \(q\geq 2\) with \(q>p\) and where for \(j=0,1,\dots,k_i-1\), the initial values \(G^{(i)}_j\) and the coefficients \(A^{(i)}_{j+1}\) are rational integers. Earlier results corresponding to special cases are discussed briefly and it is shown that, under certain conditions on the recurrences \(G^{(i)}\), on the variables \(x\), \(x_i\) and on \(q\), \(p\), the equations have no solutions for sufficiently large, effectively computable degree \(q\).