On a functional equation with polynomials (Q1124061)

The authors attack and partially solve the following problem: Find all nontrivial Q in \({\mathbb{C}}[x]\) and all S in \({\mathbb{C}}[x]\) with deg S\(=2\) for which the equation \((1.1)\quad Q(S(x))=cQ(x)Q(x+\gamma)\) holds for a suitable c in \({\mathbb{C}}\). Here \(\gamma\neq 0\) is in \({\mathbb{C}}\). Without loss of generality, (1.1) may be replaced by \((*)\quad Q(Ax^ 2+E)=cQ(x)Q(x+1),\) \(A\neq 0\). If \(E=0\), then (*) has a nontrivial solution only if \(A=\pm 1\), and then a list of all monic solutions is given. Now suppose that \(E\neq 0\), let \(Q(x)=\prod^{N}_{j=1}(x-\beta_ j),\) and let \({\mathfrak A}=\{\beta_ 1,...,\beta_ N\}\). Then Q will be a solution of (*) if and only if \({\mathfrak A}=\{1-\beta_ 1,...,1-\beta_ N)\) and \({\mathfrak A}=\{S(\beta_ 1),...,S(\beta_ N)\}\). Primitive solutions of (*) can be found under each of four spectral hypotheses on S (e.g., if \(S(1/2)=1/2,\) then \(Q(x)=x-1/2\) is a solution of (*)). In order to proceed further, the authors assume that both S and Q are in \({\mathbb{R}}[x]\). If there exists a nontrivial monic polynomial Q satisfying (*) and if Q has at least one real root, then one of the special hypotheses on S is satisfied, and all possible solutions Q can be determined. These results are applied to the problem of finding all nonconstant polynomials t in \({\mathbb{R}}[x]\) and nonzero rational functions R over \({\mathbb{R}}\) for which the equation \(R(t(x))=R(x)-R(x+1)\) is satisfied. It is necessary that deg t\(=2\), \(t(x)=x^ 2+Bx+C,\) with \(1/4=C-B^ 2/4+B/2,\) and then \(R(x)=c(x+B/2-1/2)^{-1}\) for some c in \({\mathbb{R}}\).