Diophantine equations for classical continuous orthogonal polynomials (Q1433051)

For rational numbers \(A\), \(B\), \(C\) with \(AB\neq 0\) and for arbitrary rational integers \(m> n\geq 3\), the authors study the Diophantine equation \[ Ap_m(x)+ Bp_n(y)= C\qquad (x,y\in \mathbb{Z}), \] where \(\{p_k(x)\}\) is one of the three classical continuous orthogonal polynomial families, i.e., Laguerre polynomials, Jacobi polynomials (including Gegenbauer, Legendre or Chebyshev polynomials) and Hermite polynomials. In particular, they prove that, with exception of the Chebyshev polynomials for all such polynomial families, there are at most finitely many solutions \((x,y)\in \mathbb{Z}^2\) provided that \(n\geq 4\). Their method is based, in part, on a theorem of Szegő on monotonicity of stationary points of polynomials which satisfy the following second-order Sturm-Liouville differential equation: \[ (ax^2)+ bx+ c) y^{\prime\prime}_n(x)+ (dx+ e)y_n'(x)- \lambda_n y_n(x)= 0\qquad (n\in\mathbb{Z}_{\geq 0}). \]