Eventually rational and \(m\)-sparse points of linear ordinary differential operators with polynomial coefficients (Q1591135)

Linear homogeneous ordinary differential equations with polynomial coefficients are considered to obtain some properties for the coefficients of power series solutions. The basic aim of the author here is to find all points \({\mathfrak a}\) (ordinary or singular) and all formal power series solutions \(\Sigma c_n(x- {\mathfrak a})^n\) to the given equation such that the elements of sequences \(c=(c_0,c_1, c_2,\dots)\) -- considered as a function of \(n\) -- have following properties: (1) \(c_n=R(n)\) for all large enough \(n\), where \(R(n)\) is a rational function of \(n\); (2) there exists an integer \(N\) such that \((c_n\neq 0)\Rightarrow (n\equiv N\pmod m)\) for all large enough \(n\). Some different examples are given to demonstrate the methods.