A representation formula of sum rules for zeros of polynomials
Let \(P_ N(x) = \prod^ N_{k=1}(x-x_ k)\) be an eigenfunction of the linear differential operator \(\sum^ n_{k=0}g_ k(x)f^{(k)}\), where \(g_ k(x)\) is a polynomial of degree not higher than \(k\), and suppose that all zeros of \(P_ N(x)\) are simple. The author obtains an explicit recurrence relation for the quantities \(y_ s = \sum^ N_{i=1} x^ s_ i\) by representing the so-called sum rules \(J_ s^{(i)}\), defined by \[ J_ s^{(i)} = \sum{x^ s_{\ell_ 1}\over (x_{\ell_ 1} - x_{\ell_ 2})(x_{\ell_ 1} - x_{\ell_ 3})\dots(x_{\ell_ 1} - x_{\ell_ i})}, \] in terms of the \(y_ t\) \((t \leq s-i + 1)\). The representation formula is derived using Lucas polynomials.
scientific article; zbMATH DE number 109282
scientific article