Stieltjes–Calogero–Gil’ relations associated to entire functions of finite order

Abstract

The classical Stieltjes–Calogero relations involving the zeros of the Hermite polynomials found over the years counterpart associated with virtually all the important polynomials appearing in mathematical physics. The standard method of deriving them typically rests on two different ways of looking at the Laurent expansion of the logarithmic derivative of a polynomial about a singular point, that is, a zero of the polynomial. While the first way is a straightforward formalism, the second proves to be cumbersome and is usually handled on a case by case basis, often based on Painlevé transcendent techniques, if the logarithmic derivative of the polynomial in question is solution to a suitable nonlinear differential equation. In this note we settle the problem completely and in full generality via an analysis based on Gil’s remarkable relations involving the zeros of an entire function of finite order. In addition to being elegant and efficient, the new method has wider applicability, addressing not just polynomials but also entire functions of finite order.