An Explicit Solution of the Central Connection Problem for an nth Order Linear Ordinary Differential Equation with Polynomial Coefficients

Abstract

We consider a differential equation of the form $( - 1)^n y^{(n)} (x) = (x^m + Q(x))y(x)$, where $Q(x)$ is a polynomial of degree $m - [{m / n}] - 2$. For a particular recessive solution $y(x)$ of this equation, uniquely determined by an asymptotic expansion about $x = \infty $, we derive explicit representations of its n initial values $Y^{(k)} (0)$, $k = 0, \cdots ,n - 1$. These representations have the form of multiple sums, whose coefficients themselves are hypergeometric sums of several variables. To obtain our results, we apply the Mellin transformation to our differential equation and are led to a system of difference equations, which can be solved explicitly, and whose solution provides the representation of the initial values.