An analogue of Siegel's determinant

TL;DR

This paper presents a new proof for the non-vanishing of certain determinants related to linear differential equations, extending Siegel's ideas and applying to hypergeometric equations.

Contribution

It introduces an analogue of Siegel's determinant and provides a simplified proof of its non-vanishing for specific differential equations, including some hypergeometric cases.

Findings

Established properties of determinants attached to linear forms of derivatives.

Provided a short proof of non-vanishing for a class of differential equations.

Extended Siegel's determinant theory to new contexts.

Abstract

Siegel-Shidlovskii theory of $E$-functions involves a non-vanishing proof for the determinants attached to the linear forms $D^kR(t)$, derivatives of an auxiliary function $R(t)$. Let a non-zero function $F(t)$ satisfy $m$th order linear differential equation which we shall write using the differential operator $\Delta=tD$ and let $L(t)$ be any non-zero linear form of the derivatives $\Delta^i F(t)$ $(i=0,...,m-1; m\ge 2)$. The determinants $\det\mathcal A_k$ attached to the linear forms $\Delta^kL(t)$ have certain simple properties that allow us to give a short proof for the non-vanishing of $\det\mathcal A_k$ for a class of differential equations including a subclass of hypergeometric differential equations.