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arXiv · 2209.12719

An analogue of Siegel's determinant

Tapani Matala-aho
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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.

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Tapani Matala-aho. 2022-09-26. An analogue of Siegel's determinant. https://arxiv.org/abs/2209.12719

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