On A Matrix Hypergeometric Differential Equation

In this paper we consider a matrix Hypergeometric differential equation, which are special matrix functions and solution of a specific second order linear differential equation. The aim of this work is to extend a well known theorem on Hypergeometric function in the complex plane to a matrix version, and we show that the asymptotic expansions of Hypergeometric function in the complex plane ” that are given in the literature are special members of our main result. Background and motivation are discussed.

Logarithmic growth filtrations for -modules over the bounded Robba ring

In the 1970s, Dwork defined the logarithmic growth (log-growth for short) filtrations for $p$ -adic differential equations $Dx=0$ on the $p$ -adic open unit disc $|t|<1$ , which measure the asymptotic behavior of solutions $x$ as $|t|\to 1^{-}$ . Then, Dwork calculated the log-growth filtration for $p$ -adic Gaussian hypergeometric differential equation. In the late 2000s, Chiarellotto and Tsuzuki proposed a fundamental conjecture on the log-growth filtrations for $(\varphi ,\nabla )$ -modules over $K[\![t]\!]_0$ , which can be regarded as a generalization of Dwork's calculation. In this paper, we prove a generalization of the conjecture to $(\varphi ,\nabla )$ -modules over the bounded Robba ring. As an application, we prove a generalization of Dwork's conjecture proposed by Chiarellotto and Tsuzuki on the specialization property for log-growth Newton polygons.