Calculation of local formal Mellin transforms

TL;DR

This paper introduces local Mellin transforms for connections on the punctured formal disk, establishing their properties and providing explicit formulas, analogous to the well-studied local Fourier transforms.

Contribution

It extends the theory of local Fourier transforms to local Mellin transforms, showing their categorical equivalences and deriving explicit calculation formulas.

Findings

Local Mellin transforms induce equivalences between categories.

Explicit formulas for local Mellin transforms are provided.

Analogous results to local Fourier transforms are established.

Abstract

Much recent work has been done on the local Fourier transforms for connections on the punctured formal disk. Specifically, the local Fourier transforms have been introduced, shown to induce certain equivalences of categories, and explicit formulas have been found to calculate them. In this paper, we prove analogous results in a similar situation, the local Mellin transforms for connections on the punctured formal disk. Specifically, we introduce the local Mellin transforms and show that they induce equivalences between certain categories of vector spaces with connection and vector spaces with invertible difference operators, as well as find formulas for explicit calculation in the same spirit as the calculations for the local Fourier transforms.