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.
Keywords
local integral transforms, Mellin transform, difference
operator, differential operator
