The twisted Mellin transform

The "twisted Mellin transform" is a slightly modified version of the usual classical Mellin transform on $L^2(\mathbb R)$. In this short note we investigate some of its basic properties. From the point of views of combinatorics one of its most important interesting properties is that it intertwines the differential operator, $df/dx$, with its finite difference analogue, $\nabla f= f(x)-f(x-1)$. From the point of view of analysis one of its most important properties is that it describes the asymptotics of one dimensional quantum states in Bargmann quantization.