Finite-dimensional Calogero representation of the $q$-differential operator

Abstract

A finite-dimensional matrix representation of the Jackson $q$-differential operator $D_q$, defined by $D_{q}f(x)$ $=$ $(f(qx)-f(x))/(x(q-1))$, is written down following Calogero. Such a representation of $D_q$ should have applications in $q$-analysis leading to the corresponding extensions of the numerous results of Calogero's work.

arXiv comment5 pages, LaTeX

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• Finite-dimensional representations of difference operators, and the identification of remarkable matrices