Finite-Dimensional Calculus

Details Finite-Dimensional Calculus Finite-Dimensional Calculus

2007-09-21

arxiv.org/abs/0709.3387v2

2009 J. Phys. A: Math. Theor. 42 375214

Physics

Mathematical Physics

26 pages. Added material on Krawtchouk polynomials. Additional references included

Scientific paper

10.1088/1751-8113/42/37/375214

We discuss topics related to finite-dimensional calculus in the context of finite-dimensional quantum mechanics. The truncated Heisenberg-Weyl algebra is called a TAA algebra after Tekin, Aydin, and Arik who formulated it in terms of orthofermions. It is shown how to use a matrix approach to implement analytic representations of the Heisenberg-Weyl algebra in univariate and multivariate settings. We provide examples for the univariate case. Krawtchouk polynomials are presented in detail, including a review of Krawtchouk polynomials that illustrates some curious properties of the Heisenberg-Weyl algebra, as well as presenting an approach to computing Krawtchouk expansions. From a mathematical perspective, we are providing indications as to how to implement in finite terms Rota's "finite operator calculus".

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