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arXiv:1303.6587 · Journal of Mathematical Physics

From sequences to polynomials and back, via operator orderings

Tewodros Amdeberhan, Valerio De Angelis, Atul Dixit, Victor H. Moll, Christophe Vignat

Published Dec 1, 2013• 3 citations
arXivHTMLPDF

Abstract

Bender and Dunne [“Polynomials and operator orderings,” J. Math. Phys. 29, 1727–1731 (1988)] showed that linear combinations of words qkpnqn−k, where p and q are subject to the relation qp − pq = ı, may be expressed as a polynomial in the symbol \documentclass[12pt]{minimal}\begin{document}$z = \frac{1}{2}(qp+pq)$\end{document}z=12(qp+pq). Relations between such polynomials and linear combinations of the transformed coefficients are explored. In particular, examples yielding orthogonal polynomials are provided.

DOI
10.1063/1.4836778

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Bender and Dunne [“Polynomials and operator orderings,” J. Math. Phys. 29, 1727–1731 (1988)] showed that linear combinations of words qkpnqn−k, where p and q are subject to the relation qp − pq = ı, may be expressed as a polynomial in the symbol \documentclass[12pt]{minimal}\begin{document}z=12(qp+pq)z = \frac{1}{2}(qp+pq)z=21​(qp+pq)\end{document}z=12(qp+pq). Relations between such polynomials and linear combinations of the transformed coefficients are explored. In particular, examples yielding orthogonal polynomials are provided.
An alternative and combinatorial proof is given for a connection between a system of Hahn polynomials and identities for symmetric elements in the Heisenberg algebra, which was first observed by Bender, Mead, and Pinsky [Phys. Rev. Lett. 56 (1986), J. Math. Phys. 28, 509 (1987)] and proved by Koornwinder [J. Phys. Phys. 30(4), 1989]. In the same vein two results announced by Bender and Dunne [J. Math. Phys. 29 (8), 1988] connecting a special one-parameter class of Hermitian operator orderings and the continuous Hahn polynomials are also proved.
Let  , A ^ , B ^ , C ^ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429092053/5e12ab0e-ba42-4caa-9bdf-6325e9117d3e/content/ineqn10_1.tif"/> Ĉ be arbitrary linear operators. Prove that [ A ^ B ^ , C ^ ] = [ A ^ , C ^ ] B ^ + A ^ [ B ^ , C ^ ] , https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429092053/5e12ab0e-ba42-4caa-9bdf-6325e9117d3e/content/ineqn10_2.tif"/> [ A ^ , [ B ^ , C ^ ] ] + [ B ^ , [ C ^ , A ^ ] ] + [ C ^ , [ A ^ , B ^ ] ] = 0. https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429092053/5e12ab0e-ba42-4caa-9bdf-6325e9117d3e/content/ineqn10_3.tif"/>