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International Journal of Mathematics and Mathematical Sciences
Volume 2004 (2004), Issue 59, Pages 3151-3170
http://dx.doi.org/10.1155/S016117120431210X

Hypersymmetric functions and Pochhammers of 2×2 nonautonomous matrices

Département de Physique, Université du Québec à Trois-Rivières, Québec, Trois-Rivières, Canada G9A 5H7

Received 9 December 2003

Copyright © 2004 Hindawi Publishing Corporation. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We introduce the hypersymmetric functions of 2×2 nonautonomous matrices and show that they are related, by simple expressions, to the Pochhammers (factorial polynomials) of these matrices. The hypersymmetric functions are generalizations of the associated elementary symmetric functions, and for a specific class of 2×2 matrices, having a high degree of symmetry, they reduce to these latter functions. This class of matrices includes rotations, Lorentz boosts, and discrete time generators for the harmonic oscillators. The hypersymmetric functions are defined over four sets of independent indeterminates using a triplet of interrelated binary partitions. We work out the algebra of this triplet of partitions and then make use of the results in order to simplify the expressions for the hypersymmetric functions for a special class of matrices. In addition to their obvious applications in matrix theory, in coupled difference equations, and in the theory of symmetric functions, the results obtained here also have useful applications in problems involving successive rotations, successive Lorentz transformations, discrete harmonic oscillators, and linear two-state systems.