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Models of *q*‐algebra representations: Matrix elements of the *q*‐oscillator algebra

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10.1063/1.530308

### Abstract

This article continues a study of function space models of irreducible representations of *q* analogs of Lie enveloping algebras, motivated by recurrence relations satisfied by *q*‐hypergeometric functions. Here a *q* analog of the oscillatoralgebra (not a quantum algebra) is considered. It is shown that various *q* analogs of the exponential function can be used to mimic the exponential mapping from a Lie algebra to its Lie group and the corresponding matrix elements of the ‘‘group operators’’ on these representation spaces are computed. This ‘‘local’’ approach applies to more general families of special functions, e.g., with complex arguments and parameters, than does the quantum group approach. It is shown that the matrix elements themselves transform irreducibly under the action of the algebra.*q* analogs of a formula are found for the product of two hypergeometric functions _{1} *F* _{1} and the product of a _{1} *F* _{1} and a Bessel function. They are interpreted here as expansions of the matrix elements of a ‘‘group operator’’ (via the exponential mapping) in a tensor product basis (for the tensor product of two irreducible oscillatoralgebra representations) in terms of the matrix elements in a reduced basis. As a by‐product of this analysis an interesting new orthonormal basis was found for a *q* analog of the Bargmann–Segal Hilbert space of entire functions.

© 1993 American Institute of Physics

Received 09 September 1992
Accepted 08 July 1993

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/content/aip/journal/jmp/34/11/10.1063/1.530308

1993-11-01

2014-07-29

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