In his classic book on group representations and special functions Vilenkin studied the matrix elements of irreducible representations of the Euclidean and oscillator Lie algebras with respect to countable bases of eigenfunctions of the Cartan subalgebras, and he computed the summation identities for Bessel functions and Laguerre polynomials associated with the addition theorems for these matrix elements. He also studied matrix elements of the pseudo‐Euclidean and pseudo‐oscillator algebras with respect to the continuum bases of generalized eigenfunctions of the Cartan subalgebras of these Lie algebras and this resulted in realizations of the addition theorems for the matrix elements as integral transform identities for Bessel functions and for confluent hypergeometric functions. Here we work out q analogs of these results in which the usual exponential function mapping from the Lie algebra to the Lie group is replaced by the q‐exponential mappings Eq and eq. This study of representations of the Euclidean quantum algebra and the q‐oscillator algebra (not a quantum algebra) leads to summation, integral transform, and q‐integral transform identities for q analogs of the Bessel and confluent hypergeometric functions, extending the results of Vilenkin for the q=1 case.
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April 1994
Research Article|
April 01 1994
Models of q‐algebra representations: q‐integral transforms and ‘‘addition theorems’’
E. G. Kalnins;
E. G. Kalnins
Department of Mathematics and Statistics, University of Waikato, Hamilton, New Zealand
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Willard Miller, Jr.
Willard Miller, Jr.
School of Mathematics and Institute for Mathematics and its Applications, University of Minnesota, Minneapolis, Minnesota 55455
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J. Math. Phys. 35, 1951–1975 (1994)
Article history
Received:
August 03 1993
Accepted:
November 24 1993
Citation
E. G. Kalnins, Willard Miller; Models of q‐algebra representations: q‐integral transforms and ‘‘addition theorems’’. J. Math. Phys. 1 April 1994; 35 (4): 1951–1975. https://doi.org/10.1063/1.530581
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