Polyanalytic relativistic second Bargmann transforms

We construct coherent states through special superpositions of photon number states of the relativistic isotonic oscillator. In each superposition the coefficients are chosen to be L 2 eingenfunctions of a sigma weight Maass Laplacian on the Poincare disk, which are associated with discrete eigenvalues. For each nonzero m the associated coherent states transform constitutes the m true polyanalytic extension of a relativistic version of the second Bargmann transform, whose integral kernel is expressed in terms of a special Appel Kampe de Feriet hypergeometric function. The obtained results could be used to extend the known semi classical analysis of quantum dynamics of the relativistic isotonic oscillator.

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