Generalized hypergeometric coherent states for special functions: mathematical and physical properties

TL;DR

This paper introduces a new class of generalized hypergeometric coherent states linked to special functions, exploring their mathematical properties, physical implications, and applications in quantum models, including quantization and thermodynamics.

Contribution

It extends the theory of coherent states to associated Jacobi polynomials and hypergeometric functions, providing new analytical tools and insights for quantum physics and special functions.

Findings

Resolution of the identity with Meijer's G-function weight

Reproducing kernel and its analytical expansion

Analysis of probability density and temporal stability

Abstract

In continuation of our previous works J. Phys. A: Math. Gen. 35, 9355-9365 (2002), J. Phys. A: Math. Gen. 38, 7851 (2005) and Eur. Phys. J. D 72, 172 (2018), we investigate a class of generalized coherent states for associated Jacobi polynomials and hypergeometric functions, satisfying the resolution of the identity with respect to a weight function expressed in terms of Meijer's G-function. We extend the state Hilbert space of the constructed states and discuss the property of the reproducing kernel and its analytical expansion. Further, we provide the expectation values of observables relevant to this quantum model. We also perform the quantization of the complex plane, compute and analyze the probability density and the temporal stability in these states. Using the completeness relation provided by the coherent states, we achieve the thermodynamic analysis in the diagonal $P$-representation of the density operator.