Quantum statistical properties of multiphoton hypergeometric coherent states and the discrete circle representation Arjika, S. Calixto Molina, Manuel Guerrero, Julio S.A. thanks M.C. and J.G. for their hospitality during his stay at the University of Granada where this work was done, and the Coimbra Group for the financial support. This study has been partially financed by the Consejería de Conocimiento, Investigación y Universidad, Junta de Andalucía, and European Regional Development Fund (ERDF) under projects with Ref. Nos. FQM381 and SOMM17/6105/UGR, and by the Spanish MICINN under Project No. PGC2018-097831-B-I00. J.G. thanks the Spanish MICINN for financial support (Grant No. FIS2017-84440-C2-2-P). We review the definition of hypergeometric coherent states, discussing some representative examples. Then, we study mathematical and statistical properties of hypergeometric Schrödinger cat states, defined as orthonormalized eigenstates of kth powers of nonlinear f-oscillator annihilation operators, with f of the hypergeometric type. These “k-hypercats” can be written as an equally weighted superposition of hypergeometric coherent states ∣zl⟩, l = 0, 1, …, k − 1, with zl = ze2πil/k a kth root of zk, and they interpolate between number and coherent states. This fact motivates a continuous circle representation for high k. We also extend our study to truncated hypergeometric functions (finite dimensional Hilbert spaces), and a discrete exact circle representation is provided. We also show how to generate k-hypercats by amplitude dispersion in a Kerr medium and analyze their generalized Husimi Q-function in the super- and sub-Poissonian cases at different fractions of the revival time. 2024-01-03T09:35:12Z 2024-01-03T09:35:12Z 2019 info:eu-repo/semantics/article J. Math. Phys. 60, 103506 (2019); [doi: 10.1063/1.5099683] https://hdl.handle.net/10481/86521 https://doi.org/10.1063/1.5099683 eng http://creativecommons.org/licenses/by-nc-nd/3.0/ info:eu-repo/semantics/openAccess Creative Commons Attribution-NonCommercial-NoDerivs 3.0 License American Institute of Physics