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arXiv:2602.23764

A new class of coherent states involving Fox-Wright functions and their generalization in the bicomplex framework

Snehasis Bera, Sourav Das, Abhijit Banerjee

quant-phmath-phmath.CVSubmitted Feb 27, 2026• Announced Mar 2, 2026• 0 citations

Abstract

In this work, an extensive class of coherent states is introduced by taking the Fox Wright function as the normalization function. It is demonstrated that these states satisfy the key requirements of continuity, normalizability and resolution of unity. Furthermore, coherent states associated with the continuous spectrum are obtained through a discrete to continuous limiting procedure. Moreover, FW generalized multi parameter nu function is introduced and shown to act as the normalization function for the Fox Wright coherent states in the continuous spectrum. Later the Fox Wright function with bicomplex arguments has been introduced and its existence has been investigated. Bicomplex Fox Wright coherent states are also developed for the discrete spectrum based on this new function and their properties are analyzed. Subsequently, the results regarding Fox Wright coherent states are generalized to the bicomplex setting. In addition, a bicomplex FW generalized multi-parameter nu function is defined to demonstrate that it provides the normalization for these states in the continuous spectrum.

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In this paper we extend the applicability of Fox-Wright functions beyond mathematics, specifically in quantum physics. We focused our attention on a new application, on the connection between the Fox-Wright functions and the generalized coherent states formalism. We constructed the generalized coherent states in the Barut-Girardello manner, in which the Fox-Wright functions play the role of normalization functions, and we demonstrated that the Fox-Wright coherent states satisfy all general conditions imposed on the set of coherent states. In parallel, we examined the properties of both pure and mixed (thermal) Fox-Wright coherent states. All calculations were performed within the diagonal operators ordering technique (DOOT) using the Dirac's bra-ket formalism. Finally, we introduced some (specifically, integral) feedback elements that Fox-Wright coherent states induce in the theory of special functions, including a new integral representation of Fox-Wright functions.

Abstract The purpose of this paper is to show that by using a certain type of discrete-continuous limit, a series of integral entities can be defined (Mittag-Leffler multi-index functions, associated coherent states and their properties), which are counterparts of the corresponding discrete entities. We built and examine the properties of a new aspect of generalized integral multi-index Mittag-Leffler functions and we constructed and examined the properties of coherent states associated with this new function. This approach is motivated through the fact that these functions can be connected with the coherent states of the continuous spectrum, as well as with so-called nu-function.

In the present scenario, coherent states of a quantum harmonic oscillator are generalized with positive Fox H auxiliary functions. The novel generalized coherent states provide canonical coherent states and Mittag-Leffler or Wright generalized coherent states, as particular cases, and resolve the identity operator, over the Fock space, with a weight function that is the product of a Fox H function and a Wright generalized hypergeometric function. The novel generalized coherent states, or the corresponding truncated generalized coherent states, are characterized by anomalous statistics for large values of the number of excitations: the corresponding decay laws exhibit, for determined values of the involved parameters, various behaviors that depart from exponential and inverse-power-law decays, or their product. The analysis of the Mandel Q factor shows that, for small values of the label, the statistics of the number of excitations becomes super-Poissonian, or sub-Poissonian, by simply choosing sufficiently large values of one of the involved parameters. The time evolution of a generalized coherent state interacting with a thermal reservoir and the purity are analyzed.

The present paper starts from a previously deduced result, in which the

ν

-function plays the role of the normalization function of generalized hypergeometric coherent states for quantum systems with a continuous spectrum. We have generalized this idea, obtaining a new function, the generalized

ν

-function. By defining a discrete-continuous limit, we revealed a series of interesting properties that, in the last instance, allow the formulation and solution of new integrals involving the generalized

ν

-functions which depend on both scalar arguments as well as those containing creation and annihilation operators, which generate the generalized hypergeometric coherent states. To our knowledge, the results obtained by us do not appear in the literature.

In this paper we construct a new generalization of coherent states based on the application of Wright functions. This explicit family of coherent states is based on the generalization of the classical coherent states by using a positive weight function. We analyze in detail by means of the Mandel parameter, the deviations from the conventional coherent states due to this generalization that leads to sub- or super-Poissonian behaviour. We also discuss the connection between generalized coherent states and weighted Poisson distributions. Finally, we briefly show the relation between the normalizing function here used and the solution of a fractional differential equation with variable coefficients.

In order to examine the generalized coherent states, defined in the Barut-Girardello manner, whose normalization unction is just the four-parameter generalized Mittag-Leffler function, we used a special pair of non-linear perators which generate these states. Some characteristic properties for pure, as well as mixed (thermal) oherent states are also examined. All calculations are made using the rules of the diagonal ordering of perators (DOOT). Finally, the integral counterpart of the Mittag-Leffler coherent states are examined which is onnected with nu-function. The paper is an example of the application of a mathematical entities (Mittag-Leffler function and nu-function) in quantum mechanics (coherent states formalism)