A new generalization of nonlinear coherent states for the pseudoharmonic oscillator

Abstract

We construct two-parameters family of nonlinear coherent states by replacing the factorial in coefficients $z^n/\sqrt{n!}$ of the canonical coherent states by a specific generalized factorial $x_n^{\gamma ,\sigma }!$ where parameters $\gamma$ and $\sigma$ satisfy some conditions for which the normalization condition and the resolution of identity are verified. The obtained family is a generalization of the Barut–Girardello coherent states and those of the philophase states. In the particular case of parameters $\gamma$ and $\sigma$, we attache these states to the pseudoharmonic oscillator depending on two parameters $\alpha ,\beta > 0$. The obtained nonlinear coherent states are superposition of eigenstates of this oscillator. The associated Bargmann-type transform is defined and we derive some results.

1 Introduction

Coherent states (CSs) were first discovered by Schrödinger in 1926 [55] as wavepackets having dynamics similar to that of a classical particle submitted to a quadratic potential. They have arised from the study of the quantum harmonic oscillator (HO) to become very useful in different areas of physics. In 1963, coherent states were simultaneously rediscovered by Glauber [22, 23], Klauder [32, 33] and Sudarshan [59] in quantum optics of coherent light beams emitted by lasers. Coherent states of HO have three equivalent definitions, namely: i) they are eigenfunctions of the annihilation operator; ii) they minimize the Heisenberg uncertainty relation and iii) they are obtained by a shift of the vacuum state under the action of the unitary operator of the Weyl Heisenberg group. Thanks to the properties of these states, various generalizations of CSs were proposed, see [3, 13] and references therein. Along with a generalization of CSs, an algebraic generalization called nonlinear coherent states (NLCSs) were implicitly defined by Shanta et al [56] in a compact form and introduced explicitly by de Matos Filho and Vogel [10] and Man’ko et al [37]. In 1999, Gazeau and Klauder [21] have proposed new coherent states for systems with discrete or continuous spectra. Recently, Klauder, Penson, and Sixdeniers [34] have introduced another important class of generalized CSs through the solutions of the Stieltjes and Hausdorff moment problem. More recently, Roknizadeh and Tavassoly [54] showed that all these states may be studied in the so-called NLCSs or f-deformed CSs category.

Thanks to its mathematical advantages, the HO is used to model real molecular vibrations although they are anharmonic. An anharmonic potential (a more realistic potential), which also permits an exact mathematical treatment, is the so-called “pseudoharmonic oscillator” (PHO) potential. This potential may be considered in a certain sense as an intermediate oscillator between the HO and more anharmonic oscillators, e. g. Morse oscillator, Pöschl–Teller oscillator (which are more realistic). In a series of papers Popov and his collaborators have constructed and studied some properties of the corresponding different kinds of associated CSs of PHO, i.e. Barut and Girardello [48], Gazeau and Klauder [49] and Klauder and Perelomov [50]. For other work on the PHO and further applications see the articles [1, 14, 16, 29, 42, 51], and their references.

In this paper, we construct a two-parameters family of NLCSs, denoted $|z,\gamma ,\sigma \rangle$, by replacing the factorial n! in coefficients $z^{n}/\sqrt{n!}$ of the canonical CSs by a specific generalized factorial $x_{n}^{\gamma ,\sigma }!:=x_{1}^{\gamma ,\sigma }\cdots x_{n}^{\gamma ,\sigma }$ with $x_{0}^{\gamma ,\sigma }=0$, where $x_{n}^{\gamma ,\sigma }$ is a sequence of positive numbers (given by (3.1) see below) and $\gamma ,\sigma$ are two real parameters. The obtained NLCSs are a generalization of the Barut Girardello CSs type [7] obtained when $2\gamma =1,2,3,\ldots$ and $\sigma =0$ and those of the philophase states [8] occuring when $\gamma =1/2$ and $\sigma$ being a positive integer. The case $\gamma =1/2$ and $\sigma =0$, have been considered in [2] where the authors associated to their NLCSs two set of orthogonal polynomials by following the work of T. Ali and M. Ismail [4]. Next, we construct a family of NLCSs attached to the PHO $\Delta _{\alpha ,\beta }$ [24] depending on two parameters $\alpha$ and $\beta$. These states are obtained as a superposition of the eigenstates of this oscillator. The wave functions of constructed NLCSs are obtained in the special case $\gamma =2^{-1}\mu (\alpha )$. Finally, we exploit the obtained result to define a new Bargmann-type integral transform and we derive some interesting results.

The rest of the paper is organized as follows. In Sect. 2, we summarize the construction of NLCSs. In Sect. 3, we particularize the formalism of NLCSs for the sequences $x_n^{\gamma ,\sigma }$ and we discuss the corresponding resolution of the identity. The $x_n^{\gamma ,\sigma }$-NLCSs and Bargmann-type transform attached to PHO are defined in Sect. 4. Section 5 is devoted to the conclusion.

2 Nonlinear coherent states formalism

This section is devoted to a quick review on the construction of NLCSs. Details and proofs of statements may be found in [3, pp.146-151]. The principal idea is to involve a new sequence of positive numbers in the superposition coefficients. More precisely, let us first recall the Fock basis representation of the canonical CSs [3]:

$$\begin{aligned} |z\rangle =(e^{z{\bar{z}}})^{-1/2}\sum \limits _{n=0}^{\infty }\frac{{\bar{z}}^{n}}{\sqrt{n!}}|\psi _{n}\rangle ,\quad z\in {\mathbb {C}}. \end{aligned}$$

(2.1)

The kets $\{ |\psi _{n}\rangle \}_{n=0}^{\infty }$ constitute an orthonormal basis in an arbitrary (complex, separable, infinite dimensional) Hilbert space ${\mathcal {H}}$. The related NLCSs are constructed as follows.

Let $\{x_{n}\}_{n=0}^{\infty }$, be an infinite sequence of positive numbers with $\lim _{n\rightarrow +\infty }x_{n}=R^{2}$ where $R>0$ could be finite or infinite, but not zero. We define the generalized factorial by $x_{n}!=x_{1}x_{2}\cdots x_{n}$ and $x_{0}!=1$. For each $z\in {\mathcal {D}}$ a complex domain, the NLCSs constituting a generalization of (2.1) are defined by

$$\begin{aligned} |z\rangle =({\mathcal {N}}(z{\bar{z}}))^{-1/2}\sum \limits _{n=0}^{\infty }\frac{ {\bar{z}}^{n}}{\sqrt{x_{n}!}}|\psi _{n}\rangle ,\quad z\in {\mathcal {D}} \end{aligned}$$

(2.2)

where the normalization factor

$$\begin{aligned} {\mathcal {N}}(z{\bar{z}})=\sum \limits _{n=0}^{\infty }\frac{|z|^{2n}}{x_{n}!} \end{aligned}$$

(2.3)

is chosen so that the vectors (2.2) are normalized to one and are well defined for all z for which the sum (2.3) converges, i.e. ${\mathcal {D}}=\{z\in {\mathbb {C}},|z|<R\}$. We assume that there exists a measure $d\vartheta$ on ${\mathcal {D}}$ ensuring the following resolution of the identity

$$\begin{aligned} \int _{{\mathcal {D}}}|z\rangle \langle z|d\vartheta (z,{\bar{z}})=1_{{\mathcal {H}}} . \end{aligned}$$

(2.4)

Setting $d\vartheta (z,{\bar{z}})={\mathcal {N}}(z{\bar{z}})d\eta (z,{\bar{z}})$, it is easily seen that in order for (2.4) to be satisfied, the measure $d\eta$ should be of the form $d\eta (z,{\bar{z}})=(2\pi )^{-1}d\theta dw (\rho ),\, z=\rho e^{i\theta }$ where the measure dw solves the moment problem

$$\begin{aligned} \int _{0}^{R}\rho ^{2n}dw(\rho )=x_{n}!,\quad n=0,1,2,\ldots \, . \end{aligned}$$

(2.5)

In most of the practical situations, the support of the measure $d\eta$ is the whole domain ${\mathcal {D }}$, i.e., dw is supported on the entire interval [0, R).

To illustrate this formalism, we consider the sequence of positive numbers

$$\begin{aligned} x^{\gamma }_{n}=n\left( 2\gamma +n-1\right) ,\ \ \ n=0,1,2,3,\ldots , \end{aligned}$$

(2.6)

with $2\gamma =1,2,3,\ldots$, being a fixed parameter. Here $R=\infty$ and the moment problem is

$$\begin{aligned} \int _{0}^{\infty }\rho ^{2n}dw (\rho )=n!(2\gamma )_{n} \end{aligned}$$

(2.7)

where $(a)_{n}=a(a+1)\cdots (a+n-1)$ with $(a)_{0}=1$, is the shifted factorial. The solution of this problem is

$$\begin{aligned} dw(\rho )=\frac{2}{\pi }K_{2\gamma -1}(2\rho )\rho ^{2-2\gamma }d\rho ,\quad 0\le \rho <\infty , \end{aligned}$$

(2.8)

where

$$\begin{aligned} K_{\tau }(x)=\frac{1}{2}\left( \frac{x}{2}\right) ^{\tau }\int _{0}^{\infty }\exp \left( -t-\frac{x^{2}}{4t}\right) \frac{dt}{t^{\tau +1}},\ \ \mathfrak {R}(x)>0, \end{aligned}$$

(2.9)

is the Macdonald function of order $\tau$ [61, p.183]. The associated coherent states are of Barut–Girardello type associated with the su(1, 1) Lie algebra and which can be expressed in the Fock basis as [7]:

$$\begin{aligned} |z,\gamma \rangle =\frac{|z|^{\gamma -\frac{1}{2}}}{\sqrt{I_{2\gamma -1}(2|z|)}} \sum \limits _{n=0}^{\infty }\frac{{\bar{z}}^{n}}{\sqrt{n!(2\gamma )_{n}}} |\psi _{n}\rangle ,\quad z\in {\mathbb {C}}, \end{aligned}$$

(2.10)

where $I_{\tau }(\cdot )$ being the modified Bessel function of the first kind and of order $\tau$ [61, p.172]. Note that in [7], the Hilbert space ${\mathcal {H}}$ have not been precised and the orthonormal basis $|\psi _{n}\rangle$ in (2.10) are left without explicit expression as unitary representation of su(1, 1) Lie algebra.

3 A set of NLCSs with the sequences $x_n^{\gamma ,\sigma }$

In this section, we define a new set of NLCSs attached to new sequences $x_n^{\gamma ,\sigma }$, without specifying the Hilbert space ${\mathcal {H}}$ and the basic vectors $\left| \psi _n\right\rangle$, for which we will discuss some general properties.

3.1 NLCSs attached to a sequence $x_n^{\gamma ,\sigma }$

Here, we will be dealing with two-parameters family of NLCSs on the complex plane, which generalizes the set of CSs of Barut–Girardello type [7] and those of the philophase states [8] without specifying the Hamiltonian. Precisely, let us consider $\gamma >0$ and $\sigma \in {\mathbb {R}}\backslash {\mathbb {Z}}^{*}_{-}$, two fixed parameters and let us define the infinite sequence of positive numbers:

$$\begin{aligned} \quad x_0^{\gamma ,\sigma }=0, \quad \text {and } \quad x_n^{\gamma ,\sigma }=\frac{(n+\sigma )^2(n+2\gamma -1)}{n},\ \ n=1,2,3,\ldots \ . \end{aligned}$$

(3.1)

The corresponding factorial of (3.1) reads as

$$\begin{aligned} x^{\gamma ,\sigma }_n!=\frac{(\sigma +1)_n^2(2\gamma )_n}{n!},\ \ n=0,1,2,3,\ldots \ . \end{aligned}$$

(3.2)

Now, we define a set of $x_n^{\gamma ,\sigma }$-NLCSs through the sequence $x_n^{\gamma ,\sigma }$, under some conditions on the parameters $\gamma$ and $\sigma$, as a normalized vectors of the Hilbert space ${\mathcal {H}}=\text {span }\left\{ \left| \psi _n\right\rangle \right\} _{n=0}^{\infty }$ via the superposition

$$\begin{aligned} \left| z,\gamma ,\sigma \right\rangle =\left( {\mathcal {N}}_{\gamma ,\sigma }\left( z{\bar{z}}\right) \right) ^{-1/2}\sum _{n=0}^{\infty }\sqrt{\frac{n!}{(2\gamma )_n}}\frac{{\bar{z}}^n}{(\sigma +1)_n}\left| \psi _n\right\rangle , \end{aligned}$$

(3.3)

where ${\mathcal {N}}_{\gamma ,\sigma }(z{\bar{z}})$ is the normalization factor chosen such that $\langle \sigma ,\gamma , z \left| z,\gamma ,\sigma \right\rangle =1$. These states are defined for all $z\in {\mathbb {C}}$ (since $\lim _{n\rightarrow +\infty }x_n^{\gamma ,\sigma }=+\infty$), and the normalization factor is given in the following proposition (see Appendix A for the proof).

Proposition 3.1.1

Let $\gamma >0$ and $\sigma \in {\mathbb {R}}\backslash {\mathbb {Z}}^{*}_{-}$, be fixed parameters. Then, the normalization factor in (3.3) for the set of $x_n^{\gamma ,\sigma }$-NLCSs reads

$$\begin{aligned} {\mathcal {N}}_{\gamma ,\sigma }\left( z{\bar{z}}\right) ={}_{2}F_{3}\left( \begin{array}{c} 1,1 \\ 2\gamma ,\sigma +1,\sigma +1 \end{array} \Bigg \vert z{\bar{z}}\right) , \ z\in {\mathbb {C}}, \end{aligned}$$

(3.4)

in terms of ${}_{2}F_{3}$-hypergeometric series [36, p.62]. We have two interesting particular cases:

• when $\gamma =1/2$ and $\sigma \in {\mathbb {N}}^{*}$, (3.4) reduces to

  $$\begin{aligned} {\mathcal {N}}_{\frac{1}{2},\sigma }\left( z{\bar{z}}\right) =\left[ I_{0}(2|z|)-\sum _{n=0}^{\sigma -1}\frac{|z|^{2n}}{(n!)^2}\right] \frac{\Gamma ^2(\sigma +1)}{|z|^{2\sigma }}, \end{aligned}$$

  (3.5)

• when $\sigma =0$, (3.4) reads

  $$\begin{aligned} {\mathcal {N}}_{\gamma ,0}\left( z{\bar{z}}\right) =\Gamma (2\gamma )|z|^{1-2\gamma }I_{2\gamma -1}(2|z|), \end{aligned}$$

  (3.6)

for all $z\in {\mathbb {C}}$, where $I_{\tau }(\cdot )$ is the Bessel function of the first kind and of order $\tau$.

We see that the NLCSs (3.3) do not form an orthogonal set since the entire function $\ _2F_3\left( 1,1;2\gamma ,\sigma +1,\sigma +1; z{\bar{w}}\right)$ has no zeros if $z{\bar{w}}$ is a positive number.

Remark 3.1.1

The sequence $x_n^{\gamma ,\sigma }$ can be identified in the literature as a special case of sequences of positive numbers in the construction of the generalized hypergeometric CSs [5, 52]. However, in our goal, the main reason for choosing the sequence $x_n^{\gamma ,\sigma }$ resides in the fact that the corresponding NLCSs constructed in (3.3) generalize two specific states in the literature: Barut–Girardello coherent states [7] while setting $2\gamma =1,2,3,\ldots$ and $\sigma =0$; Philophase states [8] while setting $\gamma =1/2$ and $\sigma \in {\mathbb {N}}^{*}$. A generalized approach of such construction of NLCSs is the subject of a forthcoming paper.

Remark 3.1.2

A class of NLCSs attached to the harmonic oscillator applying hypergeometric-type operators according to the representation of the Heisenberg Lie algebra are discussed in [12]. CSs for pseudo generalization of this algebra can be found in [18].

For all that follows, the general properties of $x_n^{\gamma ,\sigma }$-NLCSs reduce to those of the above specific cases.

3.2 Resolution of identity

The constructed $x_n^{\gamma ,\sigma }$-NLCSs are required to form a complete set of vectors in the Hilbert space ${\mathcal {H}}$ by providing the resolution of identity of this space. Here, we will discuss the resolution of identity under some conditions on the parameters for which the $x_n^{\gamma ,\sigma }$-NLCSs are well defined. The problem here is to find the measure $d\vartheta _{\gamma ,\sigma }$ which satisfy the following relation

$$\begin{aligned} \int _{{\mathbb {C}}}\left| z,\gamma ,\sigma \right\rangle \left\langle z,\gamma ,\sigma \right| d\vartheta _{\gamma ,\sigma }(z)=1_{{\mathcal {H}}}. \end{aligned}$$

(3.7)

To resolve this problem, we write $d\vartheta _{\gamma ,\sigma }$ as

$$\begin{aligned} d\vartheta _{\gamma ,\sigma }(z)={\mathcal {N}}_{\gamma ,\sigma }\left( z{\bar{z}}\right) m_{\gamma ,\sigma }\left( z{\bar{z}}\right) d\varrho (z), \end{aligned}$$

(3.8)

where $m_{\gamma ,\sigma }$ is a weight function to be determined and $d\varrho$ is the Lebesgue measure on ${\mathbb {C}}$. By considering the polar coordinates $z=\rho e^{i\theta } ,\ \rho >0$ and $\theta \in [0,2\pi )$, the measure can be rewritten as

$$\begin{aligned} d\vartheta _{\gamma ,\sigma }(z)={\mathcal {N}}_{\gamma ,\sigma }\left( \rho ^{2}\right) m_{\gamma ,\sigma }\left( \rho ^{2}\right) \frac{ \rho d\rho d\theta }{2\pi }. \end{aligned}$$

(3.9)

By using the expression (3.3) of $x_n^{\gamma ,\sigma }$-NLCSs, the resolution of identity (3.7) requires, after some manipulations and making the change of variable $r=\rho ^2$, that

$$\begin{aligned} \sum \limits _{n=0}^{\infty }\frac{n!}{2(2\gamma )_n}\frac{1}{(\sigma +1)^2_n}\left( \int _0^{\infty }r^{n}m_{\gamma ,\sigma }(r)dr \right) \vert \psi _n\rangle \langle \psi _n\vert =1_{{\mathcal {H}}}. \end{aligned}$$

(3.10)

Thus, we must have

$$\begin{aligned} \int _0^{\infty }r^{n}m_{\gamma ,\sigma }(r)dr=\frac{2(2\gamma )_n(\sigma +1)^2_n}{n!}. \end{aligned}$$

(3.11)

This is a Stieltjes moment problem [53]. To establish positive solutions of this problem we use the inverse Mellin transform method as well as the fact that Mellin convolution preserves positivity (see [19, 57] and [9, p.559] for more details on this method). An explicit calculation (see Appendix B) shows that positive solution of the moment problem (3.11) is given by

$$\begin{aligned} m_{\gamma ,\sigma }(r)= & {} \frac{2r^{-1}}{\Gamma (2\gamma )\Gamma ^2(\sigma +1)}G^{3 0}_{1 3} \left( r \ \Bigg \vert \ \begin{array}{c} 1\\ 2\gamma ,\sigma +1,\sigma +1 \end{array} \right) ,\ r>0, \end{aligned}$$

(3.12)

for $(\gamma ,\sigma )\in S_1 \cup S_2$ where

$$\begin{aligned} S_1= ]0,+\infty [ \times ]-1,0],\quad \text {and } \quad S_2= ]0,1/2 ] \times ]-1,+\infty [. \end{aligned}$$

(3.13)

Here $G^{3,0}_{1,3}(\cdot )$ denotes the Meijer’s G-function [9, p.533] which governs the positivity of this weight function. Some plot illustrations are given in Figs. 1, 2¹. With this weight function, equation (3.10) reduces to $\sum _{n=0}^{\infty }\left| \psi _n\right\rangle \left\langle \psi _n\right| =1_{{\mathcal {H}}}$ since $\left\{ \left| \psi _n\right\rangle \right\}$ is an orthonormal basis of the Hilbert space ${\mathcal {H}}$.

Fig. 1

Plots of $x\mapsto G^{3,0}_{1,3}(x)$ for various $(\gamma ,\sigma )\in S_1$

Fig. 2

Plots of $x\mapsto G^{3,0}_{1,3}(x)$ for various $(\gamma ,\sigma )\in S_2$

Table 1 Values of $G_{1,3}^{3,0}(x)$ for $(\gamma ,\sigma )\in \{(0.25,-0.5); (0.125,0); (0.375,9)\}$

Then, we obtain the following result.

Proposition 3.2.1

Let $(\gamma ,\sigma )\in S_1 \cup S_2$, be fixed parameters. Then, the set of NLCSs attached to the sequence $x_n^{\gamma ,\sigma }$ defined in (3.3) satisfy the resolution of identity

$$\begin{aligned} \int _{{\mathbb {C}}}\left| z,\gamma ,\sigma \right\rangle \left\langle z,\gamma ,\sigma \right| d\vartheta _{\gamma ,\sigma }(z)=1_{{\mathcal {H}}}, \end{aligned}$$

(3.14)

in terms of a suitable measure given by

$$\begin{aligned} d\vartheta _{\gamma ,\sigma }(z)= & {} \frac{2(z{\bar{z}})^{-1}}{\Gamma (2\gamma )\Gamma ^2(\sigma +1)}\ _{2}F _{3}\left( \begin{array}{c} 1,1 \\ 2\gamma ,\sigma +1,\sigma +1 \end{array} \Bigg \vert z{\bar{z}}\right) \nonumber \\&\quad G_{13}^{30}\left( z{\bar{z}}\ \Bigg \vert \begin{array}{c} 1\\ 2\gamma ,\ \sigma +1 ,\sigma +1 \end{array} \right) d\varrho (z), \end{aligned}$$

(3.15)

where $G_{13}^{30}(\cdot )$ denotes the Meijer’s G-function and $d\varrho$ being the Lebesgue measure on ${\mathbb {C}}$.

Remark 3.2.1

In the case where (3.3) describes a quantum system, the probability of finding the state $\left| \psi _n\right\rangle$ in some normalized state $\left| z,\gamma ,\sigma \right\rangle$ of the Hilbert space ${\mathcal {H}}$ is given by

$$\begin{aligned} P_n(z,\gamma ,\sigma )=\left[ \ {}_{2}F_{3}\left( \begin{array}{c} 1,1 \\ 2\gamma ,\sigma +1,\sigma +1 \end{array} \Bigg \vert z{\bar{z}}\right) \right] ^{-1}\frac{n!}{(2\gamma )_n}\frac{|z|^{2n}}{(\sigma +1)_n^2} \end{aligned}$$

(3.16)

for all z in the complex plane ${\mathbb {C}}$.

4 $x_n^{\gamma ,\sigma }$-NLCSs attached to the pseudoharmonic oscillator

In this section, we discuss the closed form of the constructed NLCSs given in (3.3) by choosing the orthonormal basis $\left| \psi _n\right\rangle$ of the Hilbert space ${\mathcal {H}}$ as the eigenfunctions of the pseudoharmonic oscillator $\Delta _{\alpha ,\beta }$. Then, we define the associated Bargmann-type transform and we derive some interesting formulas.

4.1 The pseudoharmonic oscillator $\Delta _{\alpha ,\beta }$

An anharmonic potential that can be used to calculate the vibrational energies of a diatomic molecule has the form [24, 42]

$$\begin{aligned} V_{\varrho ,\xi _{0}}(\xi )=\varrho \left( \frac{\xi }{\xi _{0}}-\frac{\xi _{0}}{\xi }\right) ^{2}, \end{aligned}$$

(4.1)

where $\xi _{0}>0$ is the equilibrium distance between the diatomic molecule nuclei, and $\varrho >0$ with $\varrho \xi _0^{-2}$ represents a constant force. The associated stationary Schrödinger equation (in the units $\hbar ^2=2\mu$) reads [24, p.52]:

$$\begin{aligned} -\frac{d^{2}}{d\xi ^{2}}\psi (\xi )+\varrho \left( \frac{\xi }{\xi _{0}}-\frac{\xi _{0}}{\xi } \right) ^{2}\psi (\xi )=E\psi (\xi ), \end{aligned}$$

(4.2)

where $\psi$ satisfies the Dirichlet boundary condition $\psi (0)=0$. It is an exactly solvable equation. To simplify the notation, we introduce the new parameters $\alpha :=\varrho \xi _{0}^{2}$ and $\beta :=\xi _{0}^{-1}\sqrt{\varrho }$. Thereby the Hamiltonian in (4.2) takes the form [28, p.97]

$$\begin{aligned} \Delta _{\alpha ,\beta }:=-\frac{d^{2}}{d\xi ^{2}}+\beta ^{2}\xi ^{2}+\frac{\alpha }{\xi ^{2}},\ \ \xi \in {\mathbb {R}}_{+},\ \ \beta >0,\ \alpha \ge 0, \end{aligned}$$

(4.3)

called Gol’dman-Krivchenkov Hamiltonian or pseudoharmonic oscillator (PHO). Its energy spectrum in the Hilbert space $L^{2}({\mathbb {R}}_{+},d\xi )$ reduces to a discrete part consisting of eigenvalues of the form

$$\begin{aligned} E_{n}^{\nu ,\beta }:=2\beta (2n+\nu ) ,\ \ n=0,1,2,\ldots , \end{aligned}$$

(4.4)

where $\nu =\nu (\alpha )=1+\frac{1}{2}\sqrt{1+4\alpha }\ge \frac{3}{2}$ and the wave functions of the corresponding normalised eigenfunctions are given by

$$\begin{aligned} \langle \xi |\psi _{n}^{\nu ,\beta }\rangle :=(-1)^n\left( \frac{2\beta ^{\nu }n!}{\Gamma (\nu +n)}\right) ^{\frac{1}{2}}\xi ^{\nu -\frac{1}{2}}e^{-\frac{1}{2}\beta \xi ^{2}}L_{n}^{(\nu -1)}(\beta \xi ^{2}),\ \ n=0,1,2,\ldots \nonumber \\ \end{aligned}$$

(4.5)

in terms of the Laguerre polynomials $L_{n}^{(\alpha )}(\cdot )$ [36, p.239]. The set of the functions (4.5) constitutes a complete orthonormal basis for the Hilbert space $L^{2}({\mathbb {R}}_{+},d\xi )$ (see [28]).

Remark 4.1.1

It must be noticed that the PHO can be obtained from the two-particle Calogero-Sutherland model (CSM), after removing the centre-of-mass motion of the latter one [26]. Thanks to this connection with the CSM, the PHO has recently generated wide interest. The CSM describes a quantum system of N kinematically similar particles in one dimension, interacting pairwise via quadratic and centrifugal potentials [1] and have been found relevant for description of various physical phenomena such as quantum Hall effect [6, 31, 46], fractional statistics [27, 35, 43, 47]. The CSM considered in [17, 40], denoted by $H^\lambda$, coincides with the Gol’dman-Krivchenkov oscillator $\Delta _{\alpha , \beta }$ in (4.3) for $\beta ^2=1$ and $\alpha =\lambda (\lambda -1)$. That is, $H^\lambda =\frac{1}{2}\Delta _{\lambda (\lambda -1), 1}$. Another related model to the PHO is the para-Bose oscillator obtained (up to a multiplicative factor) for $\alpha =(p-1)(p-3)/4$ and $\beta =1$. In [41], three kinds of NLCSs associated with this oscillator have been considered.

4.2 The $x_n^{\gamma ,\sigma }$-NLCS attached to $\Delta _{\alpha ,\beta }$

As announced in Sect. 4, choosing the Hilbert space ${\mathcal {H}}=L^2\left( {\mathbb {R}}_{+},d\xi \right)$ with its orthonormal basis $\left| \psi _n\right\rangle \equiv \left| \psi ^{\nu ,\beta }_n\right\rangle$ we discuss in the following proposition the closed form of the $x_n^{\gamma ,\sigma }$-NLCS

$$\begin{aligned} \left| z, \gamma ,\sigma \right\rangle _{\nu ,\beta }= \left( {\mathcal {N}}_{\gamma ,\sigma }\left( z{\bar{z}}\right) \right) ^{-1/2}\sum _{n=0}^{\infty }\sqrt{\frac{n!}{(2\gamma )_n}}\frac{{\bar{z}}^{n}}{(\sigma +1)_n}\left| \psi ^{\nu ,\beta }_n\right\rangle ,\quad z\in {\mathbb {C}}, \end{aligned}$$

(4.6)

where $\gamma >0$ and $\sigma \le 0; \ \sigma \ne -1, -2, -3,\ldots$ and ${\mathcal {N}}_{\gamma ,\sigma }$ is the normalization factor in (3.4).

Proposition 4.2.1

Let $2\gamma =\nu \ge \frac{3}{2}$, $\beta >0$ and $\sigma \le 0$, be fixed parameters. Then, the wave functions of the states (4.6), denoted $|z,\nu ,\sigma \rangle _{\beta }$, can be written as

$$\begin{aligned} \begin{aligned}&\left\langle \xi |z,\nu ,\sigma \right\rangle _{\beta } = \sqrt{\frac{2\beta ^{\nu }}{\Gamma (\nu )}}\left[ \ _{2}F _{3}\left( \begin{array}{c} 1,1 \\ \nu ,\sigma +1,\sigma +1 \end{array} \Bigg \vert z{\bar{z}}\right) \right] ^{-\frac{1}{2}} \xi ^{\nu -\frac{1}{2}}e^{-\frac{1}{2}\beta \xi ^2}\\&F^{{1:0;0}}_{1:0;1}\left( \begin{array}{c} 1:-;-; \\ \sigma +1;-;\nu ; \end{array} -{\bar{z}},\beta {\bar{z}}\xi ^2\right) \end{aligned} \end{aligned}$$

(4.7)

where $F^{{p:q;k}}_{l:m;n}(\cdot )$ is the generalized Kampé de Fériet function [58, p.63].

In particular, for $\sigma =0$, equation (4.7) reduces to

$$\begin{aligned} \left\langle \xi |z,\nu ,0\right\rangle _{\beta }= & {} \sqrt{2\beta \xi }\left( -\frac{|z|}{{\bar{z}}}\right) ^{\frac{\nu -1}{2}}e^{-{\bar{z}}-\frac{1}{2}\beta \xi ^2}\left( I_{\nu -1}(2|z|)\right) ^{-\frac{1}{2}}J_{\nu -1}\left( 2i\xi \sqrt{\beta {\bar{z}}}\right) ,\,\,\nonumber \\&z\in {\mathbb {C}}, \end{aligned}$$

(4.8)

where $I_{\nu }(\cdot )$ and $J_{\nu }(\cdot )$ are the modified Bessel functions of the first kind of order $\nu$.

Proof

We start by writing the wave function of states $\left| z,\gamma ,\sigma \right\rangle _{\nu ,\beta }$ according to (4.6) as

$$\begin{aligned} \left\langle \xi |z, \gamma ,\sigma \right\rangle _{\nu ,\beta }=\left( {\mathcal {N}}_{\gamma ,\sigma }\left( z{\bar{z}}\right) \right) ^{-1/2}\sum _{n=0}^{\infty }\sqrt{\frac{n!}{(2\gamma )_n}}\frac{{\bar{z}}^n}{(\sigma +1)_n} \left\langle \xi |\psi ^{\nu ,\beta }_n\right\rangle ,\ \xi \in {\mathbb {R}}_{+}.\nonumber \\ \end{aligned}$$

(4.9)

Next, we put $2\gamma =\nu$ and replace $\left\langle \xi |\psi ^{\nu ,\beta }_n\right\rangle$ by its expression given in (4.5), we obtain

$$\begin{aligned} \left\langle \xi |z,\nu ,\sigma \right\rangle _{\beta }= & {} \left( {\mathcal {N}}_{\frac{\nu }{2},\sigma }\left( z{\bar{z}}\right) \right) ^{-1/2}\sqrt{\frac{2\beta ^{\nu }}{\Gamma (\nu )}}\xi ^{\nu -\frac{1}{2}}e^{-\frac{1}{2}\beta \xi ^2}\nonumber \\&\sum _{n=0}^{\infty }\frac{n!}{\left( \nu \right) _n}\frac{(-{\bar{z}})^n}{\left( \sigma +1\right) _n} L_n^{(\nu -1)}\left( \beta \xi ^2\right) . \end{aligned}$$

(4.10)

To obtain a closed form for $\left\langle \xi |z,\nu ,\sigma \right\rangle _{\beta }$ we should compute the infinite sum in the right hand side of the above equation which we denote in the sequel by ${\mathfrak {S}}_{\sigma ,\nu }(\xi ^2)$. Making use of the identities $k!\ L_k^{(b-1)}(x)=(b)_k \ {}_{1}F _{1}\left( -k;b;x\right)$ (see [36, p.240]) and $(-n)_k=(-1)^k n!/(n-k)!$ (see [58, p.23]), straightforward calculations give successively

$$\begin{aligned} {\mathfrak {S}}_{\sigma ,\nu }(\xi ^2)= & {} \sum \limits _{n=0}^{\infty }\frac{\left( -{\bar{z}}\right) ^n}{\left( \sigma +1\right) _n}\ _{1}F_{1}\left( -n; \nu ; \beta \xi ^2\right) \nonumber \\&= \sum \limits _{n=0}^{+\infty }\sum \limits _{k=0}^{n}\frac{ (-n)_k}{\left( \sigma +1\right) _n (\nu )_k} \frac{(-{\bar{z}})^n\left( \beta \xi ^2\right) ^k}{k!} \nonumber \\&= \sum \limits _{n=0}^{+\infty }\sum \limits _{k=0}^{+\infty }\frac{ (1)_{n+k}}{\left( \sigma +1\right) _{n+k} (\nu )_k} \frac{(-{\bar{z}})^n}{n!}\frac{\left( \beta {\bar{z}}\xi ^2\right) ^k}{k!}\end{aligned}$$

(4.11)

$$\begin{aligned}&= F^{{1:0;0}}_{1:0;1}\left( \begin{array}{c} 1:-;-; \\ \sigma +1;-;\nu ; \end{array} \Bigg \vert -{\bar{z}},\beta {\bar{z}}\xi ^2\right) ,\ z,\xi \in {\mathbb {C}}, \end{aligned}$$

(4.12)

where we have applied the identity (see [58, p.101])

$$\begin{aligned} \sum _{n=0}^{+\infty } \sum \limits _{k=0}^{n}A(k,n)=\sum _{n=0}^{+\infty } \sum \limits _{k=0}^{+\infty }A(k,n+k). \end{aligned}$$

(4.13)

Here $F^{{1:0;0}}_{1:0;1}(\cdot )$ stands for the generalized Kampé de Fériet function of two variables [58, p.63]. Finally, go back to (4.10) with (4.12) and replace the expression of ${\mathcal {N}}_{\frac{\nu }{2},\sigma }\left( z{\bar{z}}\right)$ given in (3.4), to obtain the announced result in (4.7). When $\sigma =0$, we write (4.11)

$$\begin{aligned} {\mathfrak {S}}_{0,\nu }(\xi ^2)= & {} e^{-{\bar{z}}} \sum \limits _{k=0}^{+\infty } \frac{1}{(\nu )_k} \frac{(\beta {\bar{z}}\xi ^2)^k}{k!} \nonumber \\= & {} e^{-{\bar{z}}} \Gamma (\nu ) (i\xi \sqrt{\beta {\bar{z}}})^{1-\nu } J_{\nu -1}\left( 2i\xi \sqrt{\beta {\bar{z}}}\right) \end{aligned}$$

(4.14)

in terms of the Bessel function $J_{\nu }(\cdot )$ [36, p.65]. Proceeding like in the general case, we get the announced result in (4.8). This ends the proof of Proposition 4.2.1. $\square$

Remark 4.2.1

It is interesting to note that, if we take $\beta =1$ in (4.8) with $\nu =\lambda +1/2$ we recover the compact form of the Barut–Girardello CSs type obtained in [17, 40] where the authors use a group theoretical approach based on the representation of the su(1, 1) Lie algebra

$$\begin{aligned} \begin{aligned}&{[}J_{+}^{\lambda }, J_{-}^{\lambda }] = -2 J_3^{\lambda }, \; {[}J_{3}^{\lambda }, J_{\pm }^{\lambda }] = \pm J_{\pm }^{\lambda },\\&J_{\pm }^{\lambda }=\frac{1}{4}\left[ \left( x\pm \frac{d}{dx}\right) ^2-\frac{\lambda (\lambda -1)}{x^2}\right] , J_{3}^{\lambda }=\frac{1}{2}H^{\lambda }. \end{aligned} \end{aligned}$$

(4.15)

Here $H^\lambda$ is the PHO as given in remark 4.1.1. The eigenvalue equation reads $H^{\lambda }|n,\lambda \rangle =\left( 2n+\lambda +1/2\right) |n,\lambda \rangle$ with the associated eigenfunctions

$$\begin{aligned} \langle x|n,\lambda \rangle =(-1)^n\sqrt{\frac{2\Gamma (n+1)}{\Gamma (n+\lambda +\frac{1}{2})}} x^{\lambda } e^{-\frac{x^2}{2}} L_n^{(\lambda -\frac{1}{2})}(x^2), \, \lambda >-\frac{1}{2} \end{aligned}$$

which constitute a representation space of the corresponding Fock basis. In addition to the PHO paradigm, the Barut–Girardello CS type under discussion have been considered in [11] under the Laguerre associated function $|2n,\lambda \rangle$ and $|2n+1,\lambda \rangle$ as generalized even and odd coherent states of quantized fields named Wigner cat states in connection to the Wigner-Heisenberg algebra. Other su(1, 1) CSs for some quantum solvable models such as half-oscillator, and radial part of the 3D harmonic oscillator models can be found in [17].

4.3 A Bargmann-type transform

Once we have obtained the closed form of the $x_n^{\gamma ,\sigma }$-NLCSs in (4.6), we can introduce the associated Bargmann-type transform which will make a connection between the Hilbert space $L^2\left( {\mathbb {R}}_{+},d\xi \right)$ of the physical system and the space of coefficients. For this, let us recall that by the general theory of CSs [20, p.75] the scalar product of two $x_n^{\gamma ,\sigma }$-NLCS provides the reproducing kernel for some reproducing kernel Hilbert space (RKHS), denoted here by ${\mathcal {A}}_{\nu ,\sigma }({\mathbb {C}})$. Precisely, the reproducing kernel arising from this $x_n^{\gamma ,\sigma }$-NLCS with the connection parameters $\gamma =\nu /2$, reads

$$\begin{aligned} {\mathcal {K}}(z,{\overline{w}})=\ _{2}F_{3}\left( \begin{array}{c} 1,1 \\ \nu ,\sigma +1,\sigma +1 \end{array} \Bigg \vert z{\bar{w}}\right) . \end{aligned}$$

(4.16)

The corresponding RKHS ${\mathcal {A}}_{\nu ,\sigma }({\mathbb {C}})$ is a subspace of the larger Hilbert space $L^{2}({\mathbb {C}},d\eta _{\nu ,\sigma })$ consisting of functions which are holomorphic in ${\mathbb {C}}$ where the measure $d\eta _{\nu ,\sigma }\left( z,{\bar{z}}\right) :=m_{\frac{\nu }{2},\sigma }(z{\bar{z}})d\varrho (z)$. Explicitly, we have

$$\begin{aligned} d\eta _{\nu ,\sigma }\left( z,{\bar{z}}\right) =\frac{2(z{\bar{z}})^{-1}}{\Gamma (\nu )\Gamma ^{2}\left( \sigma +1\right) }G^{3 0}_{1 3} \left( z{\bar{z}} \ \Bigg \vert \ \begin{array}{c} 1\\ \nu ,\sigma +1,\sigma +1 \end{array} \right) d\varrho (z). \end{aligned}$$

(4.17)

In view of the resolution of the identity, we see that the map ${\mathcal {B}}_{\nu ,\sigma }:L^{2}({\mathbb {R}}_{+},d\xi )\longrightarrow {\mathcal {A}}_{\nu ,\sigma }({\mathbb {C}})$ defined by

$$\begin{aligned} {\mathcal {B}}_{\nu ,\sigma }[\varphi ]\left( z\right) =\left( {\mathcal {N}}_{\frac{\nu }{2},\sigma }\left( z{\bar{z}}\right) \right) ^{1/2}\langle \varphi |z,\nu ,\sigma \rangle _{\beta } \end{aligned}$$

(4.18)

is an isometric isomorphism map, embedding $L^{2}({\mathbb {R}}_{+},d\xi )$ onto the holomorphic subspace ${\mathcal {A}}_{\nu ,\sigma }({\mathbb {C}})\subset L^{2}({\mathbb {C}},d\eta _{\nu ,\sigma })$. We make use of Proposition 4.2.1, in order to express ${\mathcal {B}}_{\nu ,\sigma }$ as an integral transform.

Theorem 4.3.1

Let $\nu \ge 3/2$, $-1<\sigma \le 0$ and $\beta >0$, be fixed parameters. Then, the Bargmann-type transform is the isometric isomorphism map ${\mathcal {B}}_{\nu ,\sigma }:L^{2}({\mathbb {R}}_{+},d\xi )\longrightarrow {\mathcal {A}}_{\nu ,\sigma }({\mathbb {C}})$ defined by means of (4.7) as

$$\begin{aligned} {\mathcal {B}}_{\nu ,\sigma }[\varphi ]\left( z\right) =\sqrt{\frac{2\beta ^{\nu }}{\Gamma (\nu )}}\int _{0}^{\infty }\xi ^{\nu -\frac{1}{2}}e^{-\frac{1}{2}\beta \xi ^{2}}F^{{1:0;0}}_{1:0;1}\left( \begin{array}{c} 1:-;-; \\ \sigma +1;-;\nu ; \end{array} \Bigg \vert -z,\beta z\xi ^2\right) \varphi (\xi )d\xi \nonumber \\ \end{aligned}$$

(4.19)

for every $z\in {\mathbb {C}}$. For $\sigma =0$, it reduces to

$$\begin{aligned} {\mathcal {B}}_{\nu ,0}[\varphi ](z)=\sqrt{2\beta \Gamma (\nu )}(-z)^{\frac{1-\nu }{2}}e^{-z}\int _{0}^{\infty }\xi ^{\frac{1}{2}}e^{-\frac{1}{2}\beta \xi ^2} J_{\nu -1}\left( 2i\xi \sqrt{\beta z}\right) \varphi (\xi )d\xi .\nonumber \\ \end{aligned}$$

(4.20)

With the help of the resolution of the identity (3.7) and the Bargmann type transform (4.18), we can represent any arbitrary function $\varphi$ in $L^{2}({\mathbb {R}}_{+},d\xi )$ in terms of the NLCSs (4.6) as follows:

$$\begin{aligned} \varphi (\cdot )=\int _{{\mathbb {C}}}\left( {\mathcal {N}}_{\frac{\nu }{2},\sigma }\left( z{\bar{z}}\right) \right) ^{-\frac{1}{2}}\overline{{\mathcal {B}}_{\nu ,\sigma }[\varphi ](z)}\langle \cdot |z,\nu ,\sigma \rangle _{\beta }d\eta _{\nu ,\sigma }(z,{\bar{z}}). \end{aligned}$$

(4.21)

Therefore, the norm square of $\varphi$ also reads

$$\begin{aligned} \langle \varphi |\varphi \rangle _{L^{2}({\mathbb {R}}_{+},d\xi )}=\int _{{\mathbb {C}}}|{\mathcal {B}}_{\nu ,\sigma }[\varphi ](z)|^{2}d\eta _{\nu ,\sigma }(z,{\bar{z}}). \end{aligned}$$

(4.22)

The careful reader has certainly recognized in (4.20) the Bessel transform [45]. As a consequence we have the following result.

Corollary 4.3.1

Let $\nu >3/2$, $-1<\sigma \le 0$ and $\beta >0$, be fixed parameters. The following integral

$$\begin{aligned}&\int _{0}^{\infty }\xi ^{2\nu -1}e^{-\beta \xi ^{2}}F^{{1:0;0}}_{1:0;1}\left( \begin{array}{c} 1:-;-; \\ \sigma +1;-;\nu ; \end{array} \Bigg \vert z,-\beta \xi ^2z\right) L_n^{(\nu -1)}(\beta \xi ^2)d\xi \nonumber \\&\quad =\frac{\Gamma (\nu )}{2\beta ^{\nu }} \frac{z^n}{\left( \sigma +1\right) _n} \end{aligned}$$

(4.23)

holds true for every $z\in {\mathbb {C}}$. When $\sigma =0$, it reduces to

$$\begin{aligned} \int _{0}^{\infty }\xi ^{\nu }e^{-\beta \xi ^2} J_{\nu -1}\left( 2\xi \sqrt{\beta z}\right) L_n^{(\nu -1)}(\beta \xi ^2)d\xi =\frac{z^{n+\frac{\nu -1}{2}}}{n!}\frac{e^{-z}}{2\beta ^{\frac{\nu +1}{2}}}. \end{aligned}$$

(4.24)

Proof

According to (4.18), the range of the basis vector $\left\{ \psi _n^{\nu ,\beta }\right\}$ under the transform ${\mathcal {B}}_{\nu ,\sigma }$ should exactly be the coefficients in (4.6) with the parameter $2\gamma =\nu$. More precisely,

$$\begin{aligned} {\mathcal {B}}_{\nu ,\sigma }[\psi _n^{\nu ,\beta }]({\bar{z}})=\sqrt{\frac{n!}{(\nu )_n}}\frac{z^n}{\left( \sigma +1\right) _n}. \end{aligned}$$

(4.25)

On the other hand, applying the transform (4.19) to $\psi _n^{\nu ,\beta }$ one obtains

$$\begin{aligned} {\mathcal {B}}_{\nu ,\sigma }[\psi _n^{\nu ,\beta }]({\bar{z}})= & {} \frac{2\beta ^{\nu }(-1)^n}{\Gamma (\nu )}\sqrt{\frac{n!}{(\nu )_n}} \int _{0}^{\infty }\xi ^{2\nu -1}e^{-\beta \xi ^{2}}\nonumber \\&F^{{1:0;0}}_{1:0;1}\left( \begin{array}{c} 1:-;-; \\ \sigma +1;-;\nu ; \end{array} \Bigg \vert -z,\beta z\xi ^2\right) L_n^{(\nu -1)}(\beta \xi ^2)d\xi . \end{aligned}$$

(4.26)

Then, matching (4.25) and (4.26) results to the announced formula (4.23). Now, taking $\sigma =0$ in (4.23) it follows the formula (4.24) using (4.14). This ends the proof. $\square$

Remark 4.3.1

Note that (4.24) can be obtained by using the formula [44, p.65]:

$$\begin{aligned} \begin{aligned}&\int _{0}^{\infty }x^{\frac{1}{2}+\mu }\exp \left( -a^2x^2\right) L^{(\mu )}_n(a^2x^2)(xy)^{1/2}J_{\mu }(xy)dx\\&= \, \frac{y^{2n+\mu +1/2}}{n!2^{2n+\mu +1}} \frac{\exp \left( -\frac{y^2}{4a^2}\right) }{a^{2(\mu +n+1)}}, \end{aligned} \end{aligned}$$

(4.27)

with $\mu >-1$, for parameters $x=\xi$, $y=2\sqrt{\beta z}$, $a=\sqrt{\beta }$ and $\mu =\nu -1$.

5 Conclusion

In this work, we have constructed a new two-parameters family of nonlinear coherent states (NLCSs) for the pseudoharmonic oscillator (PHO) by replacing n! occurring in the coefficients of the canonical coherent states (CSs) with a generalized factorial $x_{n}^{\gamma ,\sigma }!$ and the Fock basis with the eigenfunctions of the PHO, denoted $\left| \psi _n^{\nu ,\beta }\right\rangle$. The latter constitute an orthonormal basis of the Hilbert space $L^2({\mathbb {R}}_+,d\xi )$. The obtained $x_n^{\gamma ,\sigma }$-NLCSs reduce to both Barut–Girardello CSs and Philophase states for particular values of $\gamma$ and $\sigma$. The positive measure in order to realize the resolution of the identity of this $x_n^{\gamma ,\sigma }$-NLCSs have been proved for $(\gamma ,\sigma )\in ]0,+\infty [ \times ]-1,0] \cup ]0,1/2] \times ]-1,+\infty [$, and also its corresponding explicit compact form have been exactly calculated while setting $2\gamma =\nu \ge \frac{3}{2}$ . Moreover, we have introduced a new integral transformation of Bargmann type establishing an isometric isomorphism map between the space of coefficients $\{{\bar{z}}^n/\sqrt{x_n^{\gamma ,\sigma }!}\}_{n=0}^{\infty }$ and that of the PHO i.e. $L^2({\mathbb {R}}_+,d\xi )$. Connections with other relevant works have been highlighted throughout the paper. As is well known, coherent states for any quantum mechanical system are the quantum mechanical states that have the property that the peak of the modulus square of their wavefunctions follows the classical trajectory without changing the form. It will be interesting to consider such a goal for the wavefunctions (4.7) or their particular case (4.8). One can achieve this with the help of the reference [1].

Appendices

Proof of the proposition 3.1.1

The inner product of two $x_n^{\gamma ,\sigma }$-NLCSs expressed in (3.3) is given by

$$\begin{aligned} \left\langle z,\gamma ,\sigma |w,\gamma ,\sigma \right\rangle= & {} \left( {\mathcal {N}}_{\gamma ,\sigma }\left( z{\bar{z}}\right) {\mathcal {N}}_{\gamma ,\sigma }\left( w{\overline{w}}\right) \right) ^{-1/2}\ _{2}F _{3}\left( \begin{array}{c} 1,1 \\ 2\gamma ,\sigma +1,\sigma +1 \end{array} \Bigg \vert z{\bar{w}}\right) .\nonumber \\ \end{aligned}$$

(A.1)

If $\sigma +1$ is zero or a negative integer, the function $\ _{2}F _{3}(\cdot )$ is not defined, since all but a finite number of the series terms become infinite. Then, $\sigma \ne -1,-2,-3,\ldots$ have to be satisfied. The normalization is deduced from (A.1) by taking $z = w$ such that $\left\langle z,\gamma ,\sigma |z,\gamma ,\sigma \right\rangle = 1$. Now, we take $\gamma =1/2$ and $\sigma \in {\mathbb {N}}^{*}$ in (3.4) then

$$\begin{aligned} {\mathcal {N}}_{\frac{1}{2},\sigma }(z{\bar{z}})= & {} {}_{1}F_{2}\left( \begin{array}{c} 1 \\ \sigma +1,\sigma +1 \end{array} \Bigg \vert z{\bar{z}}\right) \end{aligned}$$

(A.2)

$$\begin{aligned}= & {} \frac{\Gamma ^2(\sigma +1)}{|z|^{2\sigma }}\sum _{n\ge \sigma }\frac{|z|^{2n}}{\left( n!\right) ^2} \end{aligned}$$

(A.3)

$$\begin{aligned}= & {} \left[ I_{0}(2|z|)-\sum _{n=0}^{\sigma -1}\frac{|z|^{2n}}{(n!)^2}\right] \frac{\Gamma ^2(\sigma +1)}{|z|^{2\sigma }}. \end{aligned}$$

(A.4)

For $\sigma =0$, Eq.(3.4) becomes

$$\begin{aligned} {\mathcal {N}}_{\gamma ,0}(z{\bar{z}})=\ _{0}F _{1}\left( \begin{array}{c} - \\ 2\gamma \end{array} \Bigg \vert z{\bar{z}}\right) \end{aligned}$$

where the confluent hypergeometric function $_{0}F_{1}(\cdot )$ can be written [58, p.44]:

$$\begin{aligned} I_{\nu }(\xi )=\frac{\left( \frac{1}{2}\xi \right) ^{\nu }}{\Gamma (\nu +1)}\ _{0}F _{1}\left( \begin{array}{c} - \\ \nu +1 \end{array} \Bigg \vert \frac{\xi ^2}{4}\right) . \end{aligned}$$

(A.5)

For parameters $\nu =2\gamma -1$ and $\xi =2|z|$, we obtain (3.6). $\square$

Positive solutions of the moment problem (3.11)

Here, we construct positive solutions of the moment problem by the help of the Mellin and Mellin inverse transforms [9]. We start from the formula in [15, p.337]:

$$\begin{aligned}&\int _{0}^{\infty }x^{s-1}G^{m, l}_{p, q} \left( x \ \Bigg \vert \ \begin{array}{c} a_1,\cdots ,a_p\\ b_1,\cdots ,b_q \end{array} \right) dx\nonumber \\&=\frac{ \prod \nolimits _{j=1}^{m}\Gamma (b_j+s)\prod \nolimits _{j=1}^{l}\Gamma (1-a_j-s)}{ \prod \nolimits _{j=l+1}^{p}\Gamma (a_j+s)\prod \nolimits _{j=m+1}^{q}\Gamma (1-b_j-s)} \end{aligned}$$

(B.1)

involving Meijer’s G-function with conditions

$$\begin{aligned} 0\le m\le q,\ 0\le l\le p,\ p+q<2(m+l), \end{aligned}$$

(B.2)

$$\begin{aligned} -\min \limits _{1\le j\le m}\mathfrak {R}(b_j)<\mathfrak {R}(s)<1-\max \limits _{1\le k\le l}\mathfrak {R}(a_k). \end{aligned}$$

(B.3)

We often refer to the latter as $G_{p,q}^{m,l}(x)$. Here, an empty product is interpreted as unity. It is also understood that if $l=0$, that part of (B.3) involving $1-\max \nolimits _{1\le k\le l}\mathfrak {R}(a_k)$ is treated as empty. Thus (B.3) becomes $-\min \nolimits _{1\le j\le m}\mathfrak {R}(b_j)<\mathfrak {R}(s)$. For $x=r$ and parameters $m=3, \ l=0, \ p=1, \ q=3$, $a_1=1,\ b_1=2\gamma ,\ b_2=b_3=\sigma +1$ and $s=n$, equation (B.1) reduces to

$$\begin{aligned} \int _{0}^{\infty }r^{n-1}G^{3 0}_{1 3} \left( r \ \Bigg \vert \ \begin{array}{c} 1\\ 2\gamma ,\sigma +1,\sigma +1 \end{array} \right) dr=\frac{\Gamma (n+2\gamma )\Gamma ^2(n+\sigma +1)}{n!} \end{aligned}$$

(B.4)

that becomes

$$\begin{aligned} \int _{0}^{\infty }\frac{2r^{n-1}}{\Gamma (2\gamma )\Gamma ^2(\sigma +1)}G^{3 0}_{1 3} \left( r \ \Bigg \vert \ \begin{array}{c} 1\\ 2\gamma ,\sigma +1,\sigma +1 \end{array} \right) dr=\frac{2(2\gamma )_n(\sigma +1)^2_n}{n!}.\qquad \end{aligned}$$

(B.5)

Comparing (B.5) to (3.11) we obtain the searched weight function

$$\begin{aligned} m_{\gamma ,\sigma }(r)=\frac{2r^{-1}}{\Gamma (2\gamma )\Gamma ^2(\sigma +1)}G^{3 0}_{1 3} \left( r \ \Bigg \vert \ \begin{array}{c} 1\\ 2\gamma ,\sigma +1,\sigma +1 \end{array} \right) ,\ r>0. \end{aligned}$$

(B.6)

We still need to prove the positivity of $m_{\gamma ,\sigma }(r)$, that remains to prove the positivity of $G^{3 0}_{1 3}\left( r\right)$ appearing in its expression (B.6). To accomplish this, we comeback to (B.4) which can be seen as Mellin transform of $G^{3 0}_{1 3}\left( r\big \vert \begin{array}{c} 1\\ 2\gamma ,\sigma +1,\sigma +1 \end{array} \right)$. Then, the inverse Mellin transform of $\frac{\Gamma (n+2\gamma )\Gamma ^2(n+\sigma +1)}{n!}$ reads

$$\begin{aligned} G^{3 0}_{1 3} \left( r \ \Bigg \vert \ \begin{array}{c} 1\\ 2\gamma ,\sigma +1,\sigma +1 \end{array} \right) = \frac{1}{2\pi i}\int _{-i\infty }^{+i\infty }\frac{\Gamma (n+2\gamma )\Gamma ^2(n+\sigma +1)}{n!}r^{-n}dn.\nonumber \\ \end{aligned}$$

(B.7)

Now, according to the Mellin convolution property of inverse Mellin transform [9, p.559] also called the generalized Parseval formula, if for arbitrary $f^{*}(n)$ and $g^{*}(n)$ there exists an inverse Mellin transform f(r) and g(r) for $r>0$, respectively, then

$$\begin{aligned} W(r):=\int _{0}^{\infty }f\left( \frac{r}{t}\right) g(t)\frac{1}{t}dt=\frac{1}{2\pi i}\int _{-i\infty }^{+i\infty }f^{*}(n)g^{*}(n)r^{-n}dn, \quad r>0. \end{aligned}$$

(B.8)

Note that if both f(x) and g(x) in (B.8) are positive for $x>0$, then W(x) is also positive for $x>0$. Here, the choice of function $f^*(n)$ and $g^*(n)$ will yield restrictions on the parameters $\gamma$ and $\sigma$. There are four interesting cases that can be regrouped into the two following cases:

• Case 1:

  $$\begin{aligned} f^{*}(n)=\frac{\Gamma (n+\mathbf{a })\Gamma (n+\mathbf{b })}{n!}, \qquad g^{*}(n)=\Gamma (n+\mathbf{c }); \end{aligned}$$

  (B.9)

• Case 2:

  $$\begin{aligned} f^{*}(n)=\frac{\Gamma (n+\mathbf{a} )}{n!},\qquad g^{*}(n)=\Gamma (n+\mathbf{b} )\Gamma (n+\mathbf{c} ). \end{aligned}$$

  (B.10)

Thus, the regrouped case 1 consists of the two cases $\mathbf{a} =\mathbf{b} =\sigma +1$, $\mathbf{c} =2\gamma$ and $\mathbf{a} =2\gamma$, $\mathbf{b} =\mathbf{c} =\sigma +1$, while the regrouped case 2 consists of the two cases $\mathbf{a} =2\gamma$, $\mathbf{b} =\mathbf{c} =\sigma +1$ and $\mathbf{a} =\mathbf{c} =\sigma +1$, $\mathbf{b} =2\gamma$. Note first that if one of parameters $\mathbf{a},\ b$ or $\mathbf{c}$ is equal to 1, then the Meijer’s function $G^{3 0}_{1 3}\left( r\big \vert \begin{array}{c} 1\\ \mathbf{a} ,\mathbf{b} ,\mathbf{c} \end{array} \right)$ reduces to (for example $\mathbf{a} =1$) [38, p.61]:

$$\begin{aligned} G^{2 0}_{0 2}\left( r\big \vert \begin{array}{c} -\\ \mathbf{b },\mathbf{c } \end{array} \right) = 2r^{\frac{\mathbf{b }+\mathbf{c }}{2}} K_{\mathbf{c }-\mathbf{b }}(2\sqrt{r}),\ \ \ r>0, \end{aligned}$$

(B.11)

which is a positive function (see (2.9)). Otherwise, we apply the property (B.8) with $f^*(n)$, $g^*(n)$ as given in (B.9) and (B.10) to the equation (B.7), to get the following integral representations.

Lemma 1

Two integral representations of the Meijer’s function $G^{3 0}_{1 3}$ are given by

$$\begin{aligned} G^{3 0}_{1 3} \left( r \ \Bigg \vert \ \begin{array}{c} 1\\ a, b, c \end{array} \right) = \int _0^{\infty }t^{a-1}e^{-t}G^{2 0}_{1 2} \left( \frac{r}{t}\ \Bigg \vert \ \begin{array}{c} 1\\ b,c \end{array} \right) dt , \end{aligned}$$

(B.12)

valid for $\mathfrak {R}(a),\mathfrak {R}(b),\mathfrak {R}(c)>0$, and

$$\begin{aligned} G^{3 0}_{1 3} \left( r \ \Bigg \vert \ \begin{array}{c} 1\\ a,b,c \end{array} \right) =\frac{2r^{\frac{a+c}{2}}}{\Gamma (1-b)}\int _1^{\infty }t^{\frac{a+c}{2}-1}(t-1)^{-b}K_{c-a}\left( 2\sqrt{rt}\right) dt, \end{aligned}$$

(B.13)

valid for $0<\mathfrak {R}(b)<1$ and $\mathfrak {R}(a),\mathfrak {R}(c)>0$.

We will use just one of the above formulas since (B.13) can be deduced from (B.12) and vice-versa². We also need the following result.

Lemma 2

Suppose $\mathfrak {R}(\alpha -\lambda )>0$. Then the function

$$\begin{aligned} t \mapsto G^{2 0}_{1 2} \left( t \ \Bigg \vert \ \begin{array}{c} \alpha \\ \beta ,\lambda \end{array} \right) \end{aligned}$$

(B.14)

is positive on $(0,+\infty )$.

Thus, positivity of the Meijer’s function $G^{3 0}_{1 3}(r)$ in (B.6) follows from (B.11), (B.12) and Lemma 2 under conditions $\mathfrak {R}(a),\mathfrak {R}(b)>0$ and $0<\mathfrak {R}(c)<1$. As the Meijer G-function $G_{1,3}^{3,0}(x)$ is symmetric with respect to the elements a, b, c, we finally conclude that

$$\begin{aligned} r\mapsto G^{3 0}_{1 3}\left( r\big \vert \begin{array}{c} 1\\ 2\gamma ,\sigma +1,\sigma +1 \end{array} \right) , \ r>0, \end{aligned}$$

(B.15)

is positive for $(\gamma ,\sigma )\in S_1 \cup S_2$, where $S_1= ]0,+\infty [ \times ]-1,0]$ and $S_2= ]0,1/2 ] \times ]-1,+\infty [$.

We close this appendix by giving the proofs of the Lemma 1 and Lemma 2.

Proof of lemma 1

The integral representations of $G^{3, 0}_{1, 3}(r)$ in (B.12) and (B.13) are obtained by choosing $f^*(n)$ and $g^*(n)$ as given in case 1 (B.9) and case 2 (B.10), respectively. In the former case functions f(r) and g(r) are identified from (B.1) as

$$\begin{aligned} f(r)=G^{2 0}_{1 2} \left( r \ \Bigg \vert \ \begin{array}{c} 1\\ \mathbf{a} ,\mathbf{b} \end{array} \right) , \quad \mathfrak {R}(\mathbf{a} ),\mathfrak {R}(\mathbf{b} )>0 \end{aligned}$$

(B.16)

and

$$\begin{aligned} g(r)=r^\mathbf{c }e^{-r},\quad \mathfrak {R}(\mathbf{c} )>0, \end{aligned}$$

(B.17)

while in the latter case, they are identified from [44, pp.195-196] as

$$\begin{aligned} f(r)=\left( \Gamma (1-\mathbf{a} )\right) ^{-1}r^\mathbf{a }(1-r)^{-\mathbf{a} }\chi _{]0, 1[}(r),\quad 0<\mathfrak {R}(\mathbf{a} )<1 \end{aligned}$$

(B.18)

and

$$\begin{aligned} g(r)=2r^{\frac{1}{2}(\mathbf{b} +\mathbf{c} )}K_\mathbf{b -\mathbf{c} }\left( 2\sqrt{r}\right) ,\quad \mathfrak {R}(\mathbf{b} ),\mathfrak {R}(\mathbf{c} )>0 \end{aligned}$$

(B.19)

with $\chi _{]0, 1[}$ standing for the indicator function. This completes the proof. $\square$

Proof of lemma 2

We use the formula [38, p.62]

$$\begin{aligned} G^{2 0}_{1 2} \left( z \ \Bigg \vert \ \begin{array}{c} \alpha \\ \beta ,\lambda \end{array} \right) =z^{\frac{\beta +\lambda -1}{2}}e^{-\frac{z}{2}}W_{\frac{1+\beta +\lambda }{2}-\alpha ,\frac{\beta -\lambda }{2}}(z) \end{aligned}$$

(B.20)

where $W_{\nu ,\mu }(\cdot )$ is the Whittaker function and its integral representation [25, p.1025]

$$\begin{aligned} W_{\nu ,\mu }(z)=\frac{z^{\mu +\frac{1}{2}}e^{-\frac{z}{2}}}{\Gamma (\mu -\nu +\frac{1}{2})}\int _{0}^{\infty }e^{-s z}s^{\mu -\nu -\frac{1}{2}}\left( 1+s\right) ^{\mu +\nu -\frac{1}{2}}ds, \end{aligned}$$

(B.21)

under the conditions $|\arg z|<\frac{\pi }{2}$ and $\mathfrak {R}(\mu -\nu )>-\frac{1}{2}$ for $\nu =\frac{1+\beta +\lambda }{2}-\alpha$ and $\mu =\frac{\beta -\lambda }{2}$, to write the integral representation of the Meijer’s function $G^{2 0}_{1 2}(\cdot )$ as follow

$$\begin{aligned} G^{2,0}_{1 ,2} \left( z \ \Bigg \vert \ \begin{array}{c} \alpha \\ \beta ,\lambda \end{array} \right) =\frac{z^{\beta }e^{-z}}{\Gamma (\alpha -\lambda )} \int _{0}^{\infty }e^{-s z}s^{\alpha -\lambda -1}\left( 1+s\right) ^{\beta -\alpha }ds, \end{aligned}$$

(B.22)

valid for $|\arg z|<\frac{\pi }{2}$ and $\mathfrak {R}(\alpha -\lambda )>0$. Then, the positivity of the Meijer G-function in (B.22) follows from the fact that the function $s\mapsto e^{-s z}s^{\alpha -\lambda -1}\left( 1+s\right) ^{\beta -\alpha }$ is positive on $(0,+\infty )$. This ends the proof. $\square$