R(p,q)- analogs of discrete distributions: general formalism and application

TL;DR

This paper introduces a unified framework for $ (p,q)$-deformed discrete distributions, extending classical distributions and linking them to quantum algebra applications, with new formulas for moments, mean, variance, and recursion relations.

Contribution

It develops a general formalism for $ (p,q)$-deformed distributions, deriving new properties and connecting to quantum algebra, unifying and extending previous results.

Findings

Derived $ (p,q)$-deformed factorial moments and distribution formulas.

Established recursion relations for $ (p,q)$-distributions.

Connected $ (p,q)$-deformations to quantum algebra applications.

Abstract

In this paper, we define and discuss $\mathcal{R}(p,q)$- deformations of basic univariate discrete distributions of the probability theory. We mainly focus on binomial, Euler, P\'olya and inverse P\'olya distributions. We discuss relevant $\mathcal{R}(p,q)-$ deformed factorial moments of a random variable, and establish associated expressions of mean and variance. Futhermore, we derive a recursion relation for the probability distributions. Then, we apply the same approach to build main distributional properties characterizing the generalized $q-$ Quesne quantum algebra, used in physics. Other known results in the literature are also recovered as particular cases.

Equations

$$\mathcal{R}(u,v)=\sum _{s,t=-l}^{\infty }{r}_{st}{u}^s{v}^t,$$

$$[n{]}_{\mathcal{R}(p,q)}:=\mathcal{R}(p^n,q^n),n\in \mathbb{N}$$

$$\mathcal{R}(p,q):=\mathcal{R}(1,q)=\frac{\{p=1\}-q}{1-q}\text{mbox}and[n{]}_q=\frac{1-q^n}{1-q}.$$

$$\mathcal{R}(p,q):=\mathcal{R}(1,q)=\frac{\{p=1\}-q^{-1}}{1-q}\text{mbox}and[n{]}_q^Q=\frac{1-q^{-n}}{q-1}.$$

$$\mathcal{R}(p,q)=\frac{p-q}{p-q}\text{mbox}and[n{]}_{p,q}=\frac{p^n-q^n}{p-q}.$$

$$\mathcal{R}(p,q)=\frac{1-pq}{(p^{-1}-q)p}\text{mbox}and[n{]}_{p,q}=\frac{p^{-n}-q^n}{p^{-1}-q}.$$

$$\mathcal{R}(p,q)=\frac{pq-1}{(q-p^{-1})q}\text{mbox}and[n{]}_{p,q}^Q=\frac{p^n-q^{-n}}{q-p^{-1}}.$$

$$\mathcal{R}(p,q)=g(p,q)\frac{q^{\nu }}{p^{\mu }}\frac{pq-1}{(q-p^{-1})q}\text{mbox}and[n{]}_{p,q,g}^{\mu ,\nu }=g(p,q)\frac{q^{\nu n}}{p^{\mu n}}\frac{p^n-q^{-n}}{q-p^{-1}},$$

$$[n]!_{\mathcal{R}(p,q)}:=\left\{\begin{array}[]{l}1\quad\mbox{for}\quad n=0\\ \\ \mathcal{R}(p,q)\cdots\mathcal{R}(p^{n},q^{n})\quad\mbox{for}\quad n\geq 1,\end{array}\right.$$

$$\displaystyle\bigg{[}\begin{array}[]{c}m\\ n\end{array}\bigg{]}_{\mathcal{R}(p,q)}:=\frac{[m]!_{\mathcal{R}(p,q)}}{[n]!_{\mathcal{R}(p,q)}[m-n]!_{\mathcal{R}(p,q)}},\quad m,n=0,1,2,\cdots,\quad m\geq n$$

$$\bigg{[}\begin{array}[]{c}m\\ n\end{array}\bigg{]}_{\mathcal{R}(p,q)}=\bigg{[}\begin{array}[]{c}m\\ m-n\end{array}\bigg{]}_{\mathcal{R}(p,q)},\quad m,n=0,1,2,\cdots,\quad m\geq n.$$

$$Q:\Psi ⟼Q\Psi (z):$$

$$P:\Psi ⟼P\Psi (z):$$

$${\partial }_{\mathcal{R}(p,q)}:={\partial }_{p,q}\frac{p-q}{P-Q}\mathcal{R}(P,Q)=\frac{p-q}{p^P-q^Q}\mathcal{R}(p^P,q^Q){\partial }_{p,q}$$

$$\mathcal{R}(p,q):=\mathcal{R}(1,q)=1\text{mbox}and{\partial }_q\Psi (z)=\frac{\Psi (z)-\Psi (qz)}{z(1-q)}.$$

$$\mathcal{R}(p,q):=\mathcal{R}(1,q)=\frac{1-q^{-1}}{1-q}\text{mbox}and{\partial }_q\Psi (z)=\frac{\Psi (z)-\Psi (q^{-1}z)}{z(q-1)}.$$

$$\mathcal{R}(p,q)=1\text{mbox}and{\partial }_{p,q}\Psi (z)=\frac{\Psi (pz)-\Psi (qz)}{z(p-q)}.$$

$$\mathcal{R}(p,q)=\frac{1-pq}{(p^{-1}-q)p}\text{mbox}and{\partial }_{p^{-1},q}\Psi (z)=\frac{\Psi (p^{-1}z)-\Psi (qz)}{z(p^{-1}-q)}.$$

$$\mathcal{R}(p,q)=\frac{pq-1}{(q-p^{-1})q}\text{mbox}and{\partial }_{p,q}^Q\Psi (z)=\frac{\Psi (pz)-\Psi (q^{-1}z)}{z(q-p^{-1})}.$$

$$\displaystyle\,\mathcal{R}(p,q)=g(p,q){q^{\nu}\over p^{\mu}}{p\,q-1\over(q-p^{-1})q}\quad\mbox{and}\quad\partial^{\mu,\nu}_{p,q}\varPsi(z)=g(p,q)\frac{\varPsi\Big{(}{zq^{\nu}\over p^{\mu-1}}\Big{)}-\varPsi\Big{(}{zq^{\nu-1}\over p^{\mu}}\Big{)}}{z(q-p^{-1})}.$$

$$A{A}^{†}=[N+1{]}_{\mathcal{R}(p,q)},A^{†}A=[N{]}_{\mathcal{R}(p,q)}.$$

$$\left[N,A\right]=-A,\left[N,A^{†}\right]=A^{†}$$

$$A^{†}:=z,A:={\partial }_{\mathcal{R}(p,q)},N:=z{\partial }_z,$$

$$[n]^{Q}_{p,q}={p^{n}-q^{-n}\over q-p^{-1}},\quad[n]!^{Q}_{p,q}:=\left\{\begin{array}[]{l}1\quad\mbox{for}\quad n=0\\ \\ [1]^{Q}_{p,q}\cdots[n]^{Q}_{p,q}\quad\mbox{for}\quad n\geq 1,\end{array}\right.$$

$$[-y{]}_{p,q}^Q$$

$$[x+y{]}_{p,q}^Q$$

$$[x-y{]}_{p,q}^Q$$

$$[x{]}_{p,q}^Q$$

$$\displaystyle\bigg{[}\begin{array}[]{c}n\\ \kappa\end{array}\bigg{]}_{p,q}^{Q}=\frac{[n]^{Q}_{p,q}!}{[\kappa]^{Q}_{p,q}![n-\kappa]^{Q}_{p,q}!},\quad 0\leq\kappa\leq n\in\mathbb{N},$$

$$\displaystyle\bigg{[}\begin{array}[]{c}n\\ k\end{array}\bigg{]}_{p,q}^{Q}$$

$$\displaystyle\;\left[\begin{array}[]{c}n+1\\ k\end{array}\right]_{p,q}^{Q}$$

$$\displaystyle\;\bigg{[}\begin{array}[]{c}n+1\\ k\end{array}\bigg{]}_{p,q}^{Q}$$

$$p^{-1}A{A}^{†}-A^{†}A=q^{-N-1},$$

$$[N,A^{†}]=A^{†},$$

Full text

• Corresponding author

1. Introduction

Deformed quantum algebras, namely $q-$ deformed algebras [17, 21, 22] and their extensions to $(p,q)-$analogs [4, 5], continue to attract much attention in mathematics and physics. Inspired by the $q-$ deformed quantum algebras, Chung and Kang developed a notion of $q-$ permutations and $q-$ combinations [9]. The $q-$ combinatorics and $q-$ hypergeometric series were examined in [1, 3, 8, 19]. A theory of $(p,q)-$ analogs of binomial coefficients and $(p,q)-$ Stirling numbers was elaborated in [10, 23]. In the same vein, from the $q-$ combinatorics, several authors investigated the $q-$ discrete probability distributions such as $q-$ binomial, $q-$ Euler, $q-$ hypergeometric, $q-$ Pólya, $q-$ contagious, and $q-$ uniform distributions [7, 18, 20]. Recently, our research group was interested in the study of quantum algebras, and, more especially, in the generalization of the well-known $(p,q)-$ Heisenberg algebras. From Odzijewicz’s work [21], we introduced the $\mathcal{R}(p,q)-$ deformed quantum algebras as generalizations of known deformed quantum algebras [13]. In [12], we performed the $\mathcal{R}(p,q)-$ differentiation and integration, and deducted all relevant particular cases of $q-$ and $(p,q)-$ deformations. This opens a novel route for developing the theory of $\mathcal{R}(p,q)-$ analogs of special numbers, combinatorics, and probability distributions. Then, the following question naturally arises: How to construct discrete probability distributions and deduce their properties from $\mathcal{R}(p,q)-$ deformed quantum algebras?

This paper is organized as follows: In section $2,$ we briefly recall the notions of $\mathcal{R}(p,q)-$ deformed numbers, $\mathcal{R}(p,q)-$ deformed factorial, $\mathcal{R}(p,q)-$binomial coefficients, $\mathcal{R}(p,q)-$ derivative, relevant properties from our generalization of $q-$ Quesne algebra, and some basic notations and definitions. Section $3$ is focused on main results. $\mathcal{R}(p,q)-$ deformed combinatorics and $\mathcal{R}(p,q)-$ analogs of discrete distributions are defined. The mean value, or expectation, and variance of random variables of distributions are computed. Then, we apply the same approach to build main distributional properties characterizing the generalized $q-$ Quesne quantum algebra, used in physics. Other known results are also recovered as particular cases. We end with some concluding remarks in Section $4.$

2. Quick overlook on known results

2.1. $\mathcal{R}(p,q)-$ deformed numbers

Consider $p$ and $q,$ two positive real numbers such that $0<q<p<1,$ and a given meromorphic function defined on $\mathbb{C}\times\mathbb{C}$ by:

[TABLE]

where $r_{st}$ are complex numbers, $l\in\mathbb{N}\backslash\left\{0\right\},$ $\mathcal{R}(p^{n},q^{n})>0,\forall n\in\mathbb{N},$ and $\mathcal{R}(1,1)=0.$ We denote by $\mathbb{D}_{R}=\left\{z\in\mathbb{C}:|z|<R\right\}$ a complex disc, where $R$ is the radius of convergence of the series (2.1), and by $\mathcal{O}(\mathbb{D}_{R})$ the set of holomorphic functions defined on $\mathbb{D}_{R}.$ Define the $\mathcal{R}(p,q)-$ deformed numbers by [13]:

[TABLE]

generalizing known particular numbers as follows:

• •

$q-$ Arick-Coon-Kuryskin deformation [2]

[TABLE]

• •

$q-$ Quesne deformation [22]

[TABLE]

• •

$(p,q)-$ Jagannathan-Srinivasa deformation [16]

[TABLE]

• •

$(p^{-1},q)-$ Chakrabarty - Jagannathan deformation [5]

[TABLE]

• •

Hounkonnou-Ngompe generalization of $q-$ Quesne deformation [14]

[TABLE]

• •

$(p,q,\mu,\nu,g)-$ Hounkonnou-Ngompe multi-parameter deformation [15]

[TABLE]

where $0<pq<1,$ $p^{\mu}<q^{\nu-1},$ $p>1$ and $g$ is a well behaved real non-negative function of deformation parameters $p$ and $q$ such that $g(p,q)\longrightarrow 1$ as$(p,q)\longrightarrow(1,1)$.

Define now the $\mathcal{R}(p,q)-$ deformed factorials

[TABLE]

and the $\mathcal{R}(p,q)-$ deformed binomial coefficients

[TABLE]

satisfying the relation

[TABLE]

Consider the following linear operators defined on $\mathcal{O}(\mathbb{D}_{R})$ by (see [13] for more details):

[TABLE]

and the $\mathcal{R}(p,q)-$ derivative given by:

[TABLE]

extending known derivatives as follows:

• (i)

$q-$ Heine derivative [11].

[TABLE]

• (ii)

$q-$ Quesne derivative [22].

[TABLE]

• (iii)

$(p,q)-$ Jagannathan-Srinivasa derivative [16]

[TABLE]

• (iv)

$(p^{-1},q)-$ Chakrabarty - Jagannathan derivative [5].

[TABLE]

• (v)

Hounkonnou-Ngompe generalization of $q-$ Quesne derivative [14].

[TABLE]

• (vi)

$(p,q,\mu,\nu,g)-$ Hounkonnou-Ngompe multi-parameter derivative [15].

[TABLE]

The algebra associated with the $\mathcal{R}(p,q)-$ deformation is a quantum algebra, denoted $\mathcal{A}_{\mathcal{R}(p,q)},$ generated by the set of operators $\{1,A,A^{\dagger},N\}$ satisfying the following commutation relations:

[TABLE]

with its realization on ${\mathcal{O}}(\mathbb{D}_{R})$ given by:

[TABLE]

where $\partial_{z}:=\frac{\partial}{\partial z}$ is the usual derivative on $\mathbb{C}.$

2.2. Hounkonnou-Ngompe generalization of $q-$ Quesne algebra

It corresponds to $\mathcal{R}(p,q)={p-q^{-1}\over q-p^{-1}}$, with generalized $q-$Quesne number and factorial [14]:

[TABLE]

and following properties:

[TABLE]

The generalized $q-$Quesne binomial coefficient

[TABLE]

satisfies the relations

[TABLE]

Finally, the algebra $\mathcal{A}^{Q}_{p,q}$, generated by the set $\{1,A,A^{\dagger},N\}$, associated with the generalized $q-$Quesne deformation, satisfies the following commutation relations:

[TABLE]

We also have the following identities used in the sequel:

[TABLE]

[TABLE]

where $\mu,$ $\nu,$ $p,$ and $q$ are such that $0<pq<1,$ $p^{\mu}<q^{\nu-1},$ $p>1.$ $g$ is a well behaved real non-negative function of deformation parameters $p$ and $q$ such that $g(p,q)\longrightarrow 1$ as $(p,q)\longrightarrow(1,1)$.

3. Main results

3.1. $\mathcal{R}(p,q)-$ deformed combinatorics

Given the above results, we now introduce functions $\epsilon_{i}>0,$ with $i=1,2,$ characterizing particular deformations as follows:

• (i)

$q-$ Arick-Coon-Kuryskin deformation

[TABLE]

• (ii)

$q-$ Quesne deformation

[TABLE]

• (iii)

$(p,q)-$ Jagannathan-Srinivasa deformation

[TABLE]

• (iv)

$(p^{-1},q)-$ Chakrabarty -Jagannathan deformation

[TABLE]

• (v)

Hounkonnou-Ngompe generalization of $q-$ Quesne deformation

[TABLE]

• (vi)

$(p,q,\mu,\nu,g)-$ Hounkonnou-Ngompe multi-parameter deformation

[TABLE]

The $j^{th}-$ order factorial of the $\mathcal{R}(p,q)-$ number is given by :

[TABLE]

where $n\in\mathbb{N}$ and $j\in\mathbb{N}\backslash\{0\}.$

The $\mathcal{R}(p,q)-$ deformed shifted factorial is expressed as follows:

[TABLE]

where $x,y\in\mathbb{N},$ while the $\mathcal{R}(p,q)-$ deformed Euler formula is developed as:

[TABLE]

The case of $q-$factorial in [7, 20] corresponds to $\mathcal{R}(p,q)=q,$ $\epsilon_{1}=1$ and $\epsilon_{2}=q$ leading to :

• •

The $j^{th}-$ order factorial of the $q-$ number:

[TABLE]

• •

The $q-$ deformed shifted factorial:

[TABLE]

where $x,y\in\mathbb{N}.$

• •

The $q-$ deformed Euler formula:

[TABLE]

Theorem 3.1**.**

*The $\mathcal{R}(p,q)-$ deformed Vandermonde’s formula is given by: *

[TABLE]

or, equivalently,

[TABLE]

where $n$ is a positive integer.

Proof.

For $n\in\mathbb{N}\backslash\{0\},$ we consider the following expression:

[TABLE]

For $n=1,$ we have $T_{1}(u;v)_{\mathcal{R}(p,q)}=[u+v]_{\mathcal{R}(p,q)}.$ Using the recursion relation

[TABLE]

and

[TABLE]

we obtain

[TABLE]

Therefore, for $n\in\mathbb{N}\backslash\{0\},$ $T_{n}(u;v)_{\mathcal{R}(p,q)}$ satisfies the first-order recursion relation

[TABLE]

with $T_{1}(x;y)_{\mathcal{R}(p,q)}=[u+v]_{\mathcal{R}(p,q)}.$ Recursively, it follows that $T_{n}(u;v)_{\mathcal{R}(p,q)}=[u+v]_{n,\mathcal{R}(p,q)}.$ Therefore, we get (3.6). Finally, interchanging $x$ and $y,$ and replacing $\kappa$ by $n-\kappa,$ the expression (3.6) is rewritten in the form (3.9). ∎

Taking $\mathcal{R}(p,q)=q,$ $\epsilon_{1}=1$ and $\epsilon_{2}=q$, the $q-$ deformed Vandermonde’s formula obtained in [6, 7] is recovered as:

[TABLE]

where $n$ is a positive integer.

For $\mathcal{R}(p,q)=((q-p^{-1})q)^{-1}(p\,q-1),$ involving $\epsilon_{1}=p,$ $\epsilon_{2}=q^{-1}$ and $n,$ a positive integer, the generalized $q-$ Quesne Vandermonde’s formula can be derived as follows:

[TABLE]

Theorem 3.2**.**

*The negative $\mathcal{R}(p,q)-$deformed Vandermonde’s formula is given by: *

[TABLE]

or

[TABLE]

where $n$ is a positive integer.

Proof.

For $n\in\mathbb{N}\backslash\{0\},$ we consider the following expression:

[TABLE]

For $n=1,$ we have

[TABLE]

Using the relation (3.11) and

[TABLE]

we get:

[TABLE]

Therefore, for $n\in\mathbb{N}\backslash\{0\},$ the sum $H_{n}(u;v)_{\mathcal{R}(p,q)}$ satisfies the first-order recursion relation

[TABLE]

with $H_{1}(u;v)_{\mathcal{R}(p,q)}={1\over[u+v+1]_{\mathcal{R}(p,q)}}.$ Recursively, it comes that $H_{n}(u;v)_{\mathcal{R}(p,q)}={[u+v]_{-n,\mathcal{R}(p,q)}}.$ Following the steps used to prove (3.20), we obtain (3.23), and the proof is achieved. ∎

We recover the negative $q-$ Vandermonde’s formula obtained in [7] by taking $\mathcal{R}(p,q)=q,$ $\epsilon_{1}=1$ and $\epsilon_{2}=q$:

[TABLE]

where $0<q<1.$

Lemma 3.3**.**

The following relations hold:

[TABLE]

and

[TABLE]

where $u,$ $v,$ $p,$ and $q$ are real numbers such that $0<q<p\leq 1.$

Proof.

For $n$ a positive integer, the $\mathcal{R}(p,q)-$ deformed factorial of order $\kappa$ of $x=-n$ is written as:

[TABLE]

and

[TABLE]

In the same vein, the $\mathcal{R}(p,q)-$ deformed binomial coefficient of $x=-n$ is given by:

[TABLE]

Moreover, from

[TABLE]

and

[TABLE]

the relation (3.20) can be written as follows:

[TABLE]

where

[TABLE]

After re-arranging, the relation (3.29) is reduced to (3.27). Similarly, we obtain (3.28). ∎

Setting $\epsilon_{1}=1$ and $\epsilon_{2}=q$ provides the $q-$analogs of the formulae (3.27) and (3.28) of [8] as :

[TABLE]

Remark 3.4*.*

It is worth noticing that the generalized $q-$ Quesne negative Vandermonde’s formula is obtained, by taking $\mathcal{R}(p,q)=((q-p^{-1})q)^{-1}(p\,q-1),$ $\epsilon_{1}=p$ and $\epsilon_{2}=q^{-1},$ under the form:

[TABLE]

and

[TABLE]

Definition 3.5**.**

The noncentral $\mathcal{R}(p,q)-$ Stirling numbers of the first and second kinds, $s_{\mathcal{R}(p,q)}(n,\kappa,j)$ and $S_{\mathcal{R}(p,q)}(n,\kappa,j)$ are defined via the relations:

[TABLE]

and

[TABLE]

where $n\in\mathbb{N}$ and $x$ is a real number.

For $j=0$, we obtain $s_{\mathcal{R}(p,q)}(n,\kappa;0)=s_{\mathcal{R}(p,q)}(n,\kappa)$ and $S_{\mathcal{R}(p,q)}(n,\kappa;0)=S_{\mathcal{R}(p,q)}(n,\kappa),$ which are the $\mathcal{R}(p,q)-$ deformed Stirling numbers of the first and second kinds, respectively.

Remark 3.6*.*

Some remarkable particular cases deserve notification:

• •

By setting $\mathcal{R}(p,q)=q,$ $\epsilon_{1}=1$ and $\epsilon_{2}=q,$ we recover the noncentral $q-$ Stirling numbers of the first and second kinds, $s_{q}(n,\kappa,j)$ and $S_{q}(n,\kappa,j),$ derived in [6, 7]:

[TABLE]

and

[TABLE]

where $n\in\mathbb{N}$ and $x$ is a real number.

• •

For $j=0,$ we obtain $s_{q}(n,\kappa;0)=s_{q}(n,\kappa)$ and $S_{q}(n,\kappa;0)=S_{q}(n,\kappa),$ which are the $q-$ deformed Stirling numbers of the first and second kinds, respectively.

• •

Setting $\mathcal{R}(p,q)={p\,q-1\over(q-p^{-1})q}$ corresponding to $\epsilon_{1}=p$ and $\epsilon_{2}=q^{-1},$ we deduct the noncentral generalized $q-$ Quesne Stirling numbers of the first and second kinds, $s^{Q}_{p,q}(n,\kappa,j)$ and $S^{Q}_{p,q}(n,\kappa,j),$ as follows:

[TABLE]

and

[TABLE]

where $n\in\mathbb{N}$ and $x$ is a real number.

• •

For $j=0$, we obtain $s^{Q}_{p,q}(n,\kappa;0)=s^{Q}_{p,q}(n,\kappa)$ and $S^{Q}_{p,q}(n,\kappa;0)=S^{Q}_{p,q}(n,\kappa),$ which are the generalized $q-$ Quesne Stirling numbers of the first and second kinds, respectively.

3.2. $\mathcal{R}(p,q)-$ deformed factorial and binomial moments

We introduce now the $\mathcal{R}(p,q)-$ deformed factorial and binomial moments. For that, we suppose $X$ is a discrete non-negative integer valued random variable, and for $x\in\mathbb{N},$ we consider the distribution function $g$ of the variable $X$ such that $g(x)=P(X=x).$

Definition 3.7**.**

The $r^{th}-$ order $\mathcal{R}(p,q)-$ factorial moment of the random variable $X$ is given by:

[TABLE]

while the $r^{th}-$ order $\mathcal{R}(p,q)-$ binomial moment of the random variable $X$ is provided by:

[TABLE]

where $r\in\mathbb{N}\backslash\{0\}.$

Definition 3.8**.**

For $r=1,$ we obtain:

• •

The $\mathcal{R}(p,q)-$ mean value of the random variable $X$:

[TABLE]

• •

The $\mathcal{R}(p,q)-$ variance of the random variable $X$:

[TABLE]

Proposition 3.9**.**

In terms of Stirling number, the binomial moment can be re-expresed as follows:

[TABLE]

and the relation between the factorial moment and its ${\mathcal{R}(p,q)}-$ deformed counterpart is given by:

[TABLE]

where $j\in\mathbb{N}\backslash\{0,1\},$ $\tau\in\mathbb{N},$ and $s_{\mathcal{R}(p,q)}$ is the $\mathcal{R}(p,q)-$ Stirling number of the first kind.

Proof.

Multiplying the relation

[TABLE]

by the probability distribution $g,$ and summing for all $x\in\mathbb{N},$ we deduce (3.46) from (3.39). Moreover, from

[TABLE]

and the relation (3.46), we derive (3.47). ∎

Remark 3.10*.*

Some interesting results can easily be deduced as follows:

• •

The particular case of the $q-$ deformed binomial moment in [8] is retrieved as follows :

[TABLE]

and the related factorial moment is linked to its $q-$ counterpart by the relation:

[TABLE]

where $j\in\mathbb{N}\backslash\{0\},$ and $s_{q}$ is the $q-$ Stirling number of the first kind.

• •

Putting $\mathcal{R}(p,q)={p\,q-1\over(q-p^{-1})q},$ $\epsilon_{1}=p,$ and $\epsilon_{2}=q^{-1},$ we deduct the binomial moment:

[TABLE]

whose the factorial moment is expressed, in terms of the generalized $q-$ Quesne factorial moment, by:

[TABLE]

where $j\in\mathbb{N}\backslash\{0,1\},$ $\tau\in\mathbb{N},$ and $s^{Q}_{p,q}$ is the generalized $q-$ Quesne Stirling number of the first kind.

3.3. $\mathcal{R}(p,q)-$ deformed binomial distribution

Lemma 3.11**.**

The following relation holds:

[TABLE]

In particular, for $x=p_{0}$ and $y=1,$ we obtain

[TABLE]

where $x,$ $y,$ $v,$ $p_{0}$ and $n$ are integers.

The $q-$analog of Lemma in [20] is deduced by setting $\mathcal{R}(p,q)=q,$ $\epsilon_{1}=1,$ and $\epsilon_{2}=q$, as follows :

[TABLE]

In particular, for $x=p_{0}$ and $y=1,$ we obtain

[TABLE]

where $x,$ $y,$ $v,$ $p_{0}$ and $n$ are integers.

The binomial distribution comes with a random variable $X$ taking two values, [math] and $1,$ the probabilities $Pr(X=1)=p_{0}$ and $Pr(X=0)=1-p_{0},$ and by letting $S_{n}=X_{1}+\cdots+X_{n}$ be the sum of $n$ random variables $(X_{i})_{i\in\{1,2,\cdots,n\}}$ obeying the binomial law.

Definition 3.12**.**

The $\mathcal{R}(p,q)-$deformed binomial distribution, with parameters $n,$ $p_{0},$ $p,$ and $q,$ is given, for $0\leq\kappa\leq n$ and $0<q<p\leq 1,$ by:

[TABLE]

For $\mathcal{R}(p,q)=q,$ $\epsilon_{1}=1$ and $\epsilon_{2}=q$, we obtain the $q-$ binomial distribution given in [20]:

[TABLE]

Theorem 3.13**.**

The $j^{th}-$ order $\mathcal{R}(p,q)-$ deformed factorial moment is given by:

[TABLE]

*and the factorial moment is expressed by the formula: *

[TABLE]

where $\tau\in\mathbb{N},$ $i\in\{2,\cdots,n\},$ and $s_{\mathcal{R}(p,q)}$ is the $\mathcal{R}(p,q)-$ Stirling number of the first kind. The recursion relation for the $\mathcal{R}(p,q)-$ deformed binomial distributions takes the form:

[TABLE]

Proof.

The $j^{th}-$ order $\mathcal{R}(p,q)-$factorial moment of $[S_{n}]_{\mathcal{R}(p,q)}$ is

[TABLE]

Using the relation

[TABLE]

we have

[TABLE]

Moreover, the formula (3.51) is obtained using the relations (3.50) and (3.47). From the relation

[TABLE]

and after computation, we obtain (3.52). ∎

Corollary 3.14**.**

The recursion relation for the $q-$ deformed binomial distributions is given by:

[TABLE]

Proof.

It is obtained by straightforward deduction. ∎

Now, let us consider, for $v\in\mathbb{N}\backslash\{0\},$ the $l-$order differential operator

[TABLE]

where $D_{v}:={d}/{d_{\mathcal{R}(p,q)}}$ acting on $v,$ (which will be used in the sequel), giving, for $l=2,$ the following second order differential operator:

[TABLE]

Besides, for $\epsilon_{i}>0,$ with $i\in\{1,2\},$ such that $\forall p,q,\,0<q<p\leq 1,$

[TABLE]

Lemma 3.15**.**

The mean value of the random variable sum $S_{n}$ is given by:

[TABLE]

Its corresponding variance can be written as:

[TABLE]

where ${\bf X}p_{0}[n-1]_{\mathcal{R}(p,q)}>p_{0}[n]_{\mathcal{R}(p,q)}-\mathcal{R}(p,q),$ and

[TABLE]

The mean of the product $S_{n}(S_{n}-1)\cdots(S_{n}-r+1)$ is given by:

[TABLE]

Proof.

Applying the $\mathcal{R}(p,q)-$derivative on $p_{0}$ to the left and right hand sides of the relation (3.11) leads to

[TABLE]

and

[TABLE]

respectively. According to (3.60) and (3.11), we obtain (3.61). Besides,

[TABLE]

Setting {\bf X}=\frac{1}{\mathcal{R}!(p,q)}\bigg{(}\mathcal{R}(p^{2},q^{2})-\mathcal{R}(p,q)\bigg{)} yields the second order differential equation

[TABLE]

where

[TABLE]

Using the relations (3.11) and (3.3), we obtain

[TABLE]

and from the mean value of ${S_{n}},$ we deduce {\bf Var}\big{(}[S_{n}]_{\mathcal{R}(p,q)}\big{)}. Futhermore, applying the operator {{p^{r}_{0}\bigg{(}{D_{p_{0}}}\bigg{)}^{r}\bigg{|}_{u=p_{0}}}} to formula (3.11), we obtain

[TABLE]

Moreover,

[TABLE]

or equivalently,

[TABLE]

Then,

[TABLE]

gives

[TABLE]

and after computation, the result follows. ∎

The particular case of the $q-$ deformation described in [20] corresponds to $\mathcal{R}(p,q)=q,$ $\epsilon_{1}=1,$ and $\epsilon_{2}=q,$ and yields the mean value of $S_{n}:$

[TABLE]

the variance of $S_{n}:$

[TABLE]

and the mean of the product $S_{n}(S_{n}-1)\cdots(S_{n}-r+1):$

[TABLE]

Remark 3.16*.*

Deducing the above mentioned properties for the particular case of the generalized $q-$ Quesne deformation leads to:

• •

Generalized $q-$ Quesne probability distribution:

[TABLE]

• •

$j^{th}-$ order generalized $q-$ Quesne factorial moment:

[TABLE]

• •

Generalized $q-$ Quesne factorial moment:

[TABLE]

• •

Recursion relation for the generalized $q-$ Quesne distributions:

[TABLE]

• •

Mean value:

[TABLE]

• •

Variance:

[TABLE]

• •

Mean value of the product $S_{n}(S_{n}-1)\cdots(S_{n}-r+1):$

[TABLE]

3.4. $\mathcal{R}(p,q)-$ deformed Euler distribution

Definition 3.17**.**

The $\mathcal{R}(p,q)-$ deformed exponential functions, denoted $E_{\mathcal{R}(p,q)}$ and $e_{\mathcal{R}(p,q)},$ are defined as follows:

[TABLE]

with $E_{\mathcal{R}(p,q)}(-z)\,e_{\mathcal{R}(p,q)}(z)=1.$

In the particular case where $\mathcal{R}(p,q)=1,$ $\epsilon_{1}=p,$ and $\epsilon_{2}=q,$ they provide the Jagannathan-Srinivasa $q-$exponential functions [13]:

[TABLE]

with $E_{p,q}(-z)\,e_{p,q}(z)=1.$

Definition 3.18**.**

The $\mathcal{R}(p,q)-$ deformed Euler distribution, with parameters $n,$ $\theta,$ $p,$ and $q,$ is defined by:

[TABLE]

where $0<q<p\leq 1,$ $0<\theta<1$ and $\displaystyle\sum_{x=0}^{\infty}P_{r}(X=x)=1.$

For $\mathcal{R}(p,q)=q,$ involving $\epsilon_{1}=1,$ and $\epsilon_{2}=q$, we retrieve the $q-$ Euler distribution obtained in [7]:

[TABLE]

where $x\in\mathbb{N},$ $0<q<1$ and $0<\theta<{1\over 1-q}.$

Theorem 3.19**.**

The $j^{th}-$ order $\mathcal{R}(p,q)-$ factorial moment of $X$ is given by

[TABLE]

*while the factorial moment is written as: *

[TABLE]

where

[TABLE]

$\tau\in\mathbb{N},$* $i\in\mathbb{N}\backslash\{0,1\},$ and $s_{\mathcal{R}(p,q)}$ is the $\mathcal{R}(p,q)-$ deformed Stirling number of the first kind. The recursion relation for the associated $\mathcal{R}(p,q)-$ deformed Euler distributions takes the form:*

[TABLE]

Proof.

From the $j^{th}-$ order $\mathcal{R}(p,q)-$ factorial moment, we get

[TABLE]

Using the relation

[TABLE]

we obtain

[TABLE]

Furthermore, exploiting the relation (3.69) and (3.47), we obtain (3.70). Using the $\mathcal{R}(p,q)-$ factorials, and after computation, we get (3.72). ∎

Corollary 3.20**.**

The recursion relation for the $q-$ deformed Euler distributions is given by:

[TABLE]

Proof.

It stems from a straightforward computation. ∎

Particular results for the $q-$ deformation performed in [7], recovered with $\mathcal{R}(p,q)=q,$ $\epsilon_{1}=1,$ and $\epsilon_{2}=q,$ yield the $j^{th}-$ order $q-$ factorial moment of $X$:

[TABLE]

and the $q-$ factorial moments:

[TABLE]

$i\in\mathbb{N}\backslash\{0\},$ and $s_{q}$ is the $q-$ deformed Stirling number of the first kind.

Remark 3.21*.*

The generalized $q-$ Quesne Euler distribution corresponding to the choice $\mathcal{R}(p,q)=((q-p^{-1})q)^{-1}(p\,q-1),$ $\epsilon_{1}=p$ and $\epsilon_{2}=q^{-1},$ gives the following realization:

• •

Generalized $q-$ Quesne probability distribution:

[TABLE]

• •

$j^{th}-$ order generalized $q-$ Quesne factorial moment:

[TABLE]

• •

Generalized $q-$ Quesne factorial moment:

[TABLE]

where:

[TABLE]

$\tau\in\mathbb{N},$ $i\in\mathbb{N}\backslash\{0,1\},$ and $s^{Q}_{p,q}$ is the generalized $q-$ Quesne Stirling number of the first kind;

• •

Recursion relation for the generalized $q-$ Quesne distributions:

[TABLE]

3.5. $\mathcal{R}(p,q)-$ deformed Pólya distribution

We here assume that boxes are successively drawn one after the other from an urn, initially containing $r$ white and $s$ black boxes. After each drawing, the drawn box is placed back in the urn together with $x$ boxes of the same color. We suppose that the probability of drawing a white box at the $i^{th}$ drawing, given that $j-1$ white boxes are drawn in the previous $i-1$ drawings, is given as:

[TABLE]

where $j\in\{1,2,\cdots,i\},$ $i\in\mathbb{N}\backslash\{0\},$ $0<q<p\leq 1,$ $m={-r/x},$ $u={-s/x},$ and $x$ is a non zero integer. We call this model the $\mathcal{R}(p,q)-$ Pólya urn model. Setting $\mathcal{R}(p,q)=q,$ $\epsilon_{1}=1$, and $\epsilon_{2}=q$, we obtain the probability for $q-$ Pólya urn model described in [6]:

[TABLE]

Let $T_{n}$ be the number of white boxes drawn in $n$ drawings. Then, we have:

Definition 3.22**.**

The $\mathcal{R}(p,q)-$ deformed Pólya distribution, with parameters $m,$ $u,$ $n,$ $p,$ and $q,$ is defined by:

[TABLE]

where $0<q<p\leq 1,$ $\kappa\in\{0,\cdots,n\},$ $\Psi(p,q)={\epsilon^{-x\kappa(u-n+\kappa)}_{1}\,\epsilon^{-x(n-\kappa)(m-\kappa)}_{2}},$ $x$ is an integer and $\displaystyle\sum_{\kappa=0}^{n}P_{r}(T_{n}=\kappa)=1.$

For $\mathcal{R}(p,q)=q,$ $\epsilon_{1}=1,$ and $\epsilon_{2}=q,$ we recover the $q-$ Pólya distribution [6]:

[TABLE]

where $0<q<1$ and $\kappa\in\{0,\cdots,n\}$.

Theorem 3.23**.**

*The $j^{th}-$ order $\mathcal{R}(p,q)-$ deformed factorial moment is given by: *

[TABLE]

while the factorial moment is expressed by:

[TABLE]

where

[TABLE]

$\tau\in\mathbb{N},$* $i\in\{2,\cdots,n\}$ and $s_{\mathcal{R}(p^{-x},q^{-x})}$ is the $\mathcal{R}(p,q)-$ deformed Stirling number of the first kind. The recursion relation for the $\mathcal{R}(p,q)-$ Pólya distributions is given as follows:*

[TABLE]

with

[TABLE]

Proof.

The $j^{th}-$ order $\mathcal{R}(p,q)-$ factorial moment is furnished by the formula

[TABLE]

Using the $\mathcal{R}(p,q)-$ Vandermonde’s formula (3.6) yields the expression

[TABLE]

for $j\in\{1,\cdots,n\}.$ Then, since

[TABLE]

the relation (3.75) is obtained. From (3.47), with $[-x]_{\mathcal{R}(p,q)}$ instead of $\mathcal{R}(p,q),$ (3.78) is deduced. Since

[TABLE]

then, after computation, the recursion relation is found. ∎

Corollary 3.24**.**

The recursion relation for the $q-$ Pólya distributions is derived as:

[TABLE]

with

[TABLE]

Proof.

It is straightforward. ∎

Remark 3.25*.*

For $x=-1$, we obtain the $\mathcal{R}(p,q)-$ hypergeometric distribution:

[TABLE]

where $0<q<p\leq 1,$ $\kappa\in\{0,\cdots,n\},$ $\Phi(p,q)={\epsilon^{\kappa(u-n+\kappa)}_{1}\,\epsilon^{(n-\kappa)(m-\kappa)}_{2}}$ and $\displaystyle\sum_{\kappa=0}^{n}P_{r}(T_{n}=\kappa)=1.$

The particular case of $q-$ deformation obtained in [6] is characterized by the $j^{th}-$ order $q-$ deformed factorial moment:

[TABLE]

and the factorial moment:

[TABLE]

where $i\in\{1,2,\cdots,n\}$ and $s_{q^{-x}}$ is the $q-$ deformed Stirling number of the first kind.

Remark 3.26*.*

The particular case of $q-$generalized Quesne deformation is characterized by the following properties:

• •

The generalized $q-$ Quesne probability distribution:

[TABLE]

• •

The $j^{th}-$ order generalized $q-$ Quesne factorial moment:

[TABLE]

• •

The generalized $q-$ Quesne factorial moment:

[TABLE]

where

[TABLE]

$\tau\in\mathbb{N},$ $i\in\{2,\cdots,n\}$ and $s^{Q}_{p^{-x},q^{-x}}$ is the generalized $q-$ Quesne Stirling number of the first kind;

• •

The recursion relation for the generalized $q-$ Quesne distributions:

[TABLE]

with

[TABLE]

3.6. Inverse Pólya distribution

Let $Y_{n}$ be a number of black boxes drawn until the $n^{th}$ white box is drawn. Then, we have:

Definition 3.27**.**

The inverse $\mathcal{R}(p,q)-$ deformed Pólya distribution with parameters $n,$ $m,$ $u,$ and $\kappa$ is given by:

[TABLE]

where $0<q<p\leq 1,$ $y\in\{1,\cdots,n\},$ $F(p,q)=\epsilon^{n(u-x)}_{1}\,\epsilon^{-yx(m-n+1)}_{2},$ $x$ is an integer, and

[TABLE]

Note that the inverse $q-$ deformed Pólya distribution obtained in [6] can be recovered by taking $\mathcal{R}(p,q)=q,$ $\epsilon_{1}=1,$ $\epsilon_{2}=q$ and $F(p,q)=q^{-yx(m-n+1)}$ as:

[TABLE]

where $0<q<1$ and $y\in\{1,\cdots,n\}.$

Theorem 3.28**.**

The $j^{th}-$ order $\mathcal{R}(p,q)-$factorial moment is given by:

[TABLE]

where $j\in\mathbb{N}\backslash\{0\}$ and $m+j\neq 0.$ Moreover, for $i\in\{1,2,\cdots,n\},$ the factorial moment yields:

[TABLE]

where

[TABLE]

$\tau\in\mathbb{N}$* and $s_{\mathcal{R}(p^{-x},q^{-x})}$ is the $\mathcal{R}(p,q)-$ deformed Stirling number of the first kind. The recursion relation for the inverse $\mathcal{R}(p,q)-$ Pólya distibutions is provided by:*

[TABLE]

with initial condition

[TABLE]

Proof.

Using (3.96), the $j^{th}-$ order $\mathcal{R}(p,q)-$ deformed factorial moment of $Y_{n}$ is expressed as follows:

[TABLE]

From the negative $\mathcal{R}(p,q)-$deformed Vandermonde’s formula (3.27), we have the relation

[TABLE]

Since,

[TABLE]

the relation (3.97) holds. Furthermore, the relation (3.98) is deduced using the $\mathcal{R}(p,q)-$ deformed factorial moment and (3.47). ∎

Corollary 3.29**.**

The recursion relation for the inverse $q-$ Pólya distributions is given by the formula:

[TABLE]

Finally, it is worthy to note that, for the generalized $q-$Quesne quantum algebra, we have:

• •

The inverse generalized $q-$ Quesne Pólya probability distribution:

[TABLE]

• •

The $j^{th}-$ order generalized $q-$ Quesne factorial moment:

[TABLE]

with the associated factorial moment

[TABLE]

where

[TABLE]

$s^{Q}_{p^{-x},q^{-x}}$ is the generalized $q-$ Quesne Stirling number of the first kind and $\tau\in\mathbb{N};$

• •

The recursion relation for the inverse generalized $q-$ Quesne Pólya distrubtions:

[TABLE]

4. Concluding remarks

$\mathcal{R}(p,q)$-deformed univariate dicrete binomial, Euler, Pólya and inverse Pólya distributions, induced by the $\mathcal{R}(p,q)$-deformed quantum algebras, have been formulated and discussed in a general framework. Distributions and main properties related to the generalized $q-$ Quesne algebra have been considered as illustration. Results for known $q-$ deformed distributions have also been recovered as particular cases.

Acknowledgment. This work is supported by TWAS Research Grant RGA No. 17 - 542 RG / MATHS / AF / AC _G -FR3240300147. MNH thanks Professor Kalyan Sinha for his constant solicitude, and Professor Aurel I. Stan for his hospitality during the QP40 Conference.