R(p,q)-deformed combinatorics: full characterization and illustration

TL;DR

This paper develops a comprehensive R(p,q)-deformed combinatorics framework, extending classical combinatorial concepts and applying it to deformed quantum algebra, enriching the mathematical tools for discrete probability and quantum algebra applications.

Contribution

It provides a full characterization of R(p,q)-deformed combinatorics, including new formulas and properties, and illustrates its application to generalized q-Quesne deformed quantum algebra.

Findings

Derived R(p,q)-deformed factorials and binomial coefficients

Established R(p,q)-Stirling numbers and Bell numbers

Applied formalism to q-Quesne deformed quantum algebra

Abstract

This paper addresses a theory of R(p,q)-deformed combinatorics in discrete probability. It mainly focuses on R(p,q)-deformed factorials, binomial coefficients, Vandermonde's formula, Cauchy's formula, binomial and negative binomial formulae, factorial and binomial moments, and Stirling numbers. Moreover, the R(p,q)-Stirling numbers of the second kind and the R(p,q)-Bell numbers for graphs are also derived. Related relevant properties are investigated and discussed. Finally, as a concrete illustration, the developed formalism is displayed for the well known generalized q-Quesne deformed quantum algebra to construct the corresponding deformed combinatorics, as a particular case.

Equations

$$[N,a^{†}]=a^{†}\text{mbox},[N,a]=-a,$$

$$a{a}^{†}-a^{†}a=q^{-N-1}\text{mbox}orqa{a}^{†}-a^{†}a=1$$

$$[u{]}^Q=\frac{1-q^{-u}}{q-1}.$$

$$[N,a^{†}]=a^{†}\text{mbox},[N,a]=-a,$$

$$p^{-1}a{a}^{†}-a^{†}a=q^{-N-1}\text{mbox}orqa{a}^{†}-a^{†}a=p^{N+1}$$

$$[u{]}_{p,q}^Q=\frac{p^u-q^{-u}}{q-p^{-1}}.$$

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

$$Q:\Psi ⟼Q\Psi (z):=\Psi (qz),$$

$$P:\Psi ⟼P\Psi (z):=\Psi (pz).$$

$$D_{p,q}:\Psi ⟼D_{p,q}\Psi (z):=\frac{\Psi (pz)-\Psi (qz)}{z(p-q)},$$

$$[n{]}_{p,q}:=\frac{p^n-q^n}{p-q},$$

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

$$[n{]}_{\mathcal{R}(p,q)}:=\mathcal{R}(p^n,q^n)\text{mbox}forn\ge 0,$$

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

$$\left[\begin{array}[]{c}m\\ \kappa\end{array}\right]_{\mathcal{R}(p,q)}:=\frac{\mathcal{R}!(p^{m},q^{m})}{\mathcal{R}!(p^{\kappa},q^{\kappa})\mathcal{R}!(p^{m-\kappa},q^{m-\kappa})},\quad m,\kappa=0,1,2,\cdots,\quad m\geq\kappa.$$

$$\big{(}1\oplus y\big{)}^{n}_{q}:=\prod_{i=0}^{n-1}\big{(}1+yq^{i-1}\big{)}=\sum_{\kappa=0}^{n}\,q^{\kappa\choose 2}\bigg{[}\begin{array}[]{c}n\atop\\ \kappa\end{array}\bigg{]}_{q}\,y^{\kappa}.$$

$$\prod_{i=1}^{n}(p^{i-1}+u\,q^{i-1})=\sum_{\kappa=0}^{n}\,p^{{n-\kappa\choose 2}}\,q^{\kappa\choose 2}\bigg{[}\begin{array}[]{c}n\atop\\ \kappa\end{array}\bigg{]}_{p,q}\,u^{\kappa}.$$

$$\bigg{[}\begin{array}[]{c}x+y\\ n\end{array}\bigg{]}_{q}=\sum_{\kappa=0}^{n-1}q^{\kappa(y-n+\kappa)}\,\bigg{[}\begin{array}[]{c}x\\ \kappa\end{array}\bigg{]}_{q}\,\,\bigg{[}\begin{array}[]{c}y\\ n-\kappa\end{array}\bigg{]}_{q},$$

$$\bigg{[}\begin{array}[]{c}x+y\\ n\end{array}\bigg{]}_{q}=\sum_{\kappa=0}^{n-1}q^{(n-\kappa)(x-\kappa)}\,\bigg{[}\begin{array}[]{c}x\\ \kappa\end{array}\bigg{]}_{q}\,\,\bigg{[}\begin{array}[]{c}y\\ n-\kappa\end{array}\bigg{]}_{q}.$$

$$\sum_{x=\kappa}^{n}(-1)^{n-x}\,q^{{n-x\choose 2}}\bigg{[}\begin{array}[]{c}n\\ x\end{array}\bigg{]}_{q}\,\bigg{[}\begin{array}[]{c}x\\ \kappa\end{array}\bigg{]}_{q}=\delta_{n,\kappa},$$

$$\sum_{x=\kappa}^{n}(-1)^{x-\kappa}q^{{x-\kappa\choose 2}}\bigg{[}\begin{array}[]{c}n\\ x\end{array}\bigg{]}_{q}\,\bigg{[}\begin{array}[]{c}x\\ \kappa\end{array}\bigg{]}_{q}=\delta_{n,\kappa}$$

$$[x{]}_{n,q}=q^{-\left(\genfrac{}{}{0ex}{}{n}{2}\right)}\sum _{\kappa =0}^n{s}_q(n,\kappa )[x{]}_q^{\kappa },$$

$$[x{]}_q^n=\sum _{\kappa =0}^n{q}^{\left(\genfrac{}{}{0ex}{}{\kappa }{2}\right)}{S}_q(n,\kappa )[x{]}_{\kappa ,q}$$

$${\kappa\choose j}=\sum_{m=j}^{\kappa}(-1)^{m-j}(1-q)^{m-j}s_{q}(m,j)\,\bigg{[}\begin{array}[]{c}\kappa\\ m\end{array}\bigg{]}_{q}$$

$$\bigg{[}\begin{array}[]{c}\kappa\\ j\end{array}\bigg{]}_{q}=\sum_{m=j}^{\kappa}(-1)^{m-j}(1-q)^{m-j}S_{q}(m,j)\,{\kappa\choose m},$$

$$g(x)=\sum_{m=x}^{\infty}(-1)^{m-x}q^{{m-x\choose 2}}\bigg{[}\begin{array}[]{c}m\\ x\end{array}\bigg{]}_{q}{\bf E}\Big{(}\bigg{[}\begin{array}[]{c}X\\ m\end{array}\bigg{]}_{q}\Big{)},\quad x\in\mathbb{N}$$

$$[u{]}_{\kappa ,\mathcal{R}(p,q)}=\prod _{v=1}^{\kappa }\mathcal{R}(p^{u-v+1},q^{u-v+1}),$$

$$[u{]}_{\kappa +s,\mathcal{R}(p,q)}=[u{]}_{s,\mathcal{R}(p,q)}[u-s{]}_{\kappa ,\mathcal{R}(p,q)},\kappa \in \mathbb{N}\backslash \{0\}\text{mbox}ands\in \mathbb{N}\backslash \{0\}.$$

$$[x{]}_{-\kappa ,\mathcal{R}(p,q)}:=\frac{1}{[x+\kappa {]}_{\kappa ,\mathcal{R}(p,q)}}\text{mbox},\kappa \in \mathbb{N}\backslash \{0\}.$$

$$[x{]}_{0,\mathcal{R}(p,q)}:=1.$$

$$[0{]}_{-\kappa ,\mathcal{R}(p,q)}=\frac{1}{[\kappa {]}_{\mathcal{R}(p,q)}!}\text{mbox},\kappa \in \mathbb{N}\backslash \{0\}.$$

Full text

1. 1 Introduction

2. 2 Preliminaries

3. 3 $\mathcal{R}(p,q)-$ deformed combinatorics

4. 3.1 $\mathcal{R}(p,q)-$ deformed factorials and binomial coefficients

5. 3.2 $\mathcal{R}(p,q)-$ deformed Vandermonde’s and Cauchy’s formulae

6. 3.3 $\mathcal{R}(p,q)-$ deformed binomial and negative binomial formulae

7. 3.4 $\mathcal{R}(p,q)-$ deformed Stirling numbers

8. 3.5 $\mathcal{R}(p,q)-$ deformed Bell numbers

9. 3.6 Application

10. 3.7 $\mathcal{R}(p,q)-$ deformed factorial and binomial moments

11. 4 Concluding remarks

1 Introduction

Combinatorial theory is a major branch of mathematics, which has applications in many fields such as computer science (languages, graphs, intelligent computing), natural and social sciences, biomedicine, molecular biology, operational research, engineering, and business[24, 25, 28]. Combinatorial theory and discrete mathematical methods play an important role, and occupy a central position in the theory of discrete probability. The most prominent of these methods are the combinatorial enumerative methods and the basic methods of finite difference computation. A considerable number of stochastic experiments or phenomena in discrete probability theory can be described by the stochastic models of distributions [10].

Chung and Kang developed a new combinatorics called $q-$ combinatorics and investigated the significance of $q-$ permutations and $q-$ combinations[5]. The idea of $q-$ analogs can be traced back to Euler in the 1700’s who studied $q-$ series, especially specializations of theta functions. Meanwhile, in [7], Ch. A. Charalambides examined basic $q-$ combinatorics and $q-$ hypergeometric series. The $q-$power, $q-$factorial, $q-$binomial coefficient of a real number and two $q-$ Vandermonde’s ($q-$factorial convolution) formulae were derived. The $q-$ analogs of the Cauchy’s formulas were also investigated in [1].

Furthermore, the $q-$Stirling numbers of the first and second kinds, which are the coefficients of the expansions of $q-$factorials into $q-$ powers, and of $q-$powers into $q-$factorials, respectively, were presented. Moreover, the Stirling numbers of the second kind and their generalizations were studied by several authors, (see for instance [21, 22, 23] and references theiren). The $q-$ Stirling numbers of the second kind and $q-$ Bell numbers for graphs were also analyzed in [2]. Corcino and Barientos [12] established many properties for $q-$ analogs of Stirling numbers. The vertical and horizontal recursion relations and the generating function were also computed. Besides, two parameters $(p,q)-$ Stirling numbers, which are generating functions for the joint distribution of pair statistics, were described in [29]. In addition, a theory of $(p,q)-$ analogs of binomial coefficients was elaborated. Some properties and identities like the triangular, vertical, and horizontal recursion relations, the generating function, the orthogonality and inverse relations were derived in [11].

Later in 2010, $\mathcal{R}(p,q)-$ deformed quantum algebra was introduced by Hounkonnou and Bukweli [15] as a generalization of known deformed quantum algebras. The same authors also performed $\mathcal{R}(p,q)-$ differentiation and integration, and deducted all relevant particular cases of $q-$ and $(p,q)-$ deformations [14]. This opens a novel route for developing the theory of $\mathcal{R}(p,q)-$ analogs of special numbers and combinatorics.

This paper provides a general formalism, which enables an easier construction of a combinatorial theory from deformed quantum algebras existing in the literature by assigning concrete suitable expressions to the function $\mathcal{R}$ and related specific meromorphic functions of the theory. Especially, $\mathcal{R}(p,q)-$ analogs of factorials, binomial coefficients, Vandermonde’s formula, Cauchy’s formula, binomial formula and negative binomial formula, factorial moments, binomial moments, Stirling numbers and Bell numbers on a graph are investigated and discussed. Furthermore, the case of the generalized $q-$ Quesne deformed combinatorics is derived to illustrate the presented formalism. From this new generalization, developing deformed combinatorial theories associated with other particular cases of deformed quantum algebras, known and spread in the literature, appears as a matter of triviality.

The paper is organized as follows. Section $2$ is devoted to basic notations, definitions and results related to $\mathcal{R}(p,q)-$ quantum algebras and calculus, and to basic $q-$ combinatorics. In Section $3,$ the fundamentals of $\mathcal{R}(p,q)-$ deformed combinatorics and the derivation of relevant properties are exposed. The generalized $q-$ Quesne combinatorics is derived as a case study. Some concluding remarks are addressed in Section $4.$

2 Preliminaries

In this section, we briefly recall the main definitions, notations and known results used in the sequel. For more details, the reader can refer to [7, 8, 9, 10, 11], [14, 15] and [18, 19].

The coherent states introduced by Quesne[26] can be associated with the $q-$deformed algebra satisfying the relations

[TABLE]

where $0<q<1,$ and the Quesne number is defined by:

[TABLE]

The Quesne algebra is a particular case of the Kalnins-Miller-Mukherjee algebra [20] with $\ell=1,\lambda=0.$ Furthermore, Hounkonnou and Ngompe Nkouankam [16] generalized the $q-$ Quesne algebra with the generators satisfying the relations

[TABLE]

where $0<q<p\leq 1.$ Their generalized $q-$ Quesne number is given as follows:

[TABLE]

Let now $\mathcal{R}$ be a meromorphic function defined on $\mathbb{C}\times\mathbb{C}$ by

[TABLE]

converging in the complex disc $\mathbb{D}_{R}=\left\{z\in\mathbb{C}/|z|<R\right\},$ where $r_{st}$ are complex numbers, $l\in\mathbb{N}\backslash\left\{0\right\},$ and $R$ is the radius of convergence of the series (2.3). Let us consider the set of holomorphic functions $\mathcal{O}(\mathbb{D}_{R})$ defined on $\mathbb{D}_{R}$.

In the sequel, when no possible confusion arises, $p$ and $q$ will designate two real numbers satisfying $0<q<p\leq 1.$

Definition 2.1

[14]** Let $P$ and $Q$ be two linear operators on $\mathcal{O}(\mathbb{D}_{R})$. Then, for $\varPsi\in\mathcal{O}(\mathbb{D}_{R})$, we have

[TABLE]

[TABLE]

The $(p,q)-$derivative and the $(p,q)-$number are defined, respectively, by [6]:

[TABLE]

[TABLE]

while the $\mathcal{R}(p,q)-$derivative is given by [14]:

[TABLE]

Definition 2.2

[15]** The $\mathcal{R}(p,q)-$number and the $\mathcal{R}(p,q)-$ factorials are defined, respectively, as follows:

[TABLE]

[TABLE]

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

[TABLE]

More details on $\mathcal{R}(p,q)-$ deformed quantum algebras, $\mathcal{R}(p,q)-$differentiation and integration can be found in [14, 15]. To be complete, let us briefly recall some notions about known $q-$ combinatorics pertaining to our development in the sequel.

Definition 2.3

The $q-$ shifted factorial is given by

[TABLE]

The $(p,q)-$ binomial coefficients are given by [11]:

[TABLE]

For real numbers $x$ and $y,$ the $q-$ analog of the Cauchy’s formula is given by [7]

[TABLE]

or, equivalently,

[TABLE]

Furthermore, the following orthogonality relations hold [8]:

[TABLE]

and

[TABLE]

where $\delta_{n,\kappa}$ is the Kronecker delta, $n$ and $\kappa$ are positive integers. The $n^{th}-$ order of a $q-$ factorial number ${[x]_{n,q}}$ is written as a polynomial of the $q-$ number as follows [13]:

[TABLE]

or,

[TABLE]

where the coefficients $s_{q}(n,\kappa)$ and $S_{q}(n,\kappa)$ are called $q-$ Stirling numbers of the first and second kinds, respectively.

Let $\kappa$ and $j$ be positive integers. Then, the following relations also hold:

[TABLE]

and

[TABLE]

where $x$ is an integer and $0<q<1.$ Finally, recall that the $q-$deformed probability distribution $g(x)$ of a discrete random variable $X$ is given by Ch. A. Charalambides [7] as

[TABLE]

where the series is absolutely convergent, and {\bf E}\bigg{(}\bigg{[}\begin{array}[]{c}X\\ m\end{array}\bigg{]}_{q}\bigg{)} stands for the expectation value of \bigg{[}\begin{array}[]{c}X\\ m\end{array}\bigg{]}_{q}.

3 $\mathcal{R}(p,q)-$ deformed combinatorics

Our aim is to present fundamentals of a generalization of the combinatorial theory from the $\mathcal{R}(p,q)-$ deformed quantum algebra introduced in [15] as a generalization of known deformed quantum algebras. We consider $\Phi_{i}(p,q)\in\mathcal{O}(\mathbb{D}_{R}),$ with $i=1,2,$ depending on the parameters $p$ and $q.$

3.1 $\mathcal{R}(p,q)-$ deformed factorials and binomial coefficients

Definition 3.1

Let $u$ be a real number. Then, the $\mathcal{R}(p,q)-$ deformed factorial of $u$ of order $\kappa$ is defined by :

[TABLE]

where $0<q<p\leq 1$ and $\kappa\in\mathbb{N}\backslash\{0\}.$

The relation (3.1) will also be called the $\kappa^{th}-$ order factorial of the $\mathcal{R}(p,q)-$ deformed number. From the above definition, we derive the following basic property for the $\mathcal{R}(p,q)-$ deformed factorial:

[TABLE]

Lemma 3.2

Let $x,$ $q,$ and $p$ be real numbers such that $0<q<p\leq 1.$ Then, the $\mathcal{R}(p,q)-$ deformed factorial of $x$ of negative order ${-\kappa}$ is given as follows:

[TABLE]

[TABLE]

Proof: It uses the relation (3.2). $\hfill{\square}$

For $x=0,$ the equation (3.3) yields

[TABLE]

Definition 3.3

The $\mathcal{R}(p,q)-$ deformed binomial coefficient is defined by:

[TABLE]

We assume there exist $\Phi_{i}\in\mathcal{O}(\mathbb{D}_{R}),$ with $i=1,2,$ depending on the parameters $p$ and $q,$ which link the deformed numbers $\mathcal{R}(p^{u},q^{u})$ and $\mathcal{R}(p^{-u},q^{-u})-$ as follows:

[TABLE]

Then, the following relations hold:

[TABLE]

and

[TABLE]

$u$ and $v$ being real numbers.

Proposition 3.4

Let $u$ be a natural number and $\kappa$ a positive integer. Then,

[TABLE]

[TABLE]

and

[TABLE]

Proof: Using the relation (3.5), we get

[TABLE]

Furthermore,

[TABLE]

Finally, using the relation (3.4), we obtain (3.10). $\hfill{\square}$

Remark 3.5

The formula (3.10) may be expressed as

[TABLE]

For $\mathcal{R}(u,v)=(p-q)^{-1}(u-v),$ we recover the Jagannathan-Srinivassa binomial coefficient as:

[TABLE]

The generalized $q-$Quesne binomial coefficient can be obtained by putting $\mathcal{R}(x,y)=(q-p^{-1})^{-1}(x-y^{-1})$ as:

[TABLE]

Theorem 3.6

Let $x,$ $p$ and $q$ be real numbers such that $0<q<p\leq 1,$ and $\kappa$ be a positive integer. Then, the $\mathcal{R}(p,q)-$ deformed binomial coefficients satisfy the following recursion relation:

[TABLE]

or, equivalently,

[TABLE]

with the initial conditon \bigg{[}\begin{array}[]{c}x\\ 0\end{array}\bigg{]}_{\mathcal{R}(p,q)}:=1.

Proof: Since $[x]_{\kappa,\mathcal{R}(p,q)}=[x]_{\mathcal{R}(p,q)}\,[x-1]_{\kappa-1,\mathcal{R}(p,q)},$ $[x-1]_{\kappa,\mathcal{R}(p,q)}=[x-1]_{\kappa-1,\mathcal{R}(p,q)}\,[x-\kappa]_{\mathcal{R}(p,q)},$ using the relations (3.6), the $\mathcal{R}(p,q)-$ factorial of x satisfies the recursion relation:

[TABLE]

with condition $[x]^{0}_{\mathcal{R}(p,q)}:=1.$ From the expression (3.4), the relation (3.14) is deduced. Furthermore, the expression (3.1) satisfies

[TABLE]

Dividing the members of the above equation by $[\kappa]_{\mathcal{R}(p,q)}!$, we obtain (3.15), and the proof is achieved. $\hfill{\square}$

Corollary 3.7

Let $x,$ and $r$ be real numbers. Then, the following relation holds:

[TABLE]

Proof: It is straightforward by computation. $\hfill{\square}$

Taking $\mathcal{R}(u,v)=v,$ we recuperate a simpler relation under the form

[TABLE]

where $0<q<1.$

Remark 3.8

1. (1)

Note that the results obtained by Hounkonnou and Bukweli in **[14]** can be retrieved by taking $\mathcal{R}(s,t)=((p^{-1}-q)t)(st-1).$ 2. (2)

The generalized $q-$Quesne formulae are given as follows, with real numbers $p$ and $q$ such that $0<q<p\leq 1$ and $\kappa\in\mathbb{N}\backslash\{0\}:$

• (i)

[TABLE]

• (ii)

[TABLE]

• (iii)

[TABLE]

• (iv)

[TABLE]

[TABLE]

and

[TABLE]

3.2 $\mathcal{R}(p,q)-$ deformed Vandermonde’s and Cauchy’s formulae

The $\mathcal{R}(p,q)-$ deformed Vandermonde’s formula, also called $\mathcal{R}(p,q)-$ deformed factorial convolution, is contained in the theorem below, where, for the expression simplification, we set $\Phi_{1}(p,q)=\epsilon_{1}$ and $\Phi_{2}(p,q)=\epsilon_{2};$ $x,$ $y,$ $p,$ and $q$ are real numbers with $0<q<p\leq 1.$

Theorem 3.9

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 $T_{1}(x;y)_{\mathcal{R}(p,q)}=\mathcal{R}(p^{x+y},q^{x+y}).$ Using the recursion relation (3.14) and

[TABLE]

we obtain

[TABLE]

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

[TABLE]

with $T_{1}(x;y)_{\mathcal{R}(p,q)}=[x+y]_{\mathcal{R}(p,q)}.$ Recursively, it follows that $T_{n}(x;y)_{\mathcal{R}(p,q)}={[x+y]_{n,\mathcal{R}(p,q)}}.$ Therefore, we get (3.16). Finally, interchanging $x$ by $y,$ and replacing $\kappa$ by $n-\kappa,$ the expression (3.16) is rewritten in the form (3.17). $\hfill{\square}$

Remark 3.10

Note that the $q-$ deformed Vandermonde’s formula can be retrieved by taking $\mathcal{R}(u,v)=v:$

[TABLE]

or, in an alternative form,

[TABLE]

where $0<q<1.$

From the $\mathcal{R}(p,q)-$ Vandermonde’s formula (3.16), we can deduce the following remarkable $\mathcal{R}(p,q)-$ deformed identities.

Lemma 3.11

Let $x,$ $y,$ $p,$ and $q$ be real numbers such that $0<q<p\leq 1.$ Then, the following relations hold.

[TABLE]

[TABLE]

and

[TABLE]

Proof: Replacing $y$ by $y+n$ in (3.16), we obtain

[TABLE]

Multiplying both sides of this relation by $[y]_{-n,\mathcal{R}(p,q)},$ and using $[y]_{-\kappa,\mathcal{R}(p,q)}=[y]_{-n,\mathcal{R}(p,q)}[y+n]_{n-\kappa,\mathcal{R}(p,q)},$ we get

[TABLE]

and according to (3.3), we deduce the required formula

[TABLE]

Putting now $y=-x$ in (3.22), and using, respectively,

[TABLE]

and

[TABLE]

we get (3.23). Similarly, by substituting $x$ by $-y,$ we obtain (3.24). $\hfill{\square}$

Remark 3.12

Taking $\mathcal{R}(p,q)=q,$ we retrieve the $q-$ identities as particular cases:

[TABLE]

[TABLE]

and

[TABLE]

Considering two real numbers $u$ and $v$ leads to the following results:

Theorem 3.13

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

[TABLE]

or, equivalently,

[TABLE]

Proof: From (3.4) and the $\mathcal{R}(p,q)-$ deformed Vandermonde’s formula, we get the result. $\hfill{\square}$

We obtain the $q-$ deformed Cauchy’s formulae (2.6) and (2.7) by taking $\mathcal{R}(p,q)=q.$

Theorem 3.14

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 (3.14) and

[TABLE]

$H_{n}(u;v,p,q)$ takes the following form

[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 $H_{n}(u;v)_{p,q}={[u+v]_{-n,\mathcal{R}(p,q)}}.$ Following the steps used to prove (3.30), we obtain (3.31), and the proof is achieved. $\hfill{\square}$

We recover the negative $q-$ Vandermonde’s formulae by taking $\mathcal{R}(p,q)=q$ as follows:

[TABLE]

and, alternatively,

[TABLE]

where $0<q<1.$

Lemma 3.15

[TABLE]

and

[TABLE]

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

[TABLE]

and

[TABLE]

The relation (3.30) may be written as follows:

[TABLE]

where

[TABLE]

Taking into acount the fact that

[TABLE]

the relation (3.36) is reduced to (3.34). Similarly, we obtain (3.35). $\hfill{\square}$

Setting $\epsilon_{1}=1$ and $\epsilon_{2}=q$ provides the $q-$analogs of the formulae (3.34) and (3.35) as:

[TABLE]

and

[TABLE]

Remark 3.16

By taking $\mathcal{R}(u,v)=((q-p^{-1})v)^{-1}(uv-1),$ and $n-$ a positive integer, we deduct, as particular cases, the generalized $q-$Quesne deformed Vandermonde and Cauchy’s formulae given, respectively, by:

[TABLE]

and

[TABLE]

while their negative counterparts are provided by

[TABLE]

and

[TABLE]

respectively.

3.3 $\mathcal{R}(p,q)-$ deformed binomial and negative binomial formulae

In this section, we examine in detail the $\mathcal{R}(p,q)-$ deformed binomial. Here also, $x,$ $p,$ and $q$ are real numbers with $0<q<p\leq 1;$ $n$ is a positive integer. Then,

Theorem 3.17

[TABLE]

Proof: The result follows from induction on $n.$ $\hfill{\square}$

Taking $\mathcal{R}(p,q)=q,$ we recover the $q-$ binomial formula (2.4), while $\mathcal{R}(u,v)=(p-q)^{-1}(u-v)$ gives the $(p,q)-$ binomial formula (2.5).

Theorem 3.18

[TABLE]

where $|\epsilon_{2}|<|\epsilon_{1}|.$

Proof: We have

[TABLE]

Setting $\theta={\epsilon_{2}\over\epsilon_{1}}<1$ and from [7], we obtain

[TABLE]

The proof is achieved. $\hfill{\square}$

The negative $q-$ binomial coefficient can be obtained, by setting $\mathcal{R}(u,v)=v,$ in the form:

[TABLE]

A novel negative $\mathcal{R}(p,q)-$ binomial formula can be deduced as follows:

Lemma 3.19

For $0<x<\infty,$

[TABLE]

or, in an equivalent way,

[TABLE]

Proof: Since

[TABLE]

and using [7], we obtain

[TABLE]

From (3.13) and

[TABLE]

we get (3.49). Replacing $x$ by $x^{-1},$ $\epsilon_{1}$ by $\epsilon^{-1}_{1}$ and $\epsilon_{2}$ by $\epsilon^{-1}_{2}$ leads to (3.50), and the proof is achieved. $\hfill{\square}$

Remark 3.20

The negative $q-$binomial formula can be obtained by taking $\mathcal{R}(u,v)=v:$

[TABLE]

which can also be translated into the form:

[TABLE]

Theorem 3.21

The following orthogonality relations hold:

[TABLE]

and

[TABLE]

where $\delta_{n,\kappa}$ is the Kronecker delta, $n$ and $\kappa$ are positive integers.

Proof: Since

[TABLE]

then, the relation (3.54) may be expressed as:

[TABLE]

From the expression

[TABLE]

we get (3.55). Therefore, the result holds. $\hfill{\square}$

The $q-$ deformed orthogonality relations (2.8) and (2.9) can be recovered by putting $\mathcal{R}(p,q)=q.$

Corollary 3.22

The inversion of the $\mathcal{R}(p,q)-$ binomial formula is provided by:

[TABLE]

In particular,

[TABLE]

Proof: From the relation (3.10) and replacing $\epsilon_{1}$ by $\epsilon_{1}^{-1},$ $\epsilon_{2}$ by $\epsilon_{2}^{-1},$ $x$ by $-x$, $n$ by $\kappa$ and $\kappa$ by $r$ in (3.45), we get

[TABLE]

Multiplying the members of the above expression by (-1)^{\kappa}\,\epsilon_{1}^{{\kappa\choose 2}}\,\epsilon_{2}^{{\kappa\choose 2}}\bigg{[}\begin{array}[]{c}n\\ \kappa\end{array}\bigg{]}_{\mathcal{R}(p,q)}, it comes

[TABLE]

Summing the expression (3.63) for $\kappa\in\{1,2,\cdots,n\},$ using (3.55) and

[TABLE]

we obtain (3.58). Replacing $x$ by $\epsilon^{-x}_{1}\,\epsilon^{x}_{2}$ in (3.58) and using

[TABLE]

yield (3.59). $\hfill{\square}$

Remark 3.23

Putting $\epsilon_{1}=1$ and $\epsilon_{2}=q,$ we obtain the inversion of the $q-$ binomial formula [7]:

[TABLE]

In particular,

[TABLE]

The inversion of the deformed binomial formulae (3.58) and (3.59) leads to the following results:

Lemma 3.24

[TABLE]

and

[TABLE]

Proof: Replacing $x$ by $x^{-1}$ in equation (3.58), and using the expression

[TABLE]

we obtain (3.70). Also replacing $\epsilon_{1}$ by $\epsilon^{-1}_{1}$ and $\epsilon_{2}$ by $\epsilon^{-1}_{2}$ in (3.59), we get

[TABLE]

Using the relations (3.8), (3.10) and after computation, the result holds. $\hfill{\square}$

It is worth noticing the following relevant identities from $\mathcal{R}(p,q)-$ binomial and negative $\mathcal{R}(p,q)-$ binomial formulae, as exposed and proved below:

• •

Lemma 3.25

Let $r,$ $s$ and $n$ be positive integers. Then, the following relations hold:

[TABLE]

[TABLE]

and

[TABLE]

Proof: From the $\mathcal{R}(p,q)-$ binomial formula and

[TABLE]

we get

[TABLE]

where

[TABLE]

Using

[TABLE]

and

[TABLE]

we obtain

[TABLE]

giving (3.71). Setting $s=r$ and $n=r-m$ in (3.71), and using

[TABLE]

the relation (3.72) holds. Since

[TABLE]

and putting $r=n=s$ in (3.71), we get (3.73), what achieves the proof. $\hfill{\square}$

Remark 3.26

Note the following:

1. (1)

*The relation (3.71) is a particular case of the *$\mathcal{R}(p,q)-$deformed Cauchy formula (3.28), with $u=r$ and $v=s.$ 2. (2)

The particular case of $q-$ deformation is achieved from the above formulae by taking $\mathcal{R}(p,q)=q$ as follows:

[TABLE]

[TABLE]

and

[TABLE]

• •

Lemma 3.27

Let $r,$ $s,$ and $n$ be positive integers. Then,

[TABLE]

Proof: From the relation

[TABLE]

where $|x|<1,$ and from the negative $\mathcal{R}(p,q)-$binomial formula, we get

[TABLE]

where

[TABLE]

Setting $j=n-\kappa,$ and after computation, it follows

[TABLE]

yielding the result. $\hfill{\square}$

Remark 3.28

1. (a)

This formula constitutes a particular version of the $\mathcal{R}(p,q)-$ Cauchy formula (3.28), corresponding to negative integers $x=-r$ and $y=-s.$ 2. (b)

*Putting $\mathcal{R}(x,y)=y,$ we recover the $q-$ deformed analog of this Lemma as: *

[TABLE]

• •

Lemma 3.29

For $n,$ $m,$ and $\kappa,$ positive integers, such that $\kappa\leq m\leq n,$ we have:

[TABLE]

Proof: Multiplying both the members of the negative $\mathcal{R}(p,q)-$ binomial formula

[TABLE]

by $x^{m}$ and putting $n=m+j,$ we obtain

[TABLE]

Similarly, we get

[TABLE]

and

[TABLE]

Using

[TABLE]

and the above relations, we arrive at the result. $\hfill{\square}$

Note that the $q-$ deformed version can be recovered by taking $\mathcal{R}(p,q)=q:$

[TABLE]

Remark 3.30

It is worth noticing that

1. (1)

The generalized $q-$Quesne deformed binomial and negative binomial formulae can straightforwardly be retrieved from the general formalism as particular cases as follows:

• (i)

[TABLE]

• (ii)

[TABLE]

• (iii)

[TABLE]

and

[TABLE]

• (iv)

[TABLE]

In particular,

[TABLE] 2. (2)

The associated deformed binomial formulae (3.80) and (3.81) can also be rewritten as follows:

[TABLE]

and

[TABLE]

Further, the following identities are retrieved from the above Lemmas:

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

or

[TABLE]

3.4 $\mathcal{R}(p,q)-$ deformed Stirling numbers

Definition 3.31

Let $x,$ $j,$ $p,$ and $q$ be real numbers such that $0<q<p\leq 1.$ Then, the noncentral $\mathcal{R}(p,q)-$ deformed factorial of $x$ of order $n$ and of noncentrality parameter $j$ is defined by:

[TABLE]

where $n$ is a positive integer.

Taking $j=0$ in (3.84), we obtain the $\mathcal{R}(p,q)-$ deformed factorial of $x$ of order $n.$ Following the relation

[TABLE]

the equation (3.84) takes the form

[TABLE]

Further, we get a polynomial of the $\mathcal{R}(p,q)-$ deformed number of degree $n$ as follows:

[TABLE]

or

[TABLE]

Equivalently,

[TABLE]

The coefficients $s_{\mathcal{R}(p,q)}(n,\kappa;j)$ and $S_{\mathcal{R}(p,q)}(n,\kappa;j)$ are the noncentral $\mathcal{R}(p,q)-$ deformed Stirling numbers of the first and second kind, respectively.

Remark 3.32

1. (1)

For $j=0,$ these deformed numbers are reduced to $s_{\mathcal{R}(p,q)}(n,\kappa;0)=s_{\mathcal{R}(p,q)}(n,\kappa)$ and $S_{\mathcal{R}(p,q)}(n,\kappa;0)=S_{\mathcal{R}(p,q)}(n,\kappa),$ which are nothing but the $\mathcal{R}(p,q)-$ deformed Stirling numbers of the first and second kinds, respectively. 2. (2)

Taking $\mathcal{R}(x,y)=y,$ we recover the $q-$ deformed Stirling numbers of first and second kinds (2.10) and (2.11) .

Lemma 3.33

[TABLE]

and

[TABLE]

where $n$ is a positive interger, and $|s_{\mathcal{R}(p,q)}(n,\kappa,j)|$ stands for the deformed absolute noncentral $\mathcal{R}(p,q)-$ deformed Stirling number of the first kind.

Proof: Since

[TABLE]

then,

[TABLE]

is also a polynomial of the $\mathcal{R}(p,q)-$deformed number $[x]_{\mathcal{R}(p,q)}$ of degree $n,$ and after computation, for $n\in\mathbb{N}\backslash\{0\},$ we get

[TABLE]

Furthermore,

[TABLE]

[TABLE]

Replacing now $-x$ by $x$ and $\epsilon_{1}^{-1}$ by $\epsilon_{1},$ and $\epsilon_{2}^{-1}$ by $\epsilon_{2}$ in (3.85), the result follows. $\hfill{\square}$

Theorem 3.34

The $\mathcal{R}(p,q)-$ deformed Stirling numbers of the first and second kinds verify the orthogonality relations:

[TABLE]

and

[TABLE]

Proof: From the relations (3.85) and (3.86), we write

[TABLE]

giving, after computation, (3.88). Similarly, we obtain (3.89). $\hfill{\square}$

The next statement is also valid.

Theorem 3.35

*For $n\in\mathbb{N}\backslash\{0\}$ and $\kappa\in\{1,2,\cdots,n+1\},$ the noncentral $\mathcal{R}(p,q)-$ deformed Stirling numbers of the first and second kinds, $s_{\mathcal{R}(p,q)}$ and $S_{\mathcal{R}(p,q)},$ obey, respectively, the recursion relations *

[TABLE]

with initial conditions $s_{\mathcal{R}(p,q)}(0,0,j)=1,$ $s_{\mathcal{R}(p,q)}(n,0,j)=\epsilon_{2}^{{n\choose 2}+jn}\,{[-j]_{n,\mathcal{R}(p,q)}}\geq 0$ and $s_{\mathcal{R}(p,q)}(0,\kappa,j)=0,$ $j\geq 0;$ and

[TABLE]

with initial conditions $S_{\mathcal{R}(p,q)}(0,0,j)=1,$ $S_{\mathcal{R}(p,q)}(n,0,j)=[j]^{n}_{\mathcal{R}(p,q)},$ $n\geq 0,$ and $S_{\mathcal{R}(p,q)}(0,\kappa,j)=0,$ $\kappa\geq 0.$

Proof:

1. (1)

Let us consider the relation

[TABLE]

or

[TABLE]

From the relation (3.85) and the expansion of both members of the recursion relation (3.92) into powers of $\mathcal{R}(p^{x},q^{x}),$ we get

[TABLE]

which gives (3.90). We use the relation (3.85) to get the initial conditions. 2. (2)

Similarly, consider

[TABLE]

and use the relation (3.86) to obtain

[TABLE]

Thus, we get the relation (3.91). The initial conditions follow from (3.87). $\hfill{\square}$

Theorem 3.36

For fixed $\kappa$, the generating function of noncentral $\mathcal{R}(p,q)-$ deformed Stirling numbers of the second kind is given as follows:

[TABLE]

or, equivalently, in product form

[TABLE]

for $|v|<{\epsilon^{-x+j}_{1}\over[r+j]_{\mathcal{R}(p,q)}}.$

Proof: We assume that the series (3.94) converges. Multiplying the expression (3.91) by $v^{n+1}$ and summing the resulting relation for $n=\kappa-1,\kappa,\cdots,$ and $\kappa\in\mathbb{N}\backslash\{0\},$ we obtain

[TABLE]

which implies

[TABLE]

By induction and consideration that

[TABLE]

we obtain (3.94) and (3.95).$\hfill{\square}$

Remark 3.37

The generating function of the noncentral $q-$ Stirling numbers of the second kind can be obtained as:

[TABLE]

where $\kappa\in\mathbb{N}$ and $|v|<{1\over[r+j]_{q}}.$

Lemma 3.38

For $\kappa\in\mathbb{N}$ and $n=\kappa,\kappa+1,\cdots,$ the reciprocal noncentral $\mathcal{R}(p,q)-$ deformed factorial ${{1\over[t-x]_{\kappa+1,\mathcal{R}(p,q)}}}$ is expanded into the reciprocal $\mathcal{R}(p,q)-$ deformed powers ${1\over[t]^{n+1}_{\mathcal{R}(p,q)}}$ as follows:

[TABLE]

while the reciprocal $\mathcal{R}(p,q)-$ deformed powers $[t]^{-\kappa-1}_{\mathcal{R}(p,q)}$ is expanded into the reciprocal noncentral $\mathcal{R}(p,q)-$ deformed factorial ${{1\over[t-x]_{n+1,\mathcal{R}(p,q)}}}$ as below expressed:

[TABLE]

where $t>\kappa+x.$

Proof: Setting $v={\epsilon^{t-r-x}_{1}\over[t]_{\mathcal{R}(p,q)}}$ in (3.94) and (3.95), we obtain

[TABLE]

and

[TABLE]

Thus,

[TABLE]

and after rearranging, we find (3.96). Moreover, let us fix $\kappa$ in (3.96). Replacing $n$ by $m$ and $\kappa$ by $n,$ we get

[TABLE]

Multiplying the result by $\epsilon^{-{n+1\choose 2}-x(n+1)}_{2}\,s_{\mathcal{R}(p,q)}(n;\kappa,x),$ and summing for $n=\kappa,\kappa+1,\cdots,$ we find

[TABLE]

By the orthogonality relation (3.89), we have

[TABLE]

and the result follows. $\hfill{\square}$

Theorem 3.39

The noncentral $\mathcal{R}(p,q)-$ deformed Stirling numbers of the first and second kinds are given, respectively, by

[TABLE]

and

[TABLE]

*where $n\in\mathbb{N}\backslash\{0\},$ $\kappa\in\{1,2,\cdots,n\},$ and $x\in\mathbb{N}.$ *

Proof: From the relation (3.10) and replacing $p=p^{-1},$ $q=q^{-1},$ $\epsilon_{1}=\epsilon^{-1}_{1},$ $\epsilon_{2}=\epsilon^{-1}_{2},$ $x=\epsilon^{x-r}_{1},$ $y=-\epsilon^{x-r}_{2},$ $\kappa=j,$ $n=\kappa$ in the $\mathcal{R}(p,q)-$ binomial formula (3.45), we get

[TABLE]

Multipliying (3.99) by $\epsilon^{{n\choose 2}+rn}_{2},$ and using

[TABLE]

we obtain

[TABLE]

Comparing the above relation with (3.85), we obtain (3.97). Furthermore,

[TABLE]

Using (3.59), we get

[TABLE]

and from (3.87), we obtain (3.98). $\hfill{\square}$

Remark 3.40

Taking $\mathcal{R}(p,q)=q,$ we obtain the noncentral $q-$ Stirling numbers of the first and second kinds as:

[TABLE]

and

[TABLE]

*where $n\in\mathbb{N}\backslash\{0\},$ $\kappa\in\{1,2,\cdots,n\},$ and $x\in\mathbb{N}.$ *

Corollary 3.41

Let $\kappa$ and $j$ be positive integers. Then, the following relations hold:

[TABLE]

and

[TABLE]

where $x$ is an integer and $0<q<p\leq 1.$

Proof: Replacing $j$ by $i,$ $n$ by $m,$ $\kappa$ by $j$ and $r=0$ in (3.97), we get

[TABLE]

Multiplying this result by

[TABLE]

and summing for all $m=j,j+1,\cdots,\kappa,$ we obtain

[TABLE]

Similarly, in (3.98), we replace $n$ by $m,$ $j$ by $i,$ $r=0,$ multiply the resulting expression by

[TABLE]

and sum it for all $m=j,j+1,\cdots,\kappa,$ to get the result. $\hfill{\square}$

Note that we obtain the $q-$ deformed formulae (2.12) and (2.13) by taking $\mathcal{R}(x,y)=y.$

Lemma 3.42

Let $u$ be a natural number. Then, the following relations hold:

[TABLE]

and

[TABLE]

where \zeta_{u,p,q}=\displaystyle\sum_{j=1}^{u}\epsilon^{(j-1)(x+1)}_{1}\Bigg{(}\mathcal{R}(p^{j},q^{j})\Bigg{)}^{-1} and $x\in\mathbb{N}.$

Proof: From the triangular recursion relation of the $\mathcal{R}(p,q)-$ deformed Stirling numbers of the first kind (3.90), we set $\kappa=1,$ $j=0,$ and obtain the first-order recursion relation

[TABLE]

with $n\in\mathbb{N}\backslash\{0,1\}$ and $s_{\mathcal{R}(p,q)}(1,1)=s_{\mathcal{R}(p,q)}(0,0):=1.$ By iteration, we get (3.105). Setting $\kappa=2$ also leads to the recursion relation

[TABLE]

where

[TABLE]

which is solved to give the required expression. $\hfill{\square}$

Taking $\mathcal{R}(p,q)=q,$ we recover

[TABLE]

and

[TABLE]

where $\zeta_{u,p,q}=\displaystyle\sum_{j=1}^{u}[j]^{-1}_{q}!.$

Remark 3.43

The particular case of the generalized $q-$Quesne quantum algebra is here worthy of attention as matter of illustration. Indeed, the signless (or absolute) noncentral generalized $q-$ Quesne Stirling number of the first kind is given by

[TABLE]

or

[TABLE]

where $n$ is a positive interger. The related deformed Stirling numbers of the first and second kinds verify the orthogonality relations

[TABLE]

and

[TABLE]

*They obey, respectively, the following recursion relations: *

[TABLE]

with the initial conditions $s^{Q}_{p,q}(0,0,j)=1,$ $s^{Q}_{p,q}(n,0,j)={q^{-{n\choose 2}-n(j-1)}\over p^{n}}\,{[-j]^{Q}_{n,p,q}},$ $n\geq 0.$ and $s^{Q}_{p,q}(0,\kappa,j)=0,$ $j\geq 0;$ and

[TABLE]

with the initial conditions $s^{Q}_{p,q}(0,0,j)=1,$ s^{Q}_{p,q}(n,0,j)=\big{(}{q\over p}[j]^{Q}_{p,q}\big{)}^{n}, $n\geq 0,$ and $s^{Q}_{p,q}(0,\kappa,j)=0,$ $\kappa\geq 0.$ For fix $\kappa$, they are generated by the function given by

[TABLE]

developed in the product form as:

[TABLE]

for $|v|<{p^{-x+j+1}\over q\,[r+j]^{Q}_{p,q}}.$ Their reciprocal factorial ${[t-x]^{Q}_{-\kappa-1,p,q}}$ is expanded into reciprocal generalized $q-$ Quesne powers \big{(}[t]^{Q}_{p,q}\big{)}^{-n-1} as follows:

[TABLE]

while their reciprocal powers \big{(}[t]_{p,q}^{Q}\big{)}^{-\kappa-1} are spanned in the reciprocal noncentral generalized $q-$ Quesne factorial ${[t-x]_{-n-1,p,q}^{Q}}$ as:

[TABLE]

*where $t>\kappa+x.$ Moreover, *

[TABLE]

and

[TABLE]

where $n\in\mathbb{N}\backslash\{0\},$ $\kappa\in\{1,2,\cdots,n\},$ and $x\in\mathbb{N}.$ In particular,

[TABLE]

For positive integers $\kappa$ and $j,$

[TABLE]

and

[TABLE]

where $x$ is an integer and $0<q<p\leq 1.$ Furthermore, given a natural number $u$,

[TABLE]

and

[TABLE]

where $\zeta_{u,p,q}=\displaystyle\sum_{j=1}^{u}{p^{(j-1)(x+1)+1}\over q\,[j]^{Q}_{p,q}}.$

3.5 $\mathcal{R}(p,q)-$ deformed Bell numbers

Let us consider a simple finite graph $\bf G$ with $n$ vertices, and $\kappa$ independent blocks. $V(\bf G)$ is a set of vertices of $\bf G.$ The vertices are partitioned into $\kappa$ independent blocks by $\Pi=\{V_{1},V_{2},\cdots,V_{\kappa}\}.$ We define the $\mathcal{R}(p,q)-$deformed weight as follows:

[TABLE]

where $|V_{j}|$ is the cardinality of the set $V.$ The $\mathcal{R}(p,q)-$ deformed Stirling number of the second kind for the graph is expressed by

[TABLE]

where S_{\mathcal{R}(p,q)}\big{(}\bf G,0\big{)}:=0, and $\Lambda$ denotes the independent partitions $\pi$ of $V(\bf G).$ Analogously, the $\mathcal{R}(p,q)-$ deformed Bell number for the graph is defined by

[TABLE]

Theorem 3.44

Let $\bar{T}_{n}$ be the dual path graph of $\bf G.$ Then, the $\mathcal{R}(p,q)-$ deformed Stirling numbers of the second kind and the $\mathcal{R}(p,q)-$ deformed Bell numbers for the graph $\bf G$ are given, respectively, by

[TABLE]

and

[TABLE]

where $n\in\mathbb{N}.$

Proof: Let us consider the expression

[TABLE]

For $n=0,$ the relation (3.109) is true. We assume that (3.109) is true for all $n$ and prove it for $l=n+1.$ Consider

[TABLE]

where

[TABLE]

and

[TABLE]

Using Eq.(3.14) and after computation, we obtain

[TABLE]

Thus, the proof is achieved. $\hfill{\square}$

3.6 Application

We consider the dual path graph $\bar{T}_{5}.$ It has $4$ independent partitions into $4$ blocks given as follows:

[TABLE]

[TABLE]

The $\mathcal{R}(p,q)-$deformed weight is given by

[TABLE]

Hence,

[TABLE]

Similarly, we get W_{\mathcal{R}(p,q)}(\Gamma_{2})=\Big{(}{\epsilon_{2}\over\epsilon_{1}}\Big{)}^{7}, W_{\mathcal{R}(p,q)}(\Gamma_{3})=\Big{(}{\epsilon_{2}\over\epsilon_{1}}\Big{)}^{8} and W_{\mathcal{R}(p,q)}(\Gamma_{4})=\Big{(}{\epsilon_{2}\over\epsilon_{1}}\Big{)}^{9}. Finally,

[TABLE]

Remark 3.45

1. (1)

Note that the $q-$ Stirling number of the second kind and $q-$ Bell number of the graph can easily be derived by taking $\mathcal{R}(p,q)=q$ as follows:

[TABLE]

and

[TABLE]

where $n\in\mathbb{N}.$ 2. (2)

The generalized $q-$Quesne Stirling number of the second kind and the Bell number for the dual path graph are, respectively, given by

[TABLE]

and

[TABLE]

where $n\in\mathbb{N}.$

3.7 $\mathcal{R}(p,q)-$ deformed factorial and binomial moments

For a study on $q-$ factorial and $q-$ binomial moments, see [7]. We deal here with the $\mathcal{R}(p,q)-$generalization. For that, we consider a nonnegative integer-valued discrete random variable $X,$ and $g(x)=P(X=x),$ $x\in\mathbb{N}\backslash\{0\},$ the probability distribution of $X.$

[TABLE]

[TABLE]

referred to the $r^{th}-$order $\mathcal{R}(p,q)-$ factorial and $r^{th}-$order $\mathcal{R}(p,q)-$ binomial moments, respectively, of the random variable $X.$ In the particular case of $r=1,$ we define the $\mathcal{R}(p,q)-$ mean value, also called the $\mathcal{R}(p,q)-$ expectation value, of $X$ by

[TABLE]

The $\mathcal{R}(p,q)-$ variance of $X$ is then obtained as

[TABLE]

Since $[X-1]_{\mathcal{R}(p,q)}=\epsilon^{-1}_{2}[X]_{\mathcal{R}(p,q)}-\epsilon^{-1}_{2}\epsilon^{X-1}_{1},$ and ${[X]_{2,\mathcal{R}(p,q)}}=[X]_{\mathcal{R}(p,q)}[X-1]_{\mathcal{R}(p,q)},$ then

[TABLE]

Theorem 3.46

The binomial moment is given as function of the $\mathcal{R}(p,q)-$ binomial moment as follows:

[TABLE]

while the factorial moment is given in terms of the $\mathcal{R}(p,q)-$ factorial moment by

[TABLE]

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

Proof: Multiplying (3.103) by the probability distribution $g(x)$ and summing for all $x\in\mathbb{N},$ we deduce (3.115) from (3.116). Moreover, from

[TABLE]

and the relation (3.116), we derive (3.117). $\hfill{\square}$

Remark 3.47

Putting $\mathcal{R}(u,v)=v,$ we obtain the usual binomial moment as function of the $q-$ binomial moment as:

[TABLE]

while the usual factorial moment is given in terms of the $q-$ factorial moment by

[TABLE]

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

Theorem 3.48

The ${\mathcal{R}(p,q)}-$deformed probability distribution $g(x)$ of a discrete random variable $X$ is given by the absolutely convergent series

[TABLE]

Proof: Replacing $x$ by $\kappa$ and $r$ by $m$ in expression (3.115), multiplying it by

[TABLE]

and summing for all $m=x,x+1,\cdots,$ we obtain

[TABLE]

$\hfill{\square}$

Note that the probability distribution (2.14) can be retrieved by taking $\mathcal{R}(u,v)=v.$

Remark 3.49

*The particular case of the generalized $q-$Quesne factorial and binomial moment, and probability distribution is detailed as follows: *

1. (1)

Let $X$ be a nonnegative integer-valued discrete random variable and $h(x)=P(X=x),$ $x\in\mathbb{N}\backslash\{0\},$ the probability distribution of $X.$ Assume the convergence of the series:

[TABLE]

[TABLE]

*here designated by *$r^{th}-$*order generalized $q-$ Quesne factorial and *$r^{th}-$order generalized $q-$ Quesne binomial moment, respectively, of the random variable $X.$ In the particular case of $r=1,$ we deduce the generalized $q-$ Quesne mean value, also called the generalized $q-$ Quesne expectation value, of $X$ by

[TABLE]

The associated variance of $X$ is then obtained as

[TABLE]

or, equivalently,

[TABLE] 2. (2)

*Its deformed binomial moment is given by *

[TABLE]

while the factorial moment is expressed in terms of the generalized $q-$ Quesne factorial moment by

[TABLE]

where $j\in\mathbb{N}\backslash\{0\},$ $\tau\in\mathbb{N},$ and $s^{Q}_{p,q}$ is the generalized $q-$ Quesne Stirling number of the first kind. 3. (3)

The probability distribution $h(x),x\in\mathbb{N},$ of a discrete random variable $X$ is given by

[TABLE]

4 Concluding remarks

In this paper, we have developed and illustrated the fundamentals of $\mathcal{R}(p,q)-$ deformed combinatorics, with a special focus on factorials, binomial coefficients, Vandermonde’s and Cauchy’s formulae, binomial formula, Stirling numbers, and Bell numbers induced by the $\mathcal{R}(p,q)-$ deformed quantum algebra. These results have also been derived and discussed in the particular case of the so-called generalized $q-$ Quesne deformed quantum algebra. Relevant properties have been deduced and analyzed in this framework.

Acknowledgements

This work is supported by TWAS Research Grant RGA No.17 - 542 RG/ MATHS/AF/AC _G - FR3240300147. The ICMPA-UNESCO Chair is in partnership with the Association pour la Promotion Scientifique de l’Afrique (APSA), France, and Daniel Iagolnitzer Foundation (DIF), France, supporting the development of mathematical physics in Africa. MNH acknowledges his colleagues Nicholas M. J. Hall, Isabelle Dadou, Yves Morel, Catherine Jeandel, the staff of the UMR 5566 Laboratoire d’Etudes en Géophysique et Océanographique Spatiales (LEGOS) of the Faculté des Sciences et Ingénierie, Michael Toplis, Director of the Observatoire Midi-Pyrénées, and Nguyen Tien Zung of the Institut de Mathématiques de Toulouse for their hospitality during his stay, as visiting professor, at the Université Toulouse III Paul Sabatier, where this work has been completed. MNH is also grateful to Mrs Sophie Raynaud, Head of International Relations Office, and Prof. Fabrice Dumas, Vice-president in charge of International Relations, Université Toulouse III Paul Sabatier, for all their solicitude.