R(p,q)- analogs of discrete distributions: general formalism and application
Mahouton Norbert Hounkonnou Fridolin Melong


ℛ(p,q)- analogs of discrete distributions]ℛ(p,q)- analogs of discrete distributions: general formalism and application
Mahouton Norbert Hounkonnou]Mahouton Norbert Hounkonnou*
* Corresponding author
Mahouton Norbert Hounkonnou: International Chair in Mathematical Physics
	and Applications(ICMPA-UNESCO Chair), University of Abomey-Calavi, 072 B.P. 50 Cotonou, Republic of Benin
norbert.hounkonnou@cipma.uac.bj, with copy to hounkonnou@yahoo.fr 



Fridolin Melong: International Chair in Mathematical Physics
	and Applications(ICMPA-UNESCO Chair), University of Abomey-Calavi, 072 B.P. 50 Cotonou, Republic of Benin
fridomelong@gmail.com

[2010] Primary 17B37; Secondary 60E05




In this paper, we define and discuss  ℛ(p,q)- deformations of  basic univariate discrete distributions  of the probability theory. We mainly focus on binomial, Euler, Pólya and inverse Pólya   distributions. We discuss relevant 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. 


[
    Fridolin Melong
    December 30, 2023
=====================




§ INTRODUCTION

Deformed quantum algebras, namely   q- deformed  algebras <cit.> and their extensions to  (p,q)-analogs <cit.>,  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 <cit.>.
The q- combinatorics  and q- hypergeometric series were examined in
<cit.>.
A theory of (p,q)- analogs of  binomial coefficients and (p,q)- Stirling numbers was elaborated  in <cit.>.
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 <cit.>.
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 <cit.>, we introduced the ℛ(p,q)- deformed quantum  algebras  as  generalizations of known deformed quantum algebras <cit.>.
In <cit.>, we performed the 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 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  ℛ(p,q)- deformed quantum  algebras?

This paper
is
organized as follows:
In section 2, we briefly recall the notions of  ℛ(p,q)- deformed numbers,  ℛ(p,q)- deformed factorial, 
ℛ(p,q)-binomial coefficients, 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.  ℛ(p,q)- deformed combinatorics and  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. 


§ QUICK OVERLOOK ON KNOWN RESULTS




 §.§ 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 ℂ×ℂ by: 
    ℛ(u,v)= ∑_s,t=-l^∞r_stu^sv^t,

			
			where r_st are complex numbers, l∈ℕ\{ 0}, ℛ(p^n,q^n)>0,  ∀ n∈ℕ, and ℛ(1,1)=0.
			 We denote by  𝔻_R={ z∈ℂ: |z|<R} a complex disc, where  R is the radius of convergence of the series (<ref>),  and by   𝒪(𝔻_R) the set of holomorphic functions defined on 𝔻_R.
			Define the  ℛ(p,q)- deformed numbers  by <cit.>:
				
    [n]_ℛ(p,q):=ℛ(p^n,q^n),   n∈ℕ

generalizing known particular numbers  as follows:

	
  * q-  Arick-Coon-Kuryskin deformation <cit.>
	
    ℛ(p,q):=ℛ(1,q)={p=1}-q  1-q    
    	[n]_q =1-q^n 1-q.
 
	
  * q- Quesne  deformation <cit.>
	
    ℛ(p,q):=ℛ(1,q)={p=1}-q^-1 1-q    
    	[n]^Q_q =1-q^-n q-1.

	
  * (p,q)- Jagannathan-Srinivasa deformation <cit.>
	
    ℛ(p,q)=p-q  p-q    
    	[n]_p,q=p^n-q^n  p-q.

	
  * (p^-1,q)- Chakrabarty - Jagannathan deformation <cit.>
	
    ℛ(p,q)=1-p q  (p^-1-q)p    
    	[n]_p,q=p^-n-q^n  p^-1-q.

	
  * Hounkonnou-Ngompe  generalization of q- Quesne deformation <cit.>
	
    ℛ(p,q)=p q-1  (q-p^-1)q     
    	[n]^Q_p,q=p^n-q^-n q- p^-1.

	
  * (p,q,μ,ν,g)- Hounkonnou-Ngompe  multi-parameter deformation <cit.>
	
    ℛ(p,q)=g(p,q)q^ν p^μp q-1  (q-p^-1)q     
    	[n]^μ,ν_p,q,g=g(p,q)q^ν n p^μ np^n-q^-n q- p^-1,

	where 0< pq<1, p^μ<q^ν-1, p>1 and 
		g is a well behaved real   non-negative function of deformation parameters p and q such that g(p,q)⟶ 1 as(p,q)⟶ (1,1).


	Define now the
 ℛ(p,q)- deformed factorials

    [n]!_ℛ(p,q):={[                   1     n=0;                            ; ℛ(p,q)⋯ℛ(p^n,q^n)     n≥ 1, ] .

and the  ℛ(p,q)- deformed binomial coefficients

    [[ m; n ]]_ℛ(p,q) := [m]!_ℛ(p,q)/[n]!_ℛ(p,q)[m-n]!_ℛ(p,q),   m,n=0,1,2,⋯,   m≥ n

satisfying the relation

    [[ m; n ]]_ℛ(p,q)=[[   m; m-n ]]_ℛ(p,q),   m,n=0,1,2,⋯,   m≥ n.

Consider the following linear operators defined on  𝒪(𝔻_R) by (see <cit.> for more details):

    Q:⟼ Q(z):   =   (qz),
      P:⟼ P(z):   =   (pz),

and the ℛ(p,q)- derivative given by:

    ∂_ℛ( p,q):=∂_p,qp-q/P-Qℛ( P,Q)=p-q/p^P-q^Qℛ(p^P,q^Q)∂_p,q

 extending known  derivatives as follows:

	
  (i)  q- Heine derivative <cit.>.
	
    ℛ(p,q):=ℛ(1,q)=1
    	    ∂_q(z)=(z)-(qz)/z(1-q).
 
	
  (ii)  q- Quesne  derivative <cit.>.
	
    ℛ(p,q):=ℛ(1,q)=1-q^-1 1-q    ∂_q(z)=(z)-(q^-1z)/z(q-1).

	
  (iii) (p,q)- Jagannathan-Srinivasa derivative <cit.>
	
    ℛ(p,q)=1    ∂_p,q(z)=(pz)-(qz)/z(p-q).

	
  (iv) (p^-1,q)- Chakrabarty - Jagannathan derivative <cit.>.
	
    ℛ(p,q)=1-p q  (p^-1-q)p    ∂_p^-1,q(z)=(p^-1z)-(qz)/z(p^-1-q).

	
  (v) Hounkonnou-Ngompe  generalization of q- Quesne derivative <cit.>.
	
    ℛ(p,q)=p q-1  (q-p^-1)q    ∂^Q_p,q(z)=(pz)-(q^-1z)/z(q-p^-1).

	
  (vi) (p,q,μ,ν,g)- Hounkonnou-Ngompe  multi-parameter derivative <cit.>.
	
    ℛ(p,q)=g(p,q)q^ν p^μp q-1  (q-p^-1)q    ∂^μ,ν_p,q(z)=g(p,q)(zq^ν p^μ-1)-(zq^ν -1 p^μ)/z(q-p^-1).


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

    A A^†= [N+1]_ℛ(p,q),       A^†  A = [N]_ℛ(p,q).
          [N,  A] = - A,           [N, A^†] = A^†

with its realization on  𝒪(𝔻_R) given by:

    A^† := z,     A:=∂_ℛ(p,q),      N:= z∂_z,
 
where ∂_z:=∂/∂ z is the usual derivative on ℂ.


 §.§ Hounkonnou-Ngompe generalization of q- Quesne algebra

It corresponds to ℛ(p,q)=p-q^-1 q-p^-1, with generalized q-Quesne number and factorial <cit.>:
	
    [n]^Q_p,q=p^n-q^-n q-p^-1,   [n]!^Q_p,q:={[                     1     n=0;                              ; [1]^Q_p,q⋯[n]^Q_p,q     n≥ 1, ] .

and  following properties:
	
    [-y]_p,q^Q   =    -p^-yq^y[y]_p,q^Q,
     [x+y]_p,q^Q   =    q^-y[x]_p,q^Q+p^x[y]_p,q^Q= p^y[x]_p,q^Q+q^-x[y]_p,q^Q,
     [x-y]_p,q^Q   =    q^y[x]_p,q^Q-p^x-yq^y[y]_p,q^Q= p^-y[x]_p,q^Q+p^-yq^y-x[y]_p,q^Q,
     [x]_p,q^Q    =   q-p^-1/p-q^-1[2]_p,q^Q[x-1]_p,q^Q-pq^-1[x-2]_p,q^Q.

The  generalized q-Quesne binomial coefficient
	
    [[ n; κ ]]_p,q^Q=
    	[n]^Q_p,q!/[κ]^Q_p,q![n-κ]^Q_p,q!,
    	   0≤κ≤ n ∈ℕ,

satisfies the relations

	
    [[ n; k ]]_p,q^Q
    	   =   [[   n; n-k ]]_p,q^Q=
    	p^k(n-k)[[ n; k ]]_1/qp=
    	p^k(n-k)[[   n; n-k ]]_1/qp
     [[ n+1;   k ]]_p,q^Q    =   
    	p^k[[ n; k ]]_p,q^Q
    	+q^-n-1+k[[   n; k-1 ]]_p,q^Q,
     [[ n+1;   k ]]_p,q^Q    =   
    	p^k[[ n; k ]]_p,q^Q +
    	p^n+1-k[[   n; k-1 ]]_p,q^Q
    	-(p^n-q^-n)
    	[[ n-1; k-1 ]]_p,q^Q.
	

	Finally, the algebra 𝒜^Q_p,q,
	generated by the set {1, A, A^†, N },
	associated with the  generalized q-Quesne deformation, satisfies the following commutation relations:
	
    p^-1A A^†- A^† A= q^-N-1,          qA A^†- A^† A= p^N+1
    	[N, A^†]= A^†,                   [N, A]= -A.

	 We also have the following identities used in the sequel:
	
    [n]_p,q^-1= p^-1 q[n]^Q_p,q    [[ n; κ ]]_p,q^-1=[[ n; κ ]]^Q_p,q,
 
	
    [n]^μ,ν_p,q,g=g(p,q)q^ν n p^μ n[n]^Q_p,q    [[ n; κ ]]^μ,ν_p,q,g=q^νκ(n-κ) p^μκ(n-κ)[[ n; κ ]]^Q_p,q,

	where μ, ν, p, and q are such that
	0< pq<1, p^μ<q^ν-1, p>1.


 g is a well behaved real   non-negative function of deformation parameters p and q such that g(p,q)⟶ 1 as (p,q)⟶ (1,1).

	

§ MAIN RESULTS

	

 §.§ ℛ(p,q)- deformed combinatorics

	Given the above results,  we now  introduce functions ϵ_i >0, with i=1,2,  characterizing  particular deformations as follows:
	
		
  (i) q- Arick-Coon-Kuryskin  deformation
		
    ϵ_1=1     ϵ_2=q.

		
  (ii)  q- Quesne deformation
		
    ϵ_1=1     ϵ_2=q^-1.

		
  (iii) (p,q)- Jagannathan-Srinivasa deformation
		
    ϵ_1=p     ϵ_2=q.

	
  (iv) (p^-1,q)- Chakrabarty -Jagannathan deformation
	
    ϵ_1=p^-1    ϵ_2=q.

	
  (v) Hounkonnou-Ngompe  generalization of q- Quesne deformation
	
    ϵ_1=p     ϵ_2=q^-1.

	
  (vi) (p,q,μ,ν,g)- Hounkonnou-Ngompe   multi-parameter deformation
	
    ϵ_1=p     ϵ_2=q^-1.



	
	The  j^th- order factorial of the ℛ(p,q)- number  is given by : 
	
    [n]_j,ℛ(p,q)
    	:= ∏_v=0^j-1ℛ(p^n-v,q^n-v),

	where n∈ℕ and j∈ℕ\{0}.
	
	
	The  ℛ(p,q)- deformed shifted factorial is expressed as follows:
	
    (x ⊕ y)^n_ℛ(p,q): = ∏_i=1^n(x ϵ^i-1_1 + y ϵ^i-1_2),    (x ⊕ y)^0_ℛ(p,q): =1,

	where  x,y∈ℕ, while
	
	the ℛ(p,q)- deformed  Euler formula is developed as:
	
    (x ⊕ y)^n:= ∑_κ=0^n[[ n; κ ]]_ℛ(p,q) ϵ_1^n-κ 2 ϵ_2^κ 2 x^n-κ y^κ,   x,y∈ℕ.


	  The case of q-factorial in  <cit.>
			 corresponds to ℛ(p,q)=q,  ϵ_1=1 and ϵ_2=q leading to :
			 
			 	
  * The  j^th- order factorial of the q- number:
				
    [n]_j,q
    				:= ∏_v=0^j-1[n-v]_q,    n∈ℕ     j∈ℕ\{0},

				
  * The  q- deformed shifted factorial:
				
    (x ⊕ y)^n_q: = ∏_i=1^n(x + y q^i-1),    (x ⊕ y)^0_q: =1,

				where  x,y∈ℕ.
				
  * The q- deformed  Euler formula:
				
    (x ⊕ y)_q^n:= ∑_κ=0^n[[ n; κ ]]_q q^κ 2 x^n-κ y^κ,   x,y∈ℕ.

			
				
					The ℛ(p,q)- deformed Vandermonde's formula is given by:
					
					
    [u+v]_n,ℛ(p,q)
    					= ∑_κ=0^n[[ n; κ ]]_ℛ(p,q)ϵ_1^κ(v-n+κ)ϵ_2^(n-κ)(u-κ)[u]_κ,ℛ(p,q)[v]_n-κ,ℛ(p,q),

					or, equivalently,
					
    [u+v]_n,ℛ(p,q)
    					= ∑_κ=0^n[[ n; κ ]]_ℛ(p,q)ϵ_1^(n-κ)(u-κ)ϵ_2^κ(v-n+κ)[u]_κ,ℛ(p,q)[v]_n-κ,ℛ(p,q),

				
					where n is a positive integer.
				
		
			For n∈ℕ\{0}, we consider the following expression:
			
			
    T_n(u;v)_ℛ(p,q):= ∑_κ=0^n[[ n; κ ]]_ℛ(p,q)ϵ_1^κ(v-n+κ)ϵ_2^(n-κ)(u-κ) [u]_κ,ℛ(p,q)[v]_n-κ,ℛ(p,q).

			
			For n=1, we have
			T_1(u;v)_ℛ(p,q)
			= [u+v]_ℛ(p,q). Using the recursion relation 
			
			
    [[ x; κ ]]_ℛ(p,q) = ϵ^κ_1 [[ x -1;    κ ]]_ℛ(p,q) + ϵ^x-κ_2 [[ x-1; κ-1 ]]_ℛ(p,q),  κ∈ℕ

			 and 
			
			
    [u+v-n+1]_ℛ(p,q) [u]_κ,ℛ(p,q)[v]_n-κ-1,ℛ(p,q)
    			=  ϵ_2^u-κ[u]_κ,ℛ(p,q)[v]_n-κ,ℛ(p,q)                                   +
    			ϵ_1^v-n+κ+1
    			[u]_κ+1,ℛ(p,q)[v]_n-κ-1,ℛ(p,q),

			
			we obtain
			
			
    T_n(u;v)_ℛ(p,q)   =   ∑_κ=0^n-1[[ n-1;   κ ]]_ℛ(p,q)ϵ_1^κ(v-n+κ+1)ϵ_2^(n-κ)(u-κ) [u]_κ,ℛ(p,q)[v]_n-κ,ℛ(p,q)
    
       +   ∑_κ=0^n-1[[ n-1;   κ ]]_ℛ(p,q)ϵ_1^(κ+1)(v-n+κ+1)ϵ_2^(n-κ-1)(u-κ)
       ×   [u]_κ+1,ℛ(p,q)[v]_n-κ-1,ℛ(p,q)
       =   [u+v-n+1]_ℛ(p,q) T_n-1(u;v)_ℛ(p,q).

			
			Therefore, for n∈ℕ\{0},  T_n(u;v)_ℛ(p,q) satisfies the first-order recursion relation 
			
    T_n(u;v)_ℛ(p,q) =  [u+v-n+1]_ℛ(p,q) T_n-1(u;v)_ℛ(p,q),

			with T_1(x;y)_ℛ(p,q) = [u+v]_ℛ(p,q). Recursively,  
			it follows that T_n(u;v)_ℛ(p,q) =
			[u+v]_n,ℛ(p,q).
			Therefore, we get  (<ref>). Finally, interchanging x and y, and  replacing κ by n-κ, the expression (<ref>) is rewritten in the form (<ref>). 
			 
Taking ℛ(p,q)=q,   ϵ_1=1 and ϵ_2=q,  the q- deformed Vandermonde's  formula obtained in <cit.> is recovered as:

    [u+v]_n, q   =   ∑_κ=0^n[[ n; κ ]]_q q^(n-κ)(u-κ) [u]_κ, q[v]_n-κ, q
       =   ∑_κ=0^n[[ n; κ ]]_q q^κ(v-n+κ) [u]_κ, q[v]_n-κ, q,

where n is a positive integer.

For ℛ(p,q)=((q-p^-1)q)^-1(p q-1), involving ϵ_1=p,   ϵ_2=q^-1   
and n,  a positive integer, the generalized q- Quesne Vandermonde's formula can be derived as follows:

    [u+v]^Q_n,p,q
    	   =   ∑_κ=0^n[[ n; κ ]]^Q_p,qp^κ(v-n+κ)q^-(n-κ)(u-κ)[u]^Q_κ,p,q [v]^Q_n-κ,p,q
    	
       =   ∑_κ=0^n[[ n; κ ]]^Q_p,qp^(n-κ)(u-κ)q^-κ(v-n+κ)[u]^Q_κ,p,q [v]^Q_n-κ,p,q.

 
	The negative ℛ(p,q)-deformed Vandermonde's formula is given by:
	
	
    [u+v]_-n,ℛ(p,q)
    	= ∑_κ=0^∞[[ -n;  κ ]]_ℛ(p,q)ϵ_1^κ(v+n+κ)ϵ_2^(-n-κ)(u-κ)[u]_κ,ℛ(p,q)[v]_-n-κ,ℛ(p,q),

	or
	
    [u+v]_-n,ℛ(p,q)
    	= ∑_κ=0^∞[[ -n;  κ ]]_ℛ(p,q)ϵ_1^(-n-κ)(u-κ)ϵ_2^κ(v+n+κ)[u]_κ,ℛ(p,q)[v]_-n-κ,ℛ(p,q),


	where  
	n is a positive integer.


	 For n∈ℕ\{0}, we consider the following  expression: 
	 

    H_n(u;v)_ℛ(p,q)= ∑_κ=0^∞[[ -n;  κ ]]_ℛ(p,q)ϵ_1^κ(v+n+κ)ϵ_2^(-n-κ)(u-κ)[u]_κ,ℛ(p,q)[v]_-n-κ,ℛ(p,q).


For n=1, we have

    H_1(u;v)_ℛ(p,q)= 1[u+v+1]_ℛ(p,q).

Using  the relation (<ref>) and  


    [u+v+n+1]_ℛ(p,q)[u]_κ,ℛ(p,q)[v]_-n-κ-1,ℛ(p,q)
    	= ϵ_2^u-κ[u]_κ,ℛ(p,q)[v]_-n-κ,ℛ(p,q)                              + ϵ_1^v+n+κ+1[u]_κ+1,ℛ(p,q)[v]_-n-κ-1,ℛ(p,q),
 

we get:

    H_n(u;v)_ℛ(p,q)   =   ∑_κ=0^∞[[ -n-1;    κ ]]_ℛ(p,q)ϵ_1^κ(v+n+κ)+κϵ_2^(-n-κ)(u-κ)
       ×   [u]_κ,ℛ(p,q)[v]_-n-κ,ℛ(p,q)
    	+ ∑_κ=0^∞[[ -n-1;    κ ]]_ℛ(p,q)ϵ_1^κ(v+n+κ)+κ
       ×   ϵ_2^(-n-κ)(u-κ+1)[u]_κ+1,ℛ(p,q)[v]_-n-κ-1,ℛ(p,q)
       =   H_n-1(u;v)_ℛ(p,q) [u+v+n+1]_ℛ(p,q).

Therefore, for n∈ℕ\{0}, the sum H_n(u;v)_ℛ(p,q) satisfies the first-order recursion relation 

    H_n(u;v)_ℛ(p,q) =H_n-1(u;v)_ℛ(p,q)[u+v+n+1]_ℛ(p,q),   n∈ℕ\{0},

with H_1(u;v)_ℛ(p,q) =  1 [u+v+1]_ℛ(p,q). 
Recursively, it comes that H_n(u;v)_ℛ(p,q) = [u+v]_-n,ℛ(p,q). 
Following the steps used to prove (<ref>), 
we obtain (<ref>), and the proof is achieved. 

We recover the negative q- Vandermonde's formula obtained in <cit.>  by taking ℛ(p,q)=q, ϵ_1=1 and ϵ_2=q:

    [x+y]_-n,q   =   ∑_κ=0^nq^-(n-κ)(x-κ)[-n κ]_q [x]_κ,q [y]_-n-κ,q,   |q^(x+y+1)|<1
       =   ∑_κ=0^nq^κ)(y+n+κ)[-n κ]_q [x]_κ,q [y]_-n-κ,q,
       |q^-(x+y+1)|<1,

where 0<q< 1.  

	 The following relations hold:
	
    1 [v]_n,ℛ(p,q)
    	= ∑_κ=0^∞[[ n+ κ -1;       κ ]]_ℛ(p,q)ϵ^n(u-κ)_1ϵ^κ(v-n+1)_2[u]_κ,ℛ(p,q) [u+v]_n+κ,ℛ(p,q)

	and
	
    1 [v]_n,ℛ(p,q)
    	= ∑_κ=0^∞[[ n+ κ -1;       κ ]]_ℛ(p,q)ϵ^κ(v-n+1)_1ϵ^n(u-κ)_2[u]_κ,ℛ(p,q) [u+v]_n+κ,ℛ(p,q),

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


	 For n a positive integer, the ℛ(p,q)- deformed factorial  of order κ of x=-n is written as:

    [-n]_κ,ℛ(p,q) 
    = (-1)^κ (ϵ_1ϵ_2)^-nκ- κ 2 [n+κ-1]_κ,ℛ(p,q)

and 

    ℛ(p^-j,q^-j) =- (ϵ_1 ϵ_2)^-j ℛ(p^j,q^j)   j∈{n,n+1,⋯, n+κ-1}.

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

    [[ -n;  κ ]]_ℛ(p,q) =(-1)^κ(ϵ_1 ϵ_2)^-nκ -κ 2  [[ n+κ -1;      κ ]]_ℛ(p,q).

Moreover, from

    [-v-1]_-n,ℛ(p,q)   =   1[-v-1+n]_n,ℛ(p,q)
       =   1  (-1)^n (ϵ_1 ϵ_2)^-nv + n  2[v]_n,ℛ(p,q)
 
and 

    [-u-v-1]_-n-κ,ℛ(p,q) 
    	=(-1)^-n-κ(ϵ_1 ϵ_2)^-(n+κ)(u+v) + n+κ 2[u+v]_n+κ,ℛ(p,q),
 
the relation (<ref>) can be  written as follows:

    1 (ϵ_1 ϵ_2)^-nv+n 2[v]_n,ℛ(p,q)
    = ∑_κ=0^∞[[ n+κ -1;      κ ]]_ℛ(p,q)  F(p,q),
 
where

    F(p,q) = ϵ^κ(v+n+κ)_1(ϵ_1ϵ_2)^-(n+κ)(u-κ) -nκ- κ 2[u]_κ,ℛ(p,q) (ϵ_1 ϵ_2)^-(n+κ)(u+v) + n+κ 2[u+v]_n+κ,ℛ(p,q).

After re-arranging, 




the relation (<ref>) is reduced to (<ref>). Similarly, we obtain (<ref>). 


Setting ϵ_1=1 and ϵ_2=q provides the q-analogs of the formulae (<ref>) and (<ref>) of <cit.> as :

    1 [v]_n,q   =   ∑_κ=0^∞[[ n+ κ -1;       κ ]]_qq^κ(v-n+1)[u]_κ,q [u+v]_n+κ,q,
       |q^v|<1
    
    
    
    
       =   ∑_κ=0^∞[[ n+ κ -1;       κ ]]_qq^n(u-κ)[u]_κ,q [u+v]_n+κ,q,
       |q^-v|<1.


	It is worth noticing that the  generalized q- Quesne negative Vandermonde's formula is obtained,  by taking  ℛ(p,q)=((q-p^-1)q)^-1(p q-1), ϵ_1=p and   ϵ_2=q^-1,  under the form:
	
	
    [u+v]^Q_-n,p,q   =   ∑_κ=0^∞[[ -n;  κ ]]^Q_p,qp^κ(v+n+κ)q^-(-n-κ)(u-κ)[u]^Q_κ,p,q [v]^Q_-n-κ,p,q
       =   ∑_κ=0^∞[[ -n;  κ ]]^Q_p,qp^(-n-κ)(u-κ)q^-κ(v+n+κ)[u]^Q_κ,p,q [v]^Q_-n-κ,p,q


	and
	
    1 [v]^Q_n,p,q   =   ∑_κ=0^∞[[ n+ κ -1;       κ ]]^Q_p,q p^n(u-κ) q^-κ(v-n+1)[u]^Q_κ,p,q [u+v]^Q_n+κ,p,q
        =    ∑_κ=0^∞[[ n+ κ -1;       κ ]]^Q_p,q p^κ(v-n+1) q^-n(u-κ)[u]^Q_κ,p,q [u+v]^Q_n+κ,p,q.

	
	


	The noncentral ℛ(p,q)- Stirling numbers of the first and second kinds, s_ℛ(p,q)(n,κ,j) and S_ℛ(p,q)(n,κ,j) are defined via the relations:
	
    [x-j]_n,ℛ(p,q)
    	:=ϵ_2^-n 2-j n ∑_κ=0^ns_ℛ(p,q)(n,κ;j) [x]^κ_ℛ(p,q),

	and
	
    [x]^n_ℛ(p,q)
    	:=∑_κ=0^nϵ_2^κ 2+j κ S_ℛ(p,q)(n,κ;j) [x-j]_κ,ℛ(p,q),

	where  n∈ℕ and x is a real number.

For j=0, we obtain  s_ℛ(p,q)(n, κ;0) =s_ℛ(p,q)(n,κ) and S_ℛ(p,q)(n,κ; 0) =S_ℛ(p,q)(n,κ), which are  the ℛ(p,q)- deformed Stirling numbers of the first and second kinds, respectively.
		 Some remarkable particular cases deserve notification:
			
				
  * By setting ℛ(p,q)=q,   ϵ_1=1 and ϵ_2=q, we recover the noncentral q- Stirling numbers of the first and second kinds, s_q(n,κ,j) and S_q(n,κ,j), derived in  <cit.>:
			
    [x-j]_n,q
    			=q^-n 2-j n ∑_κ=0^ns_q(n,κ;j) [x]^κ_q,

			and
			
    [x]^n_q
    			=∑_κ=0^n q^κ 2+j κ S_q(n,κ;j) [x-j]_κ, q,

			where  n∈ℕ and x is a real number.
		 
  * For j = 0, we obtain  s_q(n, κ;0) = s_q(n, κ) and S_q(n, κ; 0) =
				S_q(n, κ), which are  the q- deformed Stirling numbers of the first and second kinds, respectively.
				
  * Setting ℛ(p,q)=p q-1 (q-p^-1)q corresponding to  ϵ_1=p and ϵ_2=q^-1,  we deduct the noncentral generalized  q-  Quesne Stirling numbers of the first and second kinds, s^Q_p,q(n,κ,j) and S^Q_p,q(n,κ,j), as follows:
					
    [x-j]^Q_n,p,q
    					=q^-n 2-j n (p q)^n∑_κ=0^ns^Q_p,q(n,κ;j) (q p[x]^Q_p,q)^κ,

					and 
					
    ([x]^Q_p,q)^n
    					=(p q)^n∑_κ=0^n q^κ 2+j κ(q p)^κ S^Q_p,q(n,κ;j) [x-j]^Q_κ, p,q,

					where  n∈ℕ and x is a real number.
			
  * For j = 0,  we obtain  s^Q_p,q(n, κ;0) = s^Q_p,q(n, κ) and S^Q_p,q(n, κ; 0) =
						S^Q_p,q(n, κ), which are  the generalized q- Quesne Stirling numbers of the first and second kinds, respectively.
			
		
	

 §.§ ℛ(p,q)- deformed factorial and binomial moments
 
 We introduce now the 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∈ℕ, we consider  the distribution function g of the variable X such that g(x)=P(X=x).




	
		The r^th- order ℛ(p,q)- factorial moment of the random variable X is given by: 
		
    E([X]_r,ℛ(p,q)) := ∑_x=r^∞ [x]_r,ℛ(p,q) g(x),

		while the r^th- order ℛ(p,q)- binomial moment of the random variable X is provided by:
		
    E([[ X; r ]]_ℛ(p,q))  = ∑_x=r^∞[[ x; r ]]_ℛ(p,q) g(x),

		where r∈ℕ\{0}.
	
 For   r=1, we obtain:
	
		
  * The ℛ(p,q)- mean value of the random variable X:
		
    μ_ℛ(p,q):= E([X]_ℛ(p,q))=∑_x=1^∞[x]_ℛ(p,q) g(x).

		
  * The ℛ(p,q)- variance  of the random variable X:
		
    σ^2_ℛ(p,q):= V([X]_ℛ(p,q))=  E([X]^2_ℛ(p,q)) - [ E([X]_ℛ(p,q))]^2.

	





	 In terms of Stirling number, 
		the binomial  moment can be re-expresed  as follows:
		
    E[([ X; j ])]= ∑_m=j^∞ (-1)^m-j(ϵ_1-ϵ_2)^m-jϵ^-m 2+τ(m-j)_1s_ℛ(p,q)(m,j) E([[ X; m ]]_ℛ(p,q)),


and the relation between the factorial moment and its ℛ(p,q)- deformed counterpart is given by:
		
    E[(X)_j]= j!∑_m=j^∞ (-1)^m-j(ϵ_1-ϵ_2)^m-jϵ^-m 2+τ(m-j)_1s_ℛ(p,q)(m,j) E([ X ]_m,ℛ(p,q)) [m]_ℛ(p,q)!,
 
		where j∈ℕ\{0, 1}, τ∈ℕ, and s_ℛ(p,q) is the ℛ(p,q)- Stirling number of the first kind.
	


	Multiplying the relation
	
    x  j =∑_m=j^x(-1)^m-j(ϵ_1-ϵ_2)^m-jϵ^m 2-τ(m-j)_1 s_ℛ(p,q)(m,j) [[ x; m ]]_ℛ(p,q)
 
	by the probability distribution g, and summing for all x∈ℕ, we deduce  (<ref>) from  (<ref>). Moreover, from 
	
    E[X j]=  E[(X)_j] j!,   E([[ X; m ]]_ℛ(p,q))= E([X]_m,ℛ(p,q)) [m]_ℛ(p,q)!
 
	and the relation (<ref>), we derive (<ref>).

 Some interesting results can easily be deduced as follows:
	
		
  * The particular case of the q- deformed binomial moment in <cit.>  is retrieved as follows :
	
    E[([ X; j ])]= ∑_m=j^∞ (-1)^m-j(1-q)^m-js_q(m,j) E([[ X; m ]]_q)

	and the related  factorial moment is linked to its q- counterpart by the relation:
	
    E[(X)_j]= j!∑_m=j^∞ (-1)^m-j(1-q)^m-js_q(m,j) E([ X ]_m,q) [m]_q!,

	where j∈ℕ\{0}, and s_q is the q- Stirling number of the first kind.

  * Putting ℛ(p,q)=p q-1 (q-p^-1)q, ϵ_1=p, and ϵ_2=q^-1, we deduct the binomial moment:

    E[([ X; j ])]= ∑_m=j^∞ (-1)^m-j(p-q^-1)^m-j p^-m 2+τ(m-j)s^Q_p,q(m,j) E([[ X; m ]]^Q_p,q)

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

    E[(X)_j]= j!∑_m=j^∞ (-1)^m-j(p-q^-1)^m-j p^-m 2+τ(m-j)s^Q_p,q(m,j) E([ X ]^Q_m,p,q) [m]^Q_p,q!,

where j∈ℕ\{0, 1}, τ∈ℕ, and s^Q_p,q is the generalized q- Quesne Stirling number of the first kind.




 §.§ ℛ(p,q)- deformed binomial distribution

	
		The following relation holds:
		
    ∑^n_κ = 
    			0[n κ]_ℛ(p,q) x^κ (y ⊖ v)_ℛ(p,q)^n - κ = ∑
    		^n_κ = 0[n κ]_ℛ(p,q) y^κ (x ⊖ v)_ℛ(p,q)^n -
    			κ.

		In particular, for x=p_0 and y=1, we obtain
		
    ∑^n_κ =0[n κ]_ℛ(p,q) p_0^κ
    		(1 ⊖ 
    		v)_ℛ(p,q)^n - κ 
    		= ∑^n_κ =0[n κ]_ℛ(p,q) (p_0 
    		⊖ v)_ℛ(p,q)^n-κ=1,  ∀ p_0

		where x,  y, v, p_0 and n  are integers.
		
	 The q-analog of Lemma in <cit.>
	 is deduced by setting ℛ(p,q)=q,  ϵ_1=1, and ϵ_2=q, as follows :
		
    ∑^n_κ = 
    			0[n κ]_q x^κ (y ⊖ v)_q^n - κ = ∑
    		^n_κ = 0[n κ]_q y^κ (x ⊖ v)_q^n -
    			κ.

		In particular, for x=p_0 and y=1, we obtain
		
    ∑^n_κ =0[n κ]_q p_0^κ
    		(1 ⊖ 
    		v)_q^n - κ 
    		= ∑^n_κ =0[n κ]_q (p_0 
    		⊖ v)_q^n-κ=1,  ∀ p_0,

		where x,  y, v, p_0 and n  are integers.
		
		The binomial distribution comes with


			a random variable
			X  taking two values, 0 and 1, 

the probabilities Pr(X=1)=p_0 and Pr(X=0)=1-p_0,  and by
			
letting S_n=X_1 +⋯ + X_n be the sum of n random variables (X_i)_i∈{1,2,⋯,n} obeying the binomial law.
		
	
		The ℛ(p,q)-deformed binomial distribution, with parameters n, p_0, p, and q,  is given,  for 
		0≤κ≤ n and 0<q<p≤ 1, by:
		
    P_κ:=P_r([S_n]_ℛ(p,q)=[κ]_ℛ(p,q))= [[ n; κ ]]_ℛ(p,q) p^κ_0 (1⊖ p_0)_ℛ(p,q)^n-κ.

	
	For ℛ(p,q)=q, ϵ_1=1 and ϵ_2=q,  we obtain the q- binomial distribution given in <cit.>:
	
    P_r([S_n]_q=[κ]_q)= [[ n; κ ]]_q p^κ_0 (1⊖ p_0)_q^n-κ,    0≤κ≤ n;   0<q < 1.

	
		
		
		 The j^th- order ℛ(p,q)- deformed factorial moment is given by:
		
    μ_ℛ(p,q)( [S_n]_j,ℛ(p,q)) =  [n]_j,ℛ(p,q) p^j_0   j∈{1,2,⋯,n},

		
		and the factorial moment is expressed by the formula:
		
    μ_ℛ(p,q)[( S_n)_i] = i! ∑_j =i^n(-1)^j-i[[ n; j ]]_ℛ(p,q) p_0^j (ϵ_1-ϵ_2)^j-iϵ^-j 2+τ(j-i)_1 s_ℛ(p,q)(j,i),

		where τ∈ℕ, i∈{2,⋯,n}, and s_ℛ(p,q) is the ℛ(p,q)- Stirling number of the first kind.
		
		The recursion relation  for the ℛ(p,q)- deformed binomial distributions takes the form:
		
    P_κ+1 = [n-κ]_ℛ(p,q) [κ+1]_ℛ(p,q)p_0ϵ^n-κ_1- ϵ^n-κ_2 p_0 P_κ,     	P_0= (1⊖ p_0)_ℛ(p,q)^n.

		
	

	The j^th- order  ℛ(p,q)-factorial moment of [S_n]_ℛ(p,q) is
	
    μ_ℛ(p,q)( [S_n]_j,ℛ(p,q)) 
    	=∑_κ =j^n[κ]_j,ℛ(p,q) [[ n; κ ]]_ℛ(p,q) p^κ_0 (1⊖ p_0)_ℛ(p,q)^n-κ.

	Using the  relation
	
    [κ]_j,ℛ(p,q) [[ n; κ ]]_ℛ(p,q)=[n]_j,ℛ(p,q) [[ n-j; κ-j ]]_ℛ(p,q),

	we have
	
	
    μ_ℛ(p,q)([S_n]_j,ℛ(p,q)) 
    		   =    [n]_j,ℛ(p,q)∑_κ =j^n[[ n-j; κ-j ]]_ℛ(p,q)p^κ_0 (1⊖ p_0)_ℛ(p,q)^n-κ
       =    [n]_j,ℛ(p,q) p^j_0∑_x=0^n[[ n-j;   x ]]_ℛ(p,q)p^x_0(1⊖ p_0)_ℛ(p,q)^n-j-x.

	
	Moreover, the formula (<ref>) is obtained using the relations (<ref>) and (<ref>). From the relation
	
    [[   n; κ+1 ]]_ℛ(p,q)=[n-κ]_ℛ(p,q) [κ+1]_ℛ(p,q)[[ n; κ ]]_ℛ(p,q)

	and after computation, we obtain (<ref>).


	The recursion relation for the q- deformed binomial distributions is given by:
	
    P_κ+1 = [n-κ]_q [κ+1]_qp_0 1- q^n-κ p_0 P_κ,     P_0=∏_j=1^n(1-p_0 q^j-1).







It is obtained by straightforward deduction. 

Now, let us  consider, for v∈ℕ\{0}, the  l-order differential operator


    (vD_ v)^l=∑_j=1^l1/[j-1]!_ℛ(p,q)(∑_t=0^j-1[[ j-1;   t ]]_ℛ(p,q)(-1)^tq^t  2[j-t]^l-1_ℛ(p,q))v^j(D_v)^j,


where D_v:= d/d_ℛ(p,q) acting on v, (which will be used in the sequel), 

giving, for l=2, 

the following second order differential operator:

    (v D_ v)^2 = ℛ(p,q) (v D_ v )+1 ℛ!(p,q)( ℛ(p^2,q^2) -  ℛ(p,q)) v^2( D_ v)^2 .

Besides, for ϵ_i>0, with i∈{1,2}, such that ∀ p, q,  0<q<p≤ 1, 

    ℛ(p^x-y,q^x-y) = ϵ^-y_1 ℛ(p^x,q^x) + ϵ^-y_1ϵ^x-y_2 ℛ(p^y,q^y).






	The mean value of the random variable sum  S_n is given by:
	
    μ_ℛ(p,q)([S_n]_ℛ(p,q)) = p_0 ℛ(p^n,q^n).

	Its corresponding variance 
	can be written as:
	
    Var([S_n]_ℛ(p,q))=p_0[n]_ℛ(p,q)(ℛ(p,q)+  Xp_0[n-1]_ℛ(p,q)-p_0[n]_ℛ(p,q)),

	where 
	Xp_0[n-1]_ℛ(p,q)>p_0[n]_ℛ(p,q)-ℛ(p,q),
	and
	
    X=ℛ(p,q)^-1(ℛ(p^2,q^2)-ℛ(p,q)).

	The  mean of the product  S_n(S_n-1)⋯ (S_n-r+1)  is given by:
	
    μ_ℛ(p,q)(∏_i=0^r-1ϵ^-i_2([S_n]^r_ℛ(p,q) - ϵ^r-i_1[i]_ℛ(p,q))) = p^r_0 ∏_i=0^r-1[n-i]_ℛ(p,q).



	Applying the ℛ(p,q)-derivative on p_0 to the left and right hand sides of the relation (<ref>) leads to
		
    p_0 D_p_0∑^n_κ =0[n κ]_ℛ(p,q) p_0^κ
    		(1⊖ u)_ℛ(p,q)^n - κ
    		=	μ_ℛ(p,q)([S_n]_ℛ(p,q)),

and

		
		
    p_0 D_p_0∑^n_κ =0[n κ]_ℛ(p,q) (p_0 
    		⊖ u)_ℛ(p,q)^n-κ
    		= p_0∑^n_κ =0[n κ]_ℛ(p,q)[n-κ]_ℛ(p,q)   (p_0 
    		⊖ u)_ℛ(p,q)^n-κ-1,

	
respectively.
	According to (<ref>) and (<ref>), we obtain (<ref>).



	Besides,
	
    μ_ℛ(p,q)([S_n]^2_ℛ(p,q))=(p_0 D_p_0)^2∑_κ=0^n [[ n; κ ]]_ℛ(p,q) p^κ_0 (1⊖ u)_ℛ(p,q)^n-κ.

	Setting X =1/ℛ!(p,q)(ℛ(p^2,q^2)-ℛ(p,q)  ) yields the second order differential equation
	
    (p_0 D_p_0)^2∑_κ=0^n [[ n; κ ]]_ℛ(p,q) (p_0⊖ u)_ℛ(p,q)^n-κ
    	= ℛ(p,q)μ_ℛ(p,q)([S_n]_ℛ(p,q))+
    	Y_4,

	where
	
    Y_4 
    		   =    X p_0^2∑_κ=0^nℛ(p^n-κ,q^n-κ)ℛ(p^n-κ-1,q^n-κ-1)(p_0⊖ u)_ℛ(p,q)^n-κ.

	Using the relations (<ref>) and (<ref>), we obtain
	
	
    μ_ℛ(p,q)([S_n]^2_ℛ(p,q))= ℛ(p,q) μ_ℛ(p,q)([S_n]_ℛ(p,q)) +   X p_0^2ℛ(p^n,q^n)ℛ(p^n-1,q^n-1),


	and from the mean value of S_n,  we deduce Var([S_n]_ℛ(p,q)) . Futhermore,
	applying the operator p^r_0 (D_p_0)^r |_u
				= p_0 to formula  (<ref>), we obtain
	
	
    p^r_0 (D_p_0)^r ∑^n_κ =0[n κ] p^κ
    		(1 ⊖ 
    		u)_ℛ(p,q)^n - κ   =   μ_ℛ(p,q)([S_n]^r_ℛ(p,q)).

	
	Moreover,
	
    p^r_0  (D_p_0)^r(p_0 ⊖ u)_ℛ(p,q)^n - κ   =    p^r_0 ∏_i=0^r-1[n-κ-i]_ℛ(p,q) (p_0⊖ u)_ℛ(p,q)^n - κ-r,

	or  equivalently,
	
    p^r_0  (D_p_0)^r(p_0 ⊖ u)_ℛ(p,q)^n - κ   =   	p^r_0  ∏_i=0^r-1ϵ^-κ_1[n-i]_ℛ(p,q) (p_0⊖ u)_ℛ(p,q)^n - κ-r
       +    p_0^r∏_i=0^r-1ϵ^n-i_2[-κ]_ℛ(p,q)(p_0 ⊖ u)_ℛ(p,q)^n - κ-r.

	Then,
	
    p^r_0 (D_p_0)^r∑^n_κ =0[n κ](p ⊖ u)_ℛ(p,q)^n - κ 
    	= p_0^r∏_i=0^r-1[n-i]_ℛ(p,q)

	gives
	
    μ_ℛ(p,q)([S_n]^r_ℛ(p,q)) =p_0^r∏_i=0^r-1[n-i]_ℛ(p,q),

	and after computation, 
	 the result follows.

The particular case of the  q- deformation described in <cit.>
corresponds to ℛ(p,q)=q,  ϵ_1=1, and ϵ_2=q, and yields


the mean value of S_n:
	
    μ_q([S_n]_q) = p_0 [n]_q,


the variance of S_n:
	
    Var([S_n]_q)=[n]_q p_0 (1-p_0),



and the mean of the product  S_n(S_n-1)⋯ (S_n-r+1):  
	
    μ_q(∏_i=0^r-1q^-i([S_n]^r_q - [i]_q)) = p^r_0 ∏_i=0^r-1[n-i]_q.


	
		Deducing the above mentioned properties for the particular case of the generalized q-  Quesne deformation 

  leads to:
		
			
  * Generalized q- Quesne probability distribution:
			
    p_r(S_n=[κ]^Q_p,q)
    			=[[ n; κ ]]^Q_p,q p^κ_0((1 ⊖ p_0)^Q_p,q)^n-κ,   0≤κ≤ n;

			
  * j^th- order  generalized q- Quesne factorial moment:
			
    μ^Q_p,q([ S_n]^Q_j,p,q) = [n]^Q_j,p,q (q p)^j p^j_0   j∈{1,2,⋯,n};

			
  * Generalized q- Quesne factorial moment: 
			
    μ^Q_p,q[( S_n)_i] = i! ∑_j=i^n(-1)^j-i[[ n; j ]]^Q_p,q p_0^j (p-q^-1)^j-i p^-j 2+τ(j-i)s^Q_p,q(j,i);

			
  * Recursion relation for the  generalized q- Quesne distributions:
			
    P_κ+1 = [n-κ]^Q_p,q [κ+1]^Q_p,qp_0(1⊖ p_0)^Q_p,q P_κ,   P_0= ((1⊖ p_0)^Q_p,q)^n;

			
  * Mean value:
			
    μ^Q_p,q(S_n)=p_0 q p p^n-q^-n q-p^-1;

			
  * Variance:
			
    (σ^Q_p,q)^2(S_n)=p_0q p[n]^Q_p,q(1+(p^-1+q-1) p_0q p[n-1]^Q_p,q-p_0q p[n]^Q_p,q);

			
  * Mean value of the product   S_n(S_n-1)⋯ (S_n-r+1): 
			
    μ^Q_p,q(∏_i=0^r-1p^-i([S_n]^Q_p,q)^r - q^-r+i+1p^-1[i]^Q_p,q) =q^r  p^-r p^r_0 ∏_i=0^r-1[n-i]^Q_p,q.

		
	


 §.§ ℛ(p,q)- deformed Euler distribution


	The ℛ(p,q)- deformed exponential functions, denoted  E_ℛ(p,q) and  e_ℛ(p,q), are defined as follows:
	
    E_ℛ(p,q)(z):=∑_n=0^∞ϵ^n 2_2 z^nℛ!(p^n,q^n)     e_ℛ(p,q)(z):=∑_n=0^∞ϵ^n 2_1 z^nℛ!(p^n,q^n)

	with
	E_ℛ(p,q)(-z) e_ℛ(p,q)(z)=1.
	

In the particular case where ℛ(p,q)=1, ϵ_1=p, and ϵ_2=q, they provide  the Jagannathan-Srinivasa q-exponential functions  <cit.>:

    E_p,q(z)=∑_n=0^∞q^n 2 z^n [n]_p,q!     e_p,q(z)=∑_n=0^∞p^n 2 z^n [n]_p,q!

with 
E_p,q(-z) e_p,q(z)=1.

	The  ℛ(p,q)- deformed Euler distribution, with parameters n, θ, p, and q,  is defined by:
	
    P_r( X=x)=E_ℛ(p,q)(-θ)  ϵ^x 2_1θ^xℛ!(p^x,q^x)   x∈ℕ,

	where 0<q<p≤ 1, 0<θ<1
	and ∑_x=0^∞P_r( X=x)=1.

For ℛ(p,q)=q, involving  ϵ_1=1, and ϵ_2=q, we retrieve the q- Euler distribution  obtained in <cit.>:

    P_r( X=x)=E_q(-θ)  θ^x [x]_q!,

where x∈ℕ, 0<q <1 and 0<θ <1 1-q.
 



 The j^th- order   ℛ(p,q)- factorial moment of X is given by
		
    μ_ℛ(p,q)( [X]_j,ℛ(p,q)) = θ^j ϵ^j 2_1 E_ℛ(p,q)(-θ) e_ℛ(p,q)(ϵ^j_1θ),   j∈ℕ\{0},



while the factorial moment is written as:
		
    μ_ℛ(p,q)( X)_i=i! ∑_j=i^∞ (-1)^j-i EU(j,i) s_ℛ(p,q)(j,i),
 
		where 
    EU(j,i)=θ^jϵ^j 2_1 E_ℛ(p,q)(-θ) e_ℛ(p,q)(ϵ^j_1θ) [j]_ℛ(p,q)! (ϵ_1 -ϵ_2)^j-iϵ^-j 2+τ(j-i)_1,
  τ∈ℕ, i∈ℕ\{0,1}, and s_ℛ(p,q) is the ℛ(p,q)- deformed Stirling number of the first kind.

	
 The recursion relation for the associated  ℛ(p,q)- deformed Euler distributions takes the form:
		
    P_x+1= θ ϵ^x_1 ℛ(p^x+1,q^x+1) P_x,    	P_0=E_ℛ(p,q)(-θ).




From the j^th- order ℛ(p,q)- factorial moment, we get

    μ( [X]_j,ℛ(p,q))   =   ∑_x=j^∞[x]_j,ℛ(p,q) E_ℛ(p,q)(-θ)  ϵ^x 2_1θ^xℛ!(p^x,q^x).

Using the relation

    ℛ(p^x,q^x)^j ℛ!(p^x-j,q^x-j)= ℛ!(p^x,q^x),

we obtain

    μ( [X]_j,ℛ(p,q))   =    E_ℛ(p,q)(-θ) ∑_x=j^∞ ϵ^x 2_1θ^xℛ!(p^x-j,q^x-j)
       =   θ^j ϵ^j 2_1 E_ℛ(p,q)(-θ) ∑_h=0^∞ ϵ^h 2_1(ϵ^j_1θ)^h [h]_ℛ(p,q)!
       =   θ^j ϵ^j 2_1 E_ℛ(p,q)(-θ) e_ℛ(p,q)(ϵ^j_1θ).

Furthermore, exploiting the relation (<ref>) and (<ref>), we obtain (<ref>). Using the ℛ(p,q)- factorials,  and after computation,




we get (<ref>).
	

	The recursion relation for the q- deformed Euler distributions is given by:
	
    P_x+1= θ [x+1]_q P_x,    	P_0=E_q(-θ).



	It stems from a straightforward computation.
	
	Particular results for the  q- deformation performed in <cit.>,   recovered with  ℛ(p,q)=q, ϵ_1=1, and ϵ_2=q, yield


the j^th- order  q- factorial moment of X:

    μ_q( [X]_j,q) = θ^j,   j∈ℕ\{0},


and the q- factorial moments:

    μ_q( X)_i=i! ∑_j=i^∞ (-1)^j-i θ^j [j]_q! (1 -q)^j-i s_q(j,i),
  
 i∈ℕ\{0}, and s_q is the q- deformed Stirling number of the first kind.


	The generalized q- Quesne Euler distribution corresponding to the choice ℛ(p,q)=((q-p^-1)q)^-1(p q-1),  ϵ_1=p and ϵ_2=q^-1, gives the following realization:
	
		
  * Generalized q- Quesne  probability distribution:
		
    P_r( X=x)=E^Q_p,q(-θ)  θ^x p^Xq^x 2 q^x [x]^Q_p,q!   x∈ℕ;

		
  * j^th- order  generalized q- Quesne factorial moment:
		
    μ^Q_p,q([X]^Q_j,p,q) = θ^j p^j 2E^Q_p,q(-θ)e^Q_p,q(p^j θ),   j∈ℕ\{0};

		
  * Generalized q- Quesne factorial moment: 
		
    μ^Q_p,q[(X)_i]=i! ∑_j=i^∞ (-1)^j-i (p -q^-1)^j-i p^-j 2+τ(j-i)EU(j,i) s^Q_p,q(j,i),
 
		where:
		
    EU(j,i)=θ^j p^j 2E_ℛ(p,q)(-θ)e^Q_p,q(p^j θ)[j]^Q_p,q!,

		τ∈ℕ,  i∈ℕ\{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:
		
    P_x+1= θ p^x+1 q  [x+1]^Q_p,q P_x,     	P_0=E^Q_p,q(-θ).

	



 §.§ 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:

    P_i,j=[r+x(j-1)]_ℛ(p,q) [r+s+x(i-1)]_ℛ(p,q)=[m-j+1]_ℛ(p^-x,q^-x) [m+u-i+1]_ℛ(p^-x,q^-x),

where j∈{1,2,⋯,i}, i∈ℕ\{0}, 0<q<p≤ 1,  m=-r/ x,  u=-s/ x, and x is a non zero integer.

We call this  model  the ℛ(p,q)- Pólya urn model.

Setting ℛ(p,q)=q, ϵ_1=1, and ϵ_2=q,  we obtain the probability for  q-  Pólya urn model described in <cit.>:

    P_i,j=[r+x(j-1)]_q [r+s+x(i-1)]_q=[m-j+1]_q^-x [m+u-i+1]_q^-x.


 Let T_n be the number of white boxes drawn in n drawings.

Then, we have:

	The ℛ(p,q)- deformed Pólya distribution,  with parameters m, u, n, p, and q, is defined by:
	
	
    P_κ:=P_r( T_n=κ)=Ψ(p,q)[n κ]_ℛ(p^-x,q^-x)[m]_κ,ℛ(p^-x,q^-x)[u]_n-κ,ℛ(p^-x,q^-x) [m+u]_n,ℛ(p^-x,q^-x),


	where  0<q<p≤ 1, κ∈{0,⋯,n},  Ψ(p,q)=ϵ^-xκ(u-n+κ)_1 ϵ^-x(n-κ)(m-κ)_2, x is an integer  and ∑_κ=0^nP_r( T_n=κ)=1.

For ℛ(p,q)=q,   ϵ_1=1, and ϵ_2=q,
we recover
the q- Pólya distribution <cit.>:

    P_r(T_n=κ) = q^-x(n-κ)(m-κ)[n κ]_q^-x[m]_κ,q^-x[u]_n-κ, q^-x [m+u]_n,q^-x,   x∈ℕ

where 0<q<1 and κ∈{0,⋯,n}.

		The j^th- order ℛ(p,q)- deformed factorial moment is given by:
		
		
    μ_ℛ(p,q)([T_n]_j,ℛ(p^-x,q^-x)) = [n]_j,ℛ(p^-x,q^-x) [m]_j,ℛ(p^-x,q^-x) [m+u]_j,ℛ(p^-x,q^-x),  j∈{1,⋯,n},

		
		while the  factorial moment  is expressed by:
		
    μ_ℛ(p,q)[(T_n)_i] = i!∑_j=i^n(-1)^j-i[[ n; j ]]_ℛ(p,q) s_ℛ(p^-x,q^-x)(j,i) (ϵ^-x_1-ϵ^-x_2)^i-jP(j,i),

		where 
    P(j,i)=ϵ^j 2_1 [m]_j,ℛ(p^-x,q^-x)ϵ^τ(j-i)_1 [m+u]_j,ℛ(p^-x,q^-x),
    
		τ∈ℕ, i∈{2,⋯,n} and  s_ℛ(p^-x,q^-x) is the ℛ(p,q)- deformed Stirling number of the first kind.	
		 The recursion relation for the ℛ(p,q)- Pólya distributions is given as follows:
		
    P_κ +1 =ϵ^x(n+m-2κ-1)_2ϵ^x(u-n+2κ+1)_1[n-κ]_ℛ(p^-x,q^-x) [u-n+κ+1]_ℛ(p^-x,q^-x) [m-κ]_ℛ(p^-x,q^-x) [κ+1]_ℛ(p^-x,q^-x)P_κ,

		with
		
    P_0= ϵ^-x m n_2[u]_n,ℛ(p^-x,q^-x) [m+u]_n,ℛ(p^-x,q^-x).



	 The j^th- order ℛ(p,q)- factorial moment is furnished by the formula

    μ_ℛ(p,q)([T_n]_j,ℛ(p^-x,q^-x)) =∑_κ=j^n[κ]_j,ℛ(p^-x,q^-x)Ψ(p,q)[n κ]_ℛ(p^-x,q^-x)                              ×[m]_κ,ℛ(p^-x,q^-x)[u]_n-κ,ℛ(p^-x,q^-x) [m+u]_n,ℛ(p^-x,q^-x).

Using the ℛ(p,q)- Vandermonde's formula (<ref>) yields the expression

    μ_ℛ(p,q)([T_n]_j,ℛ(p^-x,q^-x)) =[m]_j,ℛ(p^-x,q^-x)[n]_j,ℛ(p^-x,q^-x) [m+u]_n,ℛ(p^-x,q^-x)[m+u-j]_n-j,ℛ(p^-x,q^-x),

for j∈{1,⋯,n}. 
Then, since

    [m+u]_n,ℛ(p^-x,q^-x)=[m+u]_j,ℛ(p^-x,q^-x)[m+u-j]_n-j,ℛ(p^-x,q^-x),

the relation (<ref>) is obtained. From  (<ref>), with  [-x]_ℛ(p,q)  instead of ℛ(p,q), (<ref>) is deduced. Since 
    [κ+1]_ℛ(p^-x,q^-x)[m κ+1]_ℛ(p^-x,q^-x)=[m-κ]_ℛ(p^-x,q^-x)[m κ]_ℛ(p^-x,q^-x),

then, after computation, the recursion relation is found. 


	The recursion relation for the q- Pólya distributions is derived as:
	
    P_κ +1 =q^x(n+m-2κ-1)[n-κ]_q^-x [u-n+κ+1]_q^-x [m-κ]_q^-x [κ+1]_q^-xP_κ,

	with 
	
    P_0= q^-x m n[u]_n, q^-x) [m+u]_n, q^-x.



	It is straightforward.


		 For x=-1,  we obtain the ℛ(p,q)- hypergeometric distribution:
		
    P_r(T_n=κ)   =   Φ(p,q)[[ n; κ ]]_ℛ(p,q) [m]_κ,ℛ(p,q) [u]_n-κ,ℛ(p,q) [m+u]_n,ℛ(p,q),

		where  0<q<p≤ 1, κ∈{0,⋯,n}, 
		Φ(p,q)=ϵ^κ(u-n+κ)_1 ϵ^(n-κ)(m-κ)_2
		and
		∑_κ=0^nP_r( T_n=κ)=1.

The particular case of q- deformation obtained in <cit.> is characterized by


	the j^th- order q- deformed factorial moment:
	
    μ_q([T_n]_j, q^-x) = [n]_j, q^-x [m]_j, q^-x [m+u]_j, q^-x,   j∈{1,⋯,n},


and
the factorial moment:
	
    μ_q[(T_n)_i] = i!∑_j=i^n(-1)^j-i[[ n; j ]]_q s_q^-x(j,i) (1-q^-x)^i-j[m]_j,q^-x [m+u]_j,q^-x,
 
	where    
	 i∈{1,2,⋯,n}  and  s_q^-x is the q- deformed Stirling number of the first kind.

	  The particular case of q-generalized Quesne deformation is characterized by the following properties:
	 	
	 		
  * The   generalized q- Quesne probability distribution:
	 		
    P_r( T_n=κ)=q^x(m-κ)(n-κ) p^x κ(u-n+κ)[n κ]^Q_p^-x,q^-x[m]^Q_κ,p^-x,q^-x[u]^Q_n-κ,p^-x,q^-x [m+u]^Q_n,p^-x,q^-x;

	 		
  * The j^th- order  generalized q- Quesne factorial moment:
	 		
    μ^Q_p,q([T_n]^Q_j,p^-x,q^-x) =(q p)^-x j[n]^Q_j,p^-x,q^-x [m]^Q_j,p^-x,q^-x [m+u]^Q_j,p^-x,q^-x,  j∈{1,2,⋯, n};

	 		
  * The  generalized q- Quesne factorial moment:
	 		
    μ^Q_p,q[(T_n)_i] = i!∑_j=i^n(-1)^j-i[[ n; j ]]^Q_p^-x,q^-xs^Q_p^-x,q^-x(j,i)  P(j,i),

	where 
	
    P(j,i)=p^j 2 (p^-x-q^x)^j-i [m]^Q_j,p^-x,q^-x p^τ(j-i) [m+u]^Q_j,p^-x,q^-x,

	τ∈ℕ, i∈{2,⋯,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:
		
    P_κ +1 =  p^x(u-n+2κ+1) q^x(n+m-2κ-1)[n-κ]^Q_p^-x,q^-x [u-n+κ+1]^Q_p^-x,q^-x [m-κ]^Q_p^-x,q^-x [κ+1]^Q_p^-x,q^-xP_κ,

		with
		
    P_0= q^x m n[u]^Q_n, p^-x,q^-x [m+u]^Q_n,p^-x,q^-x.

	



 §.§ Inverse Pólya distribution

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

	The inverse ℛ(p,q)- deformed Pólya distribution with parameters n, m, u, and κ is given by:
	
	
    P_y:=P_r( Y_n = y)= F(p,q)[n+y-1 y]_ℛ(p^-x,q^-x)[m]_n,ℛ(p^-x,q^-x) [u]_y,ℛ(p^-x,q^-x) [m+u]_n+y,ℛ(p^-x,q^-x),

	
	where  0<q<p≤ 1, y∈{1,⋯,n},  F(p,q)=ϵ^n(u-x)_1 ϵ^-yx(m-n+1)_2,  x is an integer,
	and
	
    ∑_t=0^∞F(p,q)[n+y-1 y]_ℛ(p^-x,q^-x)[m]_n,ℛ(p^-x,q^-x) [u]_y,ℛ(p^-x,q^-x) [m+u]_n+y,ℛ(p^-x,q^-x)=1.


	 Note that 
	the inverse q- deformed Pólya distribution  obtained in <cit.>   can  be recovered by taking ℛ(p,q)=q, ϵ_1=1, ϵ_2=q and F(p,q)=q^-yx(m-n+1) as:

    P_r( Y_n = y):= q^-yx(m-n+1)[n+y-1 y]_q^-x[m]_n, q^-x [u]_y,q^-x [m+u]_n+y,q^-x

where  0<q< 1 and  y∈{1,⋯,n}.



 The j^th- order ℛ(p,q)-factorial moment is given by:
		
    μ_ℛ(p,q)( [Y]_j,ℛ(p^-x,q^-x)) = [n-j+1]_j,ℛ(p^-x,q^-x)[u]_ℛ(j,p^-x,q^-x)ϵ^jx(m-n+1)_2[m+j]_j,ℛ(p^-x,q^-x),

		where j∈ℕ\{0} and m+j≠ 0.
		Moreover, for i∈{1,2,⋯,n},  the factorial moment yields:
		
    μ_ℛ(p,q)[(Y_n)_i] = i!∑_j=i^n(-1)^j-i[[ n+j-1;     j ]]_ℛ(p^-x,q^-x)IP(j,i),

		where
		
    IP(j,i)=s_ℛ(p^-x,q^-x)(j,i) ϵ^j 2_1ϵ^τ(j-i)_1(ϵ^-x_1-ϵ^-x_2)^j-i[u]_j,ℛ(p^-x,q^-x) [m+j]_j,ℛ(p^-x,q^-x) ϵ^jx(m-n+1)_2,
 
		 τ∈ℕ and  s_ℛ(p^-x,q^-x) is the ℛ(p,q)- deformed Stirling number of the first kind.
	
		
The recursion relation for the inverse ℛ(p,q)- Pólya distibutions is provided by:
		
    P_y+1 = ϵ^-x(m-n+1)_2[n+y]_ℛ(p^-x,q^-x)[u-y]_ℛ(p^-x,q^-x) [y+1]_ℛ(p^-x,q^-x)[m+u-n-y]_ℛ(p^-x,q^-x)  P_y,

		with  initial condition
		
    P_0 =ϵ^n(u-x)_1 [m]_n,ℛ(p^-x,q^-x) [m+u]_n,ℛ(p^-x,q^-x).
 
	

 Using (<ref>), the j^th- order ℛ(p,q)- deformed factorial moment  of Y_n is expressed as follows:
	

    μ_ℛ(p,q)([Y_n]_j,ℛ(p^-x,q^-x))    =   ∑_y=j^∞[y]_j,ℛ(p^-x,q^-x)Pr( Y_n =y)
       =   [n-j+1]_j,ℛ(p^-x,q^-x)[u]_j,ℛ(p^-x,q^-x)[m]_n,ℛ(p^-x,q^-x)ϵ^-n(u-x)_1ϵ^yx(m-n+1)_2
       ×   ∑_y=j^∞[n+y-1 y-j]_ℛ(p^-x,q^-x)[u-j]_y-j,ℛ(p^-x,q^-x) [m+u]_n+y,ℛ(p^-x,q^-x).


From the negative ℛ(p,q)-deformed Vandermonde's formula (<ref>), we have  the relation

    μ_ℛ(p,q)( [Y_n]_j,ℛ(p^-x,q^-x))   =   [n-j+1]_j,ℛ(p^-x,q^-x)[u]_j,ℛ(p^-x,q^-x)ϵ^jx(m-n+1)_2
       ×   [m]_n,ℛ(p^-x,q^-x) [m+j]_n+j,ℛ(p^-x,q^-x).

Since,

    [m+j]_n+j,ℛ(p^-x,q^-x)=[m+j]_j,ℛ(p,q)[m]_n,ℛ(p^-x,q^-x),

 the relation (<ref>) holds. Furthermore, the relation (<ref>) is deduced using the ℛ(p,q)- deformed factorial moment and (<ref>). 


	The recursion relation for the inverse q- Pólya distributions is given by the formula:
	
    P_y+1 = q^-x(m-n+1)[n+y]_q^-x[u-y]_q^-x [y+1]_q^-x[m+u-n-y]_q^-xP_y,     P_0 = [m]_n,q^-x [m+u]_n,q^-x.






Finally, it is worthy to note that,  for the generalized q-Quesne quantum algebra, we have:
	
		
  * The inverse generalized q- Quesne Pólya  probability distribution: 
		
    Pr( Y_n = y)= p^n(u-x) q^xy(m-n+1)[n+y-1 y]^Q_p^-x,q^-x[m]^Q_n,p^-x,q^-x [u]^Q_y,p^-x,q^-x [m+u]^Q_n+y,p^-x,q^-x;

		
  * The j^th- order generalized q- Quesne factorial moment:
		
    μ^Q_p,q [Y]^Q_j,p^-x,q^-x = [n-j+1]^Q_j,p^-x,q^-x[u]^Q_j,p^-x,q^-x q^-j x(m-n+1)[m+j]^Q_j,p^-x,q^-x(q p)^-x j

		with the associated factorial moment
		
    μ^Q_p,q[(Y_n)_i] = i!∑_j=i^n(-1)^j-i[[ n+j-1;     j ]]^Q_p^-x,q^-xs^Q_p^-x,q^-x(j,i) IP(j,i),

		where
		
    IP(j,i)=p^j 2 p^τ(j-i)(p^-x-q^x)^j-i
    			[u]^Q_j,p^-x,q^-x
    			q^-j x(m-n+1)
    			[m+j]^Q_j,p^-x,q^-x,

		 s^Q_p^-x,q^-x is the generalized q- Quesne Stirling number of the first kind and τ∈ℕ;
		
  * The recursion relation for the inverse generalized q- Quesne Pólya distrubtions:
		
		
    P_y+1 = q^x(m-n+1)[n+y]^Q_p,q[u-y]^Q_p,q [y+1]^Q_p,q[m+u-n-y]^Q_p,qP_y,     P_0 = p^n(u-x) [m]^Q_n,p^-x,q^-x [m+u]^Q_n,p^-x,q^-x.

		 
		



§ CONCLUDING  REMARKS

 ℛ(p, q)-deformed univariate  dicrete binomial, Euler, Pólya and inverse Pólya distributions, induced by  the 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. 
amsplain
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