Continuous Stieltjes-Wigert Limiting Behaviour of a Family of Confluent q-Chu-Vandermonde Distributions

Abstract

From Kemp [1], we have a family of confluent q-Chu- Vandermonde distributions, consisted by three members I, II and III, interpreted as a family of q-steady-state distributions from Markov chains. In this article, we provide the moments of the distributions of this family and we establish a continuous limiting behavior for the members I and II, in the sense of pointwise convergence, by applying a q-analogue of the usual Stirling asymptotic formula for the factorial number of order n. Specifically, we initially give the q-factorial moments and the usual moments for the family of confluent q-Chu- Vandermonde distributions and then we designate as a main theorem the conditions under which the confluent q-Chu-Vandermonde distributions I and II converge to a continuous Stieltjes-Wigert distribution. For the member III we give a continuous analogue. Moreover, as applications of this study we present a modified q-Bessel distribution, a generalized q-negative Binomial distribution and a generalized over/underdispersed (O/U) distribution. Note that in this article we prove the convergence of a family of discrete distributions to a continuous distribution which is not of a Gaussian type.

1. Introduction and Preliminaries

From Kemp [1], we have that the confluent q-Chu-Vandermonde hypergeometric sum,

$$\begin{array}{ccc}\hfill {}_1{\varphi }_1(b;c;q,c/b)& =& \sum _{n=0}^{\infty }\frac{{(b;q)}_n}{{(c;q)}_n{(q;q)}_n}{(-c/b)}^n{q}^{\left(\genfrac{}{}{0pt}{}{n}{2}\right)}=\frac{{(c/b;q)}_{\infty }}{{(c;q)}_{\infty }}\hfill \end{array}$$

(1)

where $0<q<1$ and ${(a;q)}_x={\prod }_{j=1}^x(1-a{q}^{j-1})$, $x=0,1,2,\dots$, gives rise to a family of q-Chu-Vandermonde distributions for suitable values of c and b, interpreted as a family q-steady-state distributions from Markov chains, with probability generating function (p.g.f.)

$$G(z)=\frac{{}_1{\varphi }_1(b;c;q,c/bz)}{{}_1{\varphi }_1(b;c;q,c/b)},z\in \mathcal{R}$$

(2)

and with probability function (p.f.)

$$p(x)=P[X=x]=f_X(x)=\frac{\frac{{(b;q)}_x}{{(c;q)}_x{(q;q)}_x}{q}^{\left(\genfrac{}{}{0pt}{}{x}{2}\right)}{(-c/b)}^x}{\frac{{(c/b;q)}_{\infty }}{{(c;q)}_{\infty }}},x=0,1,\dots .$$

(3)

Note that Equation (1) is a generalization of the q-binomial theorem and gives rise to two q-confluent distributions with infinite support and one with finite support.

The members of the above family of q-Chu-Vandermonde hypergeometric series discrete distributions are listed in Table 1.

Table 1. Confluent q-Chu-Vandermonde Distributions.

Confluent q-Chu-Vandermonde Distributions	Symbol of $G(z)$	Symbol of $p(x)$	Parameters b and c	Support
q-CCV-I	$G_{qCCVI}(z)$	$p_{qCCVI}(x)$	$b=-h$,$h>0,$$0<c<1$	$x=0,1,2,\dots$
q-CCV-II	$G_{qCCVII}(z)$	$p_{qCCVII}(x)$	$0<b<1$,$c=-\eta$,$\eta >0$	$x=0,1,2,\dots$
q-CCV-III	$G_{qCCVIII}(z)$	$p_{qCCVIII}(x)$	$b=q^{-n},n=0,1,\dots$,$0<c<1$	$x=0,1,\dots ,n$

Table 1. Confluent q-Chu-Vandermonde Distributions.

Confluent q-Chu-Vandermonde Distributions	Symbol of $G(z)$	Symbol of $p(x)$	Parameters b and c	Support
q-CCV-I	$G_{qCCVI}(z)$	$p_{qCCVI}(x)$	$b=-h$,$h>0,$$0<c<1$	$x=0,1,2,\dots$
q-CCV-II	$G_{qCCVII}(z)$	$p_{qCCVII}(x)$	$0<b<1$,$c=-\eta$,$\eta >0$	$x=0,1,2,\dots$
q-CCV-III	$G_{qCCVIII}(z)$	$p_{qCCVIII}(x)$	$b=q^{-n},n=0,1,\dots$,$0<c<1$	$x=0,1,\dots ,n$

The distributions of the above table have finite mean and variance when $n\to \infty$ and we cannot conclude the asymptotic normality in the sense of the DeMoivre-Laplace classical limit theorem, as in the case of ordinary hypergeometric series discrete distributions. Also, we cannot apply asymptotic methods –central or/and local limit theorems– as in Bender [2], Canfield [3], Flajolet and Soria [4], Odlyzko [5] et al.

Thus an important question is arisen about the asymptotic behaviour for $n\to \infty$ of this family of q-Chu-Vandermonde hypergeometric series discrete distributions.

Recently, the authors investigated the asymptotic behaviour of another member of q-hypergeometric series discrete distributions, having also finite mean and variance, that of a q-Binomial one [6]. Specifically it has been established a pointwise convergence to a continuous Stieltjes-Wigert distribution.

In this article, we provide a continuous limiting behaviour of the above family of confluent q- Chu- Vandermonde discrete distributions, for $0<q<1$, in the sense of pointwise convergence. Specifically, we initially give the q-factorial moments and the usual moments of this family and then we designate as a main theorem the conditions under which the confluent q-Chu-Vandermonde distributions I and II converge to a continuous Stieltjes-Wigert distribution. For the member III we give a continuous analogue. Moreover, as applications of this study we present a modified q-Bessel distribution, a generalized q-negative Binomial distribution and a generalized over / underdispersed (O/U) distribution. Note that, the main contribution of this article is that a family of discrete distributions converges to a continuous distribution which is not of a Gaussian type.

To establish the proof of our main theorem we apply a q-analogue of the well known Stirling asymptotic formula for the n factorial ($n!$) established by Kyriakoussis and Vamvakari [6]. The authors have derived an asymptotic expansion for $n\to \infty$ of the q-factorial number of order n ,

$$\begin{array}{c}\hfill {\left[n\right]}_q!={\left[1\right]}_q{\left[2\right]}_q\dots {\left[n\right]}_q=\prod _{k=1}^n\frac{1-q^k}{{(1-q)}^n}=\frac{{(q;q)}_n}{{(1-q)}^n}\end{array}$$

(4)

where $0<q<1$ and ${\left[t\right]}_q=\frac{1-q^t}{1-q}$, the q-number t. Analytically we have

$$\begin{array}{ccc}\hfill {\left[n\right]}_q!& =& \frac{{(2\pi (1-q))}^{1/2}}{{(qlog{q}^{-1})}^{1/2}}\frac{q^{\left(\genfrac{}{}{0pt}{}{n}{2}\right)}{q}^{-n/2}{\left[n\right]}_{1/q}^{n+1/2}}{{\prod }_{j=1}^{\infty }(1+q(q^{-n}-1)q^{j-1})}\left(1+O(n^{-1})\right)\hfill \end{array}$$

(5)

For answering the main question of this study we apply our above asymptotic formula for the q-factorial number of order n to provide pointwise convergence of the family of confluent q-Chu-Vandermonde distributions to a continuous Stieltjes-Wigert distribution with probability density function

$$v_q^{SW}(x)=\frac{q^{1/8}}{\sqrt{2\pi log{q}^{-1}x}}{e}^{\frac{{(logx)}^2}{2logq}},x>0$$

(6)

with mean value ${\mu }^{SW}=q^{-1}$ and standard deviation ${\sigma }^{SW}=q^{-3/2}{(1-q)}^{1/2}$

Remark 1. We note that the corresponding to the probability measure Equation (3) orthogonal polynomials are the q-Meixner ones (see [7]). Also, we have that the q-Meixner orthogonal polynomials converge to the Stieltjes-Wigert ones, both members of the q-Askey scheme (see [7,8]). But, from the convergence of the orthogonal polynomials one cannot conclude the convergence of the corresponding probability measures (see [9,10]). So, in this paper the method of pointwise convergence is followed.

2. On Factorial Moments of the Confluent q-Chu-Vandermonde Distributions

In this section, we first transfer from the random variable X of the family of confluent q-Chu-Vandermonde distributions Equation (3) to the equal-distributed deformed random variable $Y={\left[X\right]}_{1/q}$, and we then compute the mean value and variance of the random variable Y, say ${\mu }_q$ and ${\sigma }_q^2$ respectively. We also derive all the descending factorial k-th order moments of the random variable X through the computation of all the r-th orders factorials of the random variable Y, named q-factorial moments of the r.v. X.

Proposition 1. The q-mean and q-variance of the family of confluent q-Chu-Vandermonde distributions are given respectively by

$$\begin{array}{c}\hfill {\mu }_q=-\frac{c}{b}\frac{1-b}{1-q}and{\sigma }_q^2={\left(-\frac{c}{b}\right)}^2\frac{1-b}{q(1-q)}-\frac{c}{b}\frac{1-b}{1-q}\end{array}$$

Proof. 

The q-mean of the family of confluent q-Chu-Vandermonde distributions is given by

$$\begin{array}{c}\hfill {\mu }_q=E(Y)=E({\left[X\right]}_{1/q})=\sum _{x=0}^n{\left[x\right]}_{1/q}{f}_X(x)=\frac{{(c;q)}_{\infty }}{{(c/b;q)}_{\infty }}\sum _{x=0}^{\infty }{\left[x\right]}_{1/q}\frac{{(b;q)}_x}{{(c;q)}_x{(q;q)}_x}{q}^{\left(\genfrac{}{}{0pt}{}{x}{2}\right)}{\left(-\frac{c}{b}\right)}^x\end{array}$$

(7)

and since

$${\left[x\right]}_{1/q}=q^{-x+1}{\left[x\right]}_q,q^{-x+1}{q}^{\left(\genfrac{}{}{0pt}{}{x}{2}\right)}=q^{\left(\genfrac{}{}{0pt}{}{x-1}{2}\right)},\frac{{\left[x\right]}_q}{{(q;q)}_x}=\frac{1}{(1-q){(q;q)}_{x-1}}$$

and

$${(b;q)}_x=(1-b){(bq;q)}_{x-1},{(c;q)}_x=(1-c){(cq;q)}_{x-1}$$

it is written as

$${\mu }_q=-\frac{c}{b}\frac{1-b}{(1-c)(1-q)}\frac{{(c;q)}_{\infty }}{{(c/b;q)}_{\infty }}\sum _{x=1}^{\infty }\frac{{(bq;q)}_{x-1}}{{(cq;q)}_{x-1}{(q;q)}_{x-1}}{q}^{\left(\genfrac{}{}{0pt}{}{x-1}{2}\right)}{\left(-\frac{c}{b}\right)}^{x-1}$$

(8)

Using the confluent q-Chu-Vandermonde hypergeometric sum Equation (1) we obtain the formula of the q-mean in Equation (7).

For the evaluation of the q-variance we need to find the second order moment of the r.v. $Y={\left[X\right]}_{1/q}$ which is given by

$$\begin{array}{c}\hfill E\left[Y^2\right]=E\left[{\left[X\right]}_{1/q}^2\right]=\sum _{x=0}^{\infty }{\left[x\right]}_{1/q}^2{f}_X(x)=\frac{{(c;q)}_{\infty }}{{(c/b;q)}_{\infty }}\sum _{x=0}^{\infty }{\left[x\right]}_{1/q}^2\frac{{(b;q)}_x}{{(c;q)}_x{(q;q)}_x}{q}^{\left(\genfrac{}{}{0pt}{}{x}{2}\right)}{\left(-\frac{c}{b}\right)}^x\end{array}$$

(9)

Since

$${\left[x\right]}_q={[x-1]}_q+q^{x-1},q^{-2x+2}{q}^{\left(\genfrac{}{}{0pt}{}{x}{2}\right)}=q^{-1}{q}^{\left(\genfrac{}{}{0pt}{}{x-2}{2}\right)}$$

and

$${(b;q)}_x=(1-b)(1-bq){(b{q}^2;q)}_{x-2},{(c;q)}_x=(1-c)(1-cq){(c{q}^2;q)}_{x-2}$$

Equation (9) becomes

$$\begin{array}{ccc}\hfill E\left[Y^2\right]& =& {\left(-\frac{c}{b}\right)}^2\frac{(1-b)(1-bq)}{q{(1-q)}^2(1-c)(1-cq)}\frac{{(c;q)}_{\infty }}{{(c/b;q)}_{\infty }}\sum _{x=2}^{\infty }\frac{{(b{q}^2;q)}_{x-2}}{{(c{q}^2;q)}_{x-2}{(q;q)}_{x-2}}{q}^{\left(\genfrac{}{}{0pt}{}{x-2}{2}\right)}{\left(-\frac{c}{b}\right)}^{x-2}\hfill \\ & =& {\left(-\frac{c}{b}\right)}^2\frac{(1-b)(1-bq)}{q{(1-q)}^2(1-c)(1-cq)}\frac{{(c;q)}_{\infty }}{{(c{q}^2;q)}_{\infty }}={\left(-\frac{c}{b}\right)}^2\frac{(1-b)(1-bq)}{q{(1-q)}^2}\hfill \end{array}$$

So,

$${\sigma }_q^2=V(Y)=V({\left[X\right]}_{1/q})={\left(-\frac{c}{b}\right)}^2\frac{(1-b)(1-bq)}{q{(1-q)}^2}-\frac{c}{b}\frac{1-b}{1-q}-{\left(-\frac{c}{b}\right)}^2\frac{{(1-b)}^2}{{(1-q)}^2}$$

(10)

from which we obtain the formula of the q-variance given in Equation (7).

Proposition 2. The r-th order q-factorial moments of the family of confluent q-Chu-Vandermonde distributions are given by

$$E({\left[X\right]}_{r,1/q})=\frac{{(b;q)}_r}{{(1-q)}^r}{(-\frac{c}{b})}^r,r=1,2,\dots .$$

(11)

Proof. 

The r-th order q-factorial moments of the family of confluent q-Chu-Vandermonde distributions is

$$\begin{array}{ccc}\hfill E({\left[X\right]}_{r,1/q})& =& \sum _{x=r}^{\infty }{\left[x\right]}_{r,1/q}{f}_X(x)\hfill \\ & =& \frac{{(c;q)}_{\infty }}{{(c/b;q)}_{\infty }}\sum _{x=r}^{\infty }{\left[x\right]}_{1/q}{[x-1]}_{1/q}\cdots {[x-r+1]}_{1/q}\frac{{(b;q)}_x}{{(c;q)}_x{(q;q)}_x}{q}^{\left(\genfrac{}{}{0pt}{}{x}{2}\right)}{(-c/b)}^x\hfill \end{array}$$

(12)

Since

$${\left[x\right]}_{1/q}=q^{-x+1}{\left[x\right]}_q,\left(\genfrac{}{}{0pt}{}{x}{2}\right)=\left(\genfrac{}{}{0pt}{}{x-r}{2}\right)+\left(\genfrac{}{}{0pt}{}{r}{2}\right)+r(x-r),\frac{{\left[x\right]}_{r,q}}{{(q;q)}_x}=\frac{1}{{(1-q)}^r{(q;q)}_{x-r}}$$

$${(b;q)}_x={(b;q)}_r{(b{q}^r;q)}_{x-r}and\frac{{(c;q)}_{\infty }}{{(c;q)}_x}=\frac{{(c{q}^r;q)}_{\infty }}{{(c{q}^r;q)}_{x-r}}$$

the sum Equation (12) becomes

$$E({\left[X\right]}_{r,1/q})=\frac{{(c{q}^r;q)}_{\infty }{(b;q)}_r}{{(c/b;q)}_{\infty }{(1-q)}^r}{(-c/b)}^r\sum _{x=r}^{\infty }\frac{{(b{q}^r;q)}_{x-r}}{{(c{q}^r;q)}_{x-r}{(q;q)}_{x-r}}{q}^{\left(\genfrac{}{}{0pt}{}{x-r}{2}\right)}{(-c/b)}^{x-r}$$

(13)

By the confluent q-Vandermonde sum the r-th order q-factorial moments of the family of confluent q-Chu-Vandermonde distributions, reduces to Equation (11).

Proposition 3. The descending factorial k-th order moments of the r.v. X of the family of q-Chu-Vandermonde distributions are given by

$$E({(X)}_k)=\frac{k!}{{(c/b;q)}_{\infty }}\sum _{r=k}^{\infty }\frac{s_q(r,k){(q-1)}^{r-k}{(c{q}^r;q)}_{\infty }}{{\left[r\right]}_{1/q}!}E({\left[X\right]}_{r,1/q}){}_1{\varphi }_1(b{q}^r;c{q}^r;q,c{q}^r/b)$$

(14)

Proof. 

The relation of the factorial descending moments with the q-factorial descending moments through the q-Stirling numbers of the first kind is given by the sum

$$E({(X)}_k)=k!\sum _{r=k}^{\infty }\frac{s_q(r,k){(q-1)}^{r-k}}{{\left[r\right]}_q!}E({\left[X\right]}_{r,q})$$

(15)

where $s_q(r,k)$ the q-Stirling numbers of the first kind (see Charalambides [11]).

Since

$${\left(\genfrac{}{}{0pt}{}{x}{r}\right)}_q=q^{r(x-r)}{\left(\genfrac{}{}{0pt}{}{x}{r}\right)}_{1/q}$$

the sum Equation (15) is written as

$$\begin{array}{ccc}\hfill E({(X)}_k)& =& k!\sum _{r=k}^{\infty }\frac{s_q(r,k){(q-1)}^{r-k}}{{\left[r\right]}_{1/q}!}E(q^{r(x-r)}{\left[X\right]}_{r,1/q})\hfill \\ & =& k!\sum _{r=k}^{\infty }\frac{s_q(r,k){(q-1)}^{r-k}}{{\left[r\right]}_{1/q}!}\frac{{(c{q}^r;q)}_{\infty }{(b;q)}_r{(-c/b)}^r}{{(c/b;q)}_{\infty }{(1-q)}^r}\sum _{x=r}^{\infty }\frac{{(b{q}^r;q)}_{x-r}}{{(c{q}^r;q)}_{x-r}{(q;q)}_{x-r}}{q}^{\left(\genfrac{}{}{0pt}{}{x-r}{2}\right)}{(-c{q}^r/b)}^{x-r}\hfill \end{array}$$

(16)

By Equation (11) of the previous proposition 2 and the definition of the q-hypergeometric function Equation (1), Equation (16) reduces to Equation (14).

3. Pointwise Convergence of A Family of Confluent q-Chu-Vandermonde Distributions to the Stieltjes-Wigert Distribution

In this section, we transfer from the random variable X of the family of confluent q-Chu-Vandermonde distributions Equation (3) to the equal-distributed deformed random variable $Y={\left[X\right]}_{1/q}$, and using the q-analogue Stirling asymptotic formula (5), we establish the convergence to a deformed standardized continuous Stieltjes-Wigert distribution of the members I and II of the family of q-Chu-Vandermonde distributions.

Theorem 1. Let the p.f. of the family of confluent q-Chu-Vandermonde distributions be of the form

$$\begin{array}{c}\hfill f_X(x)=\frac{\frac{{(b;q)}_x}{{(c;q)}_x{(q;q)}_x}{q}^{\left(\genfrac{}{}{0pt}{}{x}{2}\right)}{(-c/b)}^x}{\frac{{(c/b;q)}_{\infty }}{{(c;q)}_{\infty }}},x=0,1,\dots \end{array}$$

(17)

where $b=b_n,$ $c=c_n,$ $n=0,1,2,\dots ,$ such that $b_n=o(1)$ and $-c_n/b_n\to \infty ,$ as $n\to \infty$. Then, for $n\to \infty ,$ the p.f. $f_X(x),x=0,1,2,\dots$ is approximated by a deformed standardized continuous Stieltjes-Wigert distribution as follows

$$\begin{array}{ccc}\hfill f_X(x)& \cong & \frac{q^{1/8}{(log{q}^{-1})}^{1/2}}{{(2\pi )}^{1/2}}{\left(q^{-3/2}{(1-q)}^{1/2}\frac{{\left[x\right]}_{1/q}-{\mu }_q}{{\sigma }_q}+q^{-1}\right)}^{1/2}\hfill \\ & & ·exp\left(\frac{1}{2logq}{log}^2\left(q^{-3/2}{(1-q)}^{1/2}\frac{{\left[x\right]}_{1/q}-{\mu }_q}{{\sigma }_q}+q^{-1}\right)\right),x\ge 0\hfill \end{array}$$

(18)

Proof. 

Since the product ${(b;q)}_x={\prod }_{j=1}^x(1-b{q}^{j-1})=(1-b)(1-bq)\cdots (1-b{q}^{x-1})$ for $b=b_n$ with $b_n\to 0$ as $n\to \infty$ is approximated by ${(b_n;q)}_x\cong 1$ the p.f. of the family of confluent q-Chu-Vandermonde distributions is discretely approximated as

$$\begin{array}{c}\hfill f_X(x)\cong \frac{\frac{q^{\left(\genfrac{}{}{0pt}{}{x}{2}\right)}{(-c_n/b_n)}^x}{{(c_n;q)}_x{(q;q)}_x}}{\frac{{(c_n/b_n;q)}_{\infty }}{{(c_n;q)}_{\infty }}},x=0,1,\dots .\end{array}$$

(19)

By using the q-Stirling asymptotic formula (5) we get the following approximation for the p.f. $f_X(x)$ with $b=b_n,$ $c=c_n$ such that $b_n=o(1)$ and $-c_n/b_n\to \infty ,$ as $n\to \infty ,$

$$\begin{array}{ccc}\hfill f_X(x)& \cong & \frac{{(qlog{q}^{-1})}^{1/2}}{{(2\pi (1-q))}^{1/2}}\frac{{\left(-c_n/b_n\right)}^x}{{(1-q)}^x}\frac{{\prod }_{j=1}^{\infty }(1+q(q^{-x}-1)q^{j-1}){(c_n;q)}_{\infty }}{q^{-x/2}{\left[x\right]}_{1/q}^{x+1/2}{(c_n;q)}_x{(c_n/b_n;q)}_{\infty }}\hfill \end{array}$$

(20)

From the standardized r.v. $Z=\frac{{\left[X\right]}_{1/q}-{\mu }_q}{{\sigma }_q}$ with ${\mu }_q$ and ${\sigma }_q$ given in Equation (7), we get

$$\begin{array}{ccc}\hfill {\left[x\right]}_{1/q}& =& {\sigma }_{q}z+{\mu }_q={\left[{\left(-\frac{c_n}{b_n}\right)}^2\frac{1-q}{q(1-q)}-\frac{c_n}{b_n}\frac{1-b_n}{1-q}\right]}^{1/2}z-\frac{c_n}{b_n}\frac{1-b_n}{1-q}\hfill \\ & =& -\frac{c_n}{b_n}\frac{1-b_n}{1-q}\left[{\left(\frac{1-q}{q(1-b_n)}-\frac{b_n}{c_n}\frac{1-q}{1-b_n}\right)}^{1/2}z+1\right]\hfill \end{array}$$

(21)

Using the assumptions $b_n=o(1)$ and $-c_n/b_n\to \infty$ as $n\to \infty ,$ we have

$$\begin{array}{c}\hfill {\left[x\right]}_{1/q}\cong -\frac{c_n}{b_n}\frac{q}{1-q}(q^{-3/2}{(1-q)}^{1/2}z+q^{-1})\end{array}$$

(22)

Also, by the previous two equations we get

$$q^{-x}=-\frac{c_n}{b_n}\frac{1-b_n}{q}\left[{\left(\frac{1-q}{q(1-b_n)}-\frac{b_n}{c_n}\frac{1-q}{1-b_n}\right)}^{1/2}z+1\right]+1$$

(23)

and

$$q^{-x}\cong -\frac{c_n}{b_n}\left(q^{-3/2}(1-q)z+q^{-1}\right)$$

(24)

Moreover, by the Equation (23) we find

$$x=\frac{1}{log{q}^{-1}}log\left(-\frac{c_n}{b_n}\frac{1-b_n}{q}\left[{\left(\frac{1-q}{q(1-b_n)}-\frac{b_n}{c_n}\frac{1-q}{1-b_n}\right)}^{1/2}z+1\right]+1\right)$$

(25)

and

$$x\cong \frac{1}{log{q}^{-1}}log\left(-\frac{c_n}{b_n}\left(q^{-3/2}(1-q)z+q^{-1}\right)\right)$$

(26)

Finally, by the Equation (21) we get

$$\begin{array}{ccc}\hfill {\left[x\right]}_{1/q}^x& =& {\left(-\frac{c_n}{b_n}\right)}^x{\left(\frac{1-b_n}{1-q}\right)}^x{\left[{\left(\frac{1-q}{q(1-b_n)}-\frac{b_n}{c_n}\frac{1-q}{1-b_n}\right)}^{1/2}z+1\right]}^x\hfill \\ & =& {\left(-\frac{c_n}{b_n}\right)}^x{\left(\frac{1-b_n}{1-q}\right)}^{x}exp\left(xlog\left[{\left(\frac{1-q}{q(1-b_n)}-\frac{b_n}{c_n}\frac{1-q}{1-b_n}\right)}^{1/2}z+1\right]\right)\hfill \end{array}$$

(27)

and

$$\begin{array}{ccc}\hfill {\left[x\right]}_{1/q}^x& \cong & {\left(-\frac{c_n}{b_n}\right)}^x{\left(\frac{1}{1-q}\right)}^x\hfill \\ & ·& exp\left(\frac{1}{log{q}^{-1}}log\left(-\frac{c_n}{b_n}\left(q^{-3/2}(1-q)z+q^{-1}\right)\right)log\left(q\left(q^{-3/2}{(1-q)}^{1/2}z+q^{-1}\right)\right)\right)\hfill \end{array}$$

(28)

with $z=\frac{{\left[x\right]}_{1/q}-{\mu }_q}{{\sigma }_q}$

Substituting all the previous approximations Equations (22),(24),(26),(28) to the p.f. $f_X(x)$ we get the approximation

$$\begin{array}{ccc}\hfill f_X(x)& \cong & \frac{{(q(1-q)log{q}^{-1})}^{1/2}}{{(2\pi q(1-q))}^{1/2}}\frac{{\prod }_{j=1}^{\infty }\left(1-\frac{c_n}{b_n}\left(q^{-3/2}{(1-q)}^{1/2}z+q^{-1}\right)q^{j-1}\right){(c_n;q)}_{\infty }}{{(c_n;q)}_x{(c_n/b_n;q)}_{\infty }}\hfill \\ \hfill \\ & ·& exp\left(\frac{1}{logq}log\left(-\frac{c_n}{b_n}\left(q^{-3/2}{(1-q)}^{1/2}z+q^{-1}\right)\right)log\left(q\left(q^{-3/2}{(1-q)}^{1/2}z+q^{-1}\right)\right)\right)\hfill \\ & ·& {\left(-\frac{c_n}{b_n}\frac{q^{1/2}}{{(1-q)}^{1/2}}\left(q^{-3/2}{(1-q)}^{1/2}z+q^{-1}\right)\right)}^{-1},z=\frac{{\left[x\right]}_{1/q}-{\mu }_q}{{\sigma }_q}\hfill \end{array}$$

(29)

As a last step, we need to estimate the products $\prod _{j=1}^{\infty }\left(1-\frac{c_n}{b_n}\left(q^{-3/2}{(1-q)}^{1/2}z+q^{-1}\right)q^{j-1}\right),$ ${(c_n/b_n;q)}_x={\prod }_{j=1}^{\infty }(1-c_n/b_n{q}^{j-1})$ and ${(c_n;q)}_{\infty }/{(c_n;q)}_x={(c_n{q}^x;q)}_{\infty }={\prod }_{j=1}^{\infty }(1-c_n{q}^x{q}^{j-1})$ by integrals. Since the first product is written as

$$\begin{array}{c}\hfill \prod _{j=1}^{\infty }\left(1-\frac{c_n}{b_n}\left(q^{-3/2}{(1-q)}^{1/2}z+q^{-1}\right)q^{j-1}\right)=exp\left(\sum _{j=1}^{\infty }log\left(1-\frac{c_n}{b_n}\left(q^{-3/2}{(1-q)}^{1/2}z+q^{-1}\right)q^{j-1}\right)\right)\end{array}$$

(30)

and the function

$$h(x)=log\left(1-\frac{c_n}{b_n}\left(q^{-3/2}{(1-q)}^{1/2}z+q^{-1}\right)q^{x-1}\right)$$

has all orders continuous derivatives in $[1,\infty )$, we can apply the Euler-Maclaurin summation formula (see [5], p. 1090) in the sum of the Equation (30).

So,

$$\begin{array}{ccc}\hfill & & \sum _{j=1}^{\infty }log\left(1-\frac{c_n}{b_n}\left(q^{-3/2}{(1-q)}^{1/2}z+q^{-1}\right)q^{j-1}\right)=\underset{1}{\overset{\infty }{\int }}log\left(1-\frac{c_n}{b_n}\left(q^{-3/2}{(1-q)}^{1/2}z+q^{-1}\right)q^{u-1}\right)du\hfill \\ & & +\frac{1}{2}log\left(1-\frac{c_n}{b_n}\left(q^{-3/2}{(1-q)}^{1/2}z+q^{-1}\right)\right)+\sum _{k=1}^m\frac{{\beta }_{2k}}{(2k)!}{h}^{(2k-1)}\left(1-\frac{c_n}{b_n}\left(q^{-3/2}{(1-q)}^{1/2}z+q^{-1}\right)\right)+R_k\hfill \end{array}$$

(31)

where

$$|R_k|\le \frac{|{\beta }_{2k}|}{(2k)!}\underset{1}{\overset{\infty }{\int }}|g^{(2k)}\left(1-\frac{c_n}{b_n}\left(q^{-3/2}{(1-q)}^{1/2}z+q^{-1}\right)q^{u-1}\right)|du$$

(32)

with ${\beta }_k$ the Bernoulli numbers.

Now, expressing the integral appearing in Equation (31) through the dilogarithm function we get

$$\begin{array}{c}\hfill \underset{1}{\overset{\infty }{\int }}log\left(1-\frac{c_n}{b_n}\left(q^{-3/2}{(1-q)}^{1/2}z+q^{-1}\right)q^{u-1}\right)du=\frac{1}{logq}{Li}_2\left(-\left(-\frac{c_n}{b_n}\right)\left(q^{-3/2}{(1-q)}^{1/2}z+q^{-1}\right)\right)\end{array}$$

(33)

where ${Li}_2(y)={\sum }_{k\ge 1}\frac{y^k}{k^2}$ the dilogarithm function. The dilogarithm satisfies the Landen’s identity

$${Li}_2(-y)={Li}_2\left(\frac{y}{y+1}\right)-\frac{1}{2}{log}^2(1+y)$$

(34)

Applying the Landen’s identity to Equation (33) we obtain

$$\begin{array}{ccc}\hfill \underset{1}{\overset{\infty }{\int }}log\left(1-\frac{c_n}{b_n}\left(q^{-3/2}{(1-q)}^{1/2}z+q^{-1}\right)q^{u-1}\right)du& =& \frac{1}{2log{q}^{-1}}{log}^2\left(-\frac{c_n}{b_n}\left(q^{-3/2}{(1-q)}^{1/2}z+q^{-1}\right)\right)\hfill \\ & +& {Li}_2\left(\frac{-\frac{c_n}{b_n}\left(q^{-3/2}{(1-q)}^{1/2}z+q^{-1}\right)}{-\frac{c_n}{b_n}\left(q^{-3/2}{(1-q)}^{1/2}z+q^{-1}\right)+1}\right)\hfill \end{array}$$

(35)

Next, we estimate the sum and the quantity $R_k$ appearing in Equation (31)

$$\begin{array}{ccc}\hfill & & \sum _{k=1}^m\frac{{\beta }_{2k}}{(2k)!}{h}^{(2k-1)}\left(1-\frac{c_n}{b_n}\left(q^{-3/2}{(1-q)}^{1/2}z+q^{-1}\right)\right)+R_k\hfill \\ & & =\frac{{\beta }_2}{2}{h}^{{}^{\prime }}\left(1-\frac{c_n}{b_n}\left(q^{-3/2}{(1-q)}^{1/2}z+q^{-1}\right)\right)+R_1+O\left({\left(-\frac{c_n}{b_n}\right)}^{-2}\right)\hfill \\ & & =\frac{{\beta }_{2}logq}{2}\frac{\left(-\frac{c_n}{b_n}\right)\left(q^{-3/2}{(1-q)}^{1/2}z+q^{-1}\right)}{1-\frac{c_n}{b_n}\left(q^{-3/2}{(1-q)}^{1/2}z+q^{-1}\right)}+O\left(-\frac{b_n}{c_n}\right)\hfill \end{array}$$

(36)

So, by applying Equations (35) and (36) to Equation (31) we obtain

$$\begin{array}{ccc}\hfill & & \sum _{j=1}^{\infty }log\left(1-\frac{c_n}{b_n}\left(q^{-3/2}{(1-q)}^{1/2}z+q^{-1}\right)q^{j-1}\right)=\frac{1}{2log{q}^{-1}}{log}^2\left(-\frac{c_n}{b_n}\left(q^{-3/2}{(1-q)}^{1/2}z+q^{-1}\right)\right)\hfill \\ & & +{Li}_2\left(\frac{-\frac{c_n}{b_n}\left(q^{-3/2}{(1-q)}^{1/2}z+q^{-1}\right)}{-\frac{c_n}{b_n}\left(q^{-3/2}{(1-q)}^{1/2}z+q^{-1}\right)+1}\right)+\frac{1}{2}log\left(1-\frac{c_n}{b_n}\left(q^{-3/2}{(1-q)}^{1/2}z+q^{-1}\right)\right)\hfill \\ & & +\frac{{\beta }_{2}logq}{2}\frac{\left(-\frac{c_n}{b_n}\right)\left(q^{-3/2}{(1-q)}^{1/2}z+q^{-1}\right)}{1-\frac{c_n}{b_n}\left(q^{-3/2}{(1-q)}^{1/2}z+q^{-1}\right)}+O\left(-\frac{b_n}{c_n}\right)\hfill \end{array}$$

(37)

Working similarly for the sum appearing in the product

$$\begin{array}{c}\hfill \prod _{j=1}^{\infty }\left(1-\frac{c_n}{b_n}{q}^{j-1}\right)=exp\left(\sum _{j=1}^{\infty }log\left(1-\frac{c_n}{b_n}{q}^{j-1}\right)\right)\end{array}$$

(38)

we obtain

$$\begin{array}{ccc}\hfill \sum _{j=1}^{\infty }log\left(1-\frac{c_n}{b_n}{q}^{j-1}\right)& =& \frac{1}{2log{q}^{-1}}{log}^2\left(-\frac{c}{b_n}\right)+{Li}_2\left(\frac{-\frac{c}{b_n}}{-\frac{c}{b_n}+1}\right)\hfill \\ & +& \frac{1}{2}log\left(1-\frac{c}{b_n}\right)+\frac{{\beta }_{2}logq}{2}\frac{\left(-\frac{c}{b_n}\right)}{1-\frac{c}{b_n}}+O\left(-\frac{b_n}{c}\right)\hfill \end{array}$$

(39)

We need now to estimate the sum appearing in the last product

$$\begin{array}{ccc}\hfill {(c_n;q)}_{\infty }/{(c_n;q)}_x& =& {(c_n{q}^x;q)}_{\infty }=\prod _{j=1}^{\infty }(1-c_n{q}^x{q}^{j-1})\hfill \\ & =& exp\left(\sum _{j=1}^{\infty }log\left(1+b_n{\left(q^{-3/2}{(1-q)}^{1/2}z+q^{-1}\right)}^{-1}{q}^{j-1}\right)\right)\hfill \end{array}$$

(40)

and working analogously as previous we get

$$\begin{array}{ccc}\hfill & & \sum _{j=1}^{\infty }log\left(1+b_n{\left(q^{-3/2}{(1-q)}^{1/2}z+q^{-1}\right)}^{-1}{q}^{j-1}\right)=\frac{1}{2log{q}^{-1}}{log}^2\left(b_n{\left(q^{-3/2}{(1-q)}^{1/2}z+q^{-1}\right)}^{-1}\right)\hfill \\ & & +{Li}_2\left({\frac{b_n{\left(q^{-3/2}{(1-q)}^{1/2}z+q^{-1}\right)}^{-1}}{b_n\left(q^{-3/2}{(1-q)}^{1/2}z+q^{-1}\right)+1}}^{-1}\right)+\frac{1}{2}log\left(1+b_n{\left(q^{-3/2}{(1-q)}^{1/2}z+q^{-1}\right)}^{-1}\right)\hfill \\ & & +\frac{{\beta }_{2}logq}{2}\frac{b_n{\left(q^{-3/2}{(1-q)}^{1/2}z+q^{-1}\right)}^{-1}}{1+b_n{\left(q^{-3/2}{(1-q)}^{1/2}z+q^{-1}\right)}^{-1}}+O\left(b_n\right)\hfill \end{array}$$

(41)

Applying the estimations Equations (37),(39),(41) to the approximation Equation (29), carrying out all the necessary manipulations and by the assumptions $b_n=o(1)$ and $-c_n/b_n\to \infty ,$ as $n\to \infty ,$ we derive

$$\begin{array}{ccc}\hfill f_X(x)& \cong & \frac{q^{1/8}{(log{q}^{-1})}^{1/2}}{{(2\pi )}^{1/2}}{\left(q^{-3/2}{(1-q)}^{1/2}z+q^{-1}\right)}^{1/2}exp\left(\frac{1}{2logq}{log}^2\left(q^{-3/2}{(1-q)}^{1/2}z+q^{-1}\right)\right)\hfill \\ & & z=\frac{{\left[x\right]}_{1/q}-{\mu }_q}{{\sigma }_q},x\ge 0\hfill \end{array}$$

(42)

that is Equation (18).

Moreover, by standardizing the continuous Stieltjes-Wigert distribution Equation (6) and then deforming this by the random variable $\frac{{\left[X\right]}_{1/q}-{\mu }_q}{{\sigma }_q}$, we obtain Equation (18) and our proof is completed.

Remark 1. Under the assumptions of the theorem 1 the probability functions I and II of the Table 1 have the asymptotic approximation Equation (18). Note that we do not have the same conclusion for the p.f. III of the table 1 since the assumption $b_n=o(1)$ does not hold.

Remark 2. From the proof of the theorem 1 we have that the p.f. of the family of confluent q- Chu -Vandermonde discrete distributions is discretely approximated by

$$\begin{array}{c}\hfill f_X(x)\cong \frac{\frac{q^{\left(\genfrac{}{}{0pt}{}{x}{2}\right)}{(-c_n/b_n)}^x}{{(c_n;q)}_x{(q;q)}_x}}{\frac{{(c_n/b_n;q)}_{\infty }}{{(c_n;q)}_{\infty }}},x=0,1,\dots .\end{array}$$

From the q-CCVI of the table 1, for $b_n=-h_n=o(1),$$h_n>0$ and $c_n=c$ constant with $0<c<1,$ $n=0,1,2,\dots ,$ we get

$$\begin{array}{c}\hfill p_{qCCVI}(x)\cong \frac{\frac{q^{\left(\genfrac{}{}{0pt}{}{x}{2}\right)}{(c/h_n)}^x}{{(c;q)}_x{(q;q)}_x}}{\frac{{(-c/h_n;q)}_{\infty }}{{(c;q)}_{\infty }}},x=0,1,\dots .\end{array}$$

(43)

Consequently, $p_{qCCVI}(x)$ is discretely approximated by a modified q-Bessel distribution.

Applications

1. A modified q-Bessel distribution: From the remarks 1 and 2 we get an asymptotic expression as in Equation (18) for the modified q-Bessel distribution with p.f.

$$\begin{array}{c}\hfill p_{MB}(x)=\frac{\frac{q^{\left(\genfrac{}{}{0pt}{}{x}{2}\right)}{(c/h_n)}^x}{{(c;q)}_x{(q;q)}_x}}{\frac{{(-c/h_n;q)}_{\infty }}{{(c;q)}_{\infty }}},x=0,1,\dots ,\end{array}$$

(44)

and with

$${\mu }_q=\frac{c}{1-q}{h}_n^{-1}and{\sigma }_q^2=\frac{c^2}{q(1-q)}{h}_n^{-2}-\frac{c}{1-q}{h}_n^{-1}$$

where $0<c<1,$$h_n>0,$$n=0,1,2,\dots$ with $h_n=o(1)$.

2. A Generalized q-Negative Binomial Distribution: The q-CCVII for $b=q^n,n=1,2,\dots$ and $\eta =q$ becomes a generalized q-negative binomial distribution with p.f.

$$p(x)=P(X=x)=\frac{\frac{{\left(\genfrac{}{}{0pt}{}{n+x-1}{x}\right)}_q}{{(-q;q)}_x}{q}^{\left(\genfrac{}{}{0pt}{}{x}{2}\right)}{q}^{-nx+x}}{\frac{{(-q^{-n+1};q)}_{\infty }}{{(-q;q)}_{\infty }}},x=0,1,\dots \mathrm{with}{\left(\genfrac{}{}{0pt}{}{n}{x}\right)}_q=\frac{{\left[n\right]}_q!}{{\left[x\right]}_q!{[n-x]}_q!}$$

(45)

From the proposition 3 the mean value and the variance of the r.v. X of this q-negative binomial distribution are given respectively by

$$E(X)=\sum _{r=1}^{\infty }{a}_1^{II}{(r)}_1{\varphi }_1(q^{n+1};-q^{r+1};q,-q^{r+1-n})$$

(46)

where

$$a_1^{II}(r)=\frac{{(-q^{r+1};q)}_{\infty }}{{(-q^{1-n};q)}_{\infty }}\frac{|s_q(r,1)|{(q^n;q)}_r{q}^r}{(1-q)q^{nr}{\left[r\right]}_{1/q}!}$$

(47)

and

$$V(X)=E(X(X-1))+E(X)-E{(X)}^2$$

(48)

with

$$E(X(X-1))=\sum _{r=2}^{\infty }{a}_2^{II}{(r)}_1{\varphi }_1(q^{n+1};-q^{r+1};q,-q^{r+1-n})$$

(49)

where

$$a_2^{II}(r)=\frac{{(-q^{r+1};q)}_{\infty }}{{(-q^{1-n};q)}_{\infty }}\frac{2|s_q(r,2)|{(q^n;q)}_r{q}^r}{{(1-q)}^2{q}^{nr}{\left[r\right]}_{1/q}!}$$

(50)

By remark 1 the above q-negative binomial distribution has the Stieltjes-Wigert asymptotic behavior for $n\to \infty$ as in Equation (18) with ${\mu }_q$ and ${\sigma }_q^2$ given by proposition 1

$${\mu }_q=q^{-n+1}\frac{1-q^n}{1-q}and{\sigma }_q^2=q^{-2n+1}\frac{1-q^n}{q(1-q)}+q^{-n+1}\frac{1-q^n}{1-q}$$

3. The Generalized Over / Underdispersed (O/U) Distribution: The q-CCVII for $b=q^n$ and $\eta =\lambda q^n$ becomes a generalized O/U distribution with p.f.

$$p(x)=P(X=x)=\frac{{(-\lambda q^n;q)}_{\infty }}{{(-\lambda ;q)}_{\infty }}\frac{{(q^n;q)}_x{q}^{\left(\genfrac{}{}{0pt}{}{x}{2}\right)}{\lambda }^x}{{(-\lambda q;q)}_x{(q;q)}_x},x=0,1,\dots (\mathrm{see}\mathrm{kemp}[1])$$

(51)

From the proposition 3 the mean value and the variance of the r.v. X of the generalized O/U distribution are given respectively by

$$E(X)=\sum _{r=1}^{\infty }{a}_1^{II}{(r)}_1{\varphi }_1(q^{r+n};-\lambda q^{r+n};q,-\lambda q^r)$$

(52)

where

$$a_1^{II}(r)=\frac{{(-\lambda q^{r+n};q)}_{\infty }}{{(-\lambda ;q)}_{\infty }}\frac{|s_q(r,1)|{(q^n;q)}_r{\lambda }^r}{(1-q){\left[r\right]}_{1/q}!}$$

(53)

and

$$V(X)=E(X(X-1))+E(X)-E{(X)}^2$$

(54)

with

$$E(X(X-1))=\sum _{r=2}^{\infty }{a}_2^{II}{(r)}_1{\varphi }_1(q^{r+n};-\lambda q^{r+n};q,-\lambda q^r)$$

(55)

where

$$a_2^{II}(r)=\frac{{(-\lambda q^{r+n};q)}_{\infty }}{{(-\lambda ;q)}_{\infty }}\frac{2|s_q(r,2)|{(q^n;q)}_r{\lambda }^r}{{(1-q)}^2{\left[r\right]}_{1/q}!}$$

(56)

By remark 1 the generalized O/U distribution for $\lambda ={\lambda }_n\to \infty ,$ has the Stieltjes-Wigert asymptotic behavior for $n\to \infty$ as in Equation (18) with ${\mu }_q={\lambda }_n/(1-q)$ and ${\sigma }_q^2=\frac{{\lambda }_n^2}{q(1-q)}+\frac{{\lambda }_n}{1-q}$ given by proposition 1.

Remark 3. As it was noted in remark 1, theorem 1 is not sufficed for the confluent q-Chu-Vandermonde hypergeometric series discrete distribution III with p.f.

$$\begin{array}{c}\hfill p_{qCCVI}(x)=P[X=x]=\frac{\frac{{(q^{-n};q)}_x}{{(c;q)}_x{(q;q)}_x}{q}^{\left(\genfrac{}{}{0pt}{}{x}{2}\right)}{(-c{q}^n)}^x}{\frac{{(c{q}^n;q)}_{\infty }}{{(c;q)}_{\infty }}}=\frac{{(c;q)}_{\infty }}{{(c{q}^n;q)}_{\infty }}\frac{{\left(\genfrac{}{}{0pt}{}{n}{x}\right)}_q{q}^{2\left(\genfrac{}{}{0pt}{}{x}{2}\right)}{c}^x}{{(c;q)}_x{(q;q)}_x},x=0,1,\dots ,n\end{array}$$

(57)

where c constant. However, the discrete approximation of the above q-CCV-III distribution for $n\to \infty$, is given by

$$\begin{array}{c}\hfill p_{qCCVIII}(x)\cong {(c;q)}_{\infty }\frac{q^{2\left(\genfrac{}{}{0pt}{}{x}{2}\right)}{c}^x}{{(c;q)}_x{(q;q)}_x},x=0,1,\dots .\end{array}$$

(58)

Berg and Valent [12], have proved that for $q<a<1/q$, the above discrete probability measure Equation (58) has a continuous analogue counterpart family of absolutely continuous probability measures on $(0,\infty )$ defined by

$$v^{SC}(dx)=\frac{p}{\pi }{\{{(\frac{a}{a-1}\frac{{(x/a;q)}_{\infty }^2}{{(q/a;q)}_{\infty }})}^2+p^2{(\frac{{(x;q)}_{\infty }^2}{{(q;q)}_{\infty }{(qa;q)}_{\infty }})}^2\}}^{-1}dx$$

(59)

where the parameter $p>0$ is given by $p=\gamma /(t^2+{\gamma }^2)$ with ${\gamma }^2=-t(1/\psi (a)+t),$ where t belongs to the interval with endpoints 0 and $-1/\psi (a)$ and is given by $\psi (a)={(q;q)}_{\infty }{\sum }_{j=0}^{\infty }\frac{q^j}{(a-q^j){(q;q)}_j}$ with $\psi (q^{+})=\infty .$

Remark 4. Gould and Srivastava [13] have presented a unification of some combinatorial identities associated with ordinary Gauss’s summation theorem and their basic (or $q-$) extension associated with the $q-$ analogue Gauss’ s theorem. They have also shown a generalization of their unification for the ordinary case involving a bilateral series and have posed as an open problem the q-extension of their bilateral result. In our work it is considered a family of confluent q-Chu-Vnadermonde distributions which can be associated with the q-analogue of Gauss’s theorem and it would be an interesting closlely-related open problem to study a bilateral family of the considered distributions.

4. Concluding Remarks

In this article, we have provided a continuous limiting behavior of a family of confluent q -Chu- Vandermonde distributions, for $0<q<1$, in the sense of pointwise convergence, by applying a q-analogue of the usual Stirling asymptotic formula for the factorial number of order n. Specifically, we have designated as a main theorem the conditions under which the confluent q-Chu-Vandermonde discrete distributions q-CCVI and II converge to a continuous Stieltjes-Wigert distribution. Moreover, as applications for this study we present a modified q-Bessel distribution, a generalized q- negative Binomial distribution and a generalized over/underdispersed O/U distribution, converging to a continuous Stieltjes-Wigert distribution. Note that the main contribution of this article is that a discrete distribution congerges to a continuous one, which is not of a Gaussian type distribution.