A Class of Power Series q-Distributions

Abstract

A class of power series q-distributions, generated by considering a q-Taylor expansion of a parametric function into powers of the parameter, is discussed. Its q-factorial moments are obtained in terms of q-derivatives of its series (parametric) function. Also, it is shown that the convolution of power series q-distributions is also a power series q-distribution. Furthermore, the q-Poisson (Heine and Euler), q-binomial of the first kind, negative q-binomial of the second kind, and q-logarithmic distributions are shown to be members of this class of distributions and their q-factorial moments are deduced. In addition, the convolution properties of these distributions are examined.

1. Introduction

Benkherouf and Bather (1988) [1] derived the Heine and Euler distributions, which constitute the q-analogs of the Poisson distribution, as feasible priors in a simple Bayesian model for oil exploration. The probability (mass) function of the q-Poisson distributions is given by (Charalambides (2016), p. 107) [2]

$$\begin{array}{c}\hfill p_x(\lambda ;q)=E_q(-\lambda )\frac{{\lambda }^x}{{\left[x\right]}_q!},x=0,1,\dots ,\end{array}$$

(1)

where $0<\lambda <1/(1-q)$ and $0<q<1$ (Euler distribution) or $0<\lambda <\infty$ and $1<q<\infty$ (Heine distribution). Also, $E_q\left(t\right)={\prod }_{i=1}^{\infty }(1+t(1-q)q^{i-1})$ is a q-exponential function, satisfying the relation $E_q\left(t\right)E_{q^{-1}}(-t)=1$, where $\left|t\right|<b\left(q\right)$ and $\left|q\right|<1$ or $\left|q\right|>1$, with the bound $b\left(q\right)\le \infty$ depending on q. It should be noted that $e_q\left(t\right)={\prod }_{i=1}^{\infty }{(1-t(1-q)q^{i-1})}^{-1}$ is another q-exponential function, connected to the first one by $e_q\left(t\right)=E_{q^{-1}}\left(t\right)$.

Kemp and Kemp (1991) [3], in their study of Weldon’s classical dice data, introduced a q-binomial distribution. It is the distribution of the number of successes in a sequence of n independent Bernoulli trials, with the odds of success at a trial varying geometrically with the number of trials. Kemp and Newton (1990) [4] further studied it as a stationary distribution of a birth and death process. The probability function of this q-binomial distribution of the first kind is given by

$$\begin{array}{c}\hfill p_x(\theta ;q)={\left[\genfrac{}{}{0.0pt}{}{n}{x}\right]}_q\frac{{\theta }^x{q}^{\left(\genfrac{}{}{0pt}{}{x}{2}\right)}}{{\prod }_{i=1}^n(1+\theta q^{i-1})},x=0,1,\dots ,n,\end{array}$$

(2)

where $0<\theta <\infty$ and $0<q<1$ or $1<q<\infty$.

Charalambides (2010) [5] in his study of the q-Bernstein polynomials as a q-binomial distribution of the second kind, introduced the negative q-binomial distribution of the second kind. It is the distribution of the number of failures until the occurrence of the nth success in a sequence of independent Bernoulli trials, with the probability of success at a trial varying geometrically with the number of successes. The probability function of this negative q-binomial distribution of the second kind is given by

$$\begin{array}{c}\hfill p_x(\theta ;q)={\left[\genfrac{}{}{0.0pt}{}{n+x-1}{x}\right]}_q{\theta }^x\prod _{i=1}^n(1-\theta q^{i-1}),x=0,1,\dots ,\end{array}$$

(3)

where $0<\theta <1$ and $0<q<1$.

A q-logarithmic distribution was studied by C. D. Kemp (1997) [6] as a group size distribution. Its probability function is given by

$$\begin{array}{c}\hfill p_x(\theta ;q)={[-l_q(1-\theta )]}^{-1}\frac{{\theta }^x}{{\left[x\right]}_q},x=1,2,\dots ,\end{array}$$

(4)

where $0<\theta <1$, $0<q<1$, and

$$-l_q(1-\theta )=\underset{t\to 0}{lim}\frac{1}{{\left[t\right]}_q}\left(\prod _{i=1}^{\infty }\frac{1-\theta q^{t+i-1}}{1-\theta q^{i-1}}-1\right)=\sum _{j=1}^{\infty }\frac{{\theta }^j}{{\left[j\right]}_q}$$

is a q-logarithmic function.

A class of power series q-distributions, which is introduced in Section 2 by considering a q-Taylor expansion of a parametric function, provides a unified approach to the study of these distributions. Its q-factorial moments, for $0<q<1$ and $1<q<\infty$, are obtained in terms of q-derivatives of its series function. As essentially noted by Dunkl (1981) [7] and formally expressed in Charalambides and Papadatos (2005) [8] and Charalambides (2016) [2], the usual factorial (and binomial) moments are given in terms of the q-factorial (and q-binomial) moments through the q-Stirling numbers of the first kind. Moreover, it is proved that a power series q-distribution is completely determined from its first two q-cumulants (or q-moments). Also, the convolution of power series q-distributions, using probability-generating functions, is shown to be a power series q-distribution. Furthermore, in Section 3, demonstrating this approach, the q-factorial moments for $0<q<1$ and $1<q<\infty$ of the q-Poisson (Heine and Euler) distributions, q-binomial distribution of the first kind, negative q-binomial distribution of the second kind, and q-logarithmic distribution are obtained as members of this class of distributions. In addition, interesting and useful structural information about these distributions is obtained through their probability-generating functions.

2. Power Series q-Distributions

Consider a positive function $g\left(\theta \right)$ of a positive parameter $\theta$ and assume that it is analytic with a q-Taylor expansion about zero (Jackson (1909) [9], Ernst (2012) [10], p. 103)

$$\begin{array}{c}\hfill g\left(\theta \right)=\sum _{x=0}^{\infty }{a}_{x,q}{\theta }^x,0<\theta <\rho ,\rho >0,\end{array}$$

(5)

where the coefficient

$$\begin{array}{c}\hfill a_{x,q}=\frac{1}{{\left[x\right]}_q!}{\left[D_q^{x}g\left(t\right)\right]}_{t=0}\ge 0,x=0,1,\dots ,0<q<1,\mathrm{or}1<q<\infty ,\end{array}$$

(6)

with $D_q=d_q/d_{q}t$ the q-derivative operator (Ernst (2012) [10], p. 200),

$$D_{q}g\left(t\right)=\frac{d_{q}g\left(t\right)}{d_{q}t}=\left\{\begin{array}{cc}\frac{{\displaystyle g\left(t\right)-g\left(qt\right)}}{{\displaystyle (1-q)t}},\hfill & q\ne 1,t\ne 0\hfill \\ Dg\left(t\right)=\frac{{\displaystyle dg\left(t\right)}}{{\displaystyle dt}},\hfill & q=1,\hfill \\ Dg\left(0\right)=\frac{{\displaystyle dg\left(0\right)}}{{\displaystyle dt}},\hfill & t=0,\hfill \end{array}\right.$$

does not involve the parameter $\theta$. Clearly, the function

$$\begin{array}{c}\hfill p_x(\theta ;q)=\frac{a_{x,q}{\theta }^x}{g\left(\theta \right)},x=0,1,\dots ,\end{array}$$

(7)

with $0<\theta <\rho$ and $0<q<1$ or $1<q<\infty$, satisfies the properties of a probability (mass) function.

Definition 1. 

A family of discrete q-distributions $p_x(\theta ;q)$, $\theta \in \Theta$, $q\in Q$, is said to be a class of power series q-distributions, with parameters θ and q and series function $g\left(\theta \right)$ if it has the representation (7), with series function satisfying condition (5).

Remark 1. 

The q-Taylor expansion (5) may be equivalently expressed as expressed as Jackson (1942) [11], Ernst (2012) [10], p. 103

$$\begin{array}{c}\hfill g\left(\theta \right)=\sum _{x=0}^{\infty }{b}_{x,q}{\theta }^x,0<\theta <\rho ,\rho >0,\end{array}$$

where the coefficient

$$\begin{array}{c}\hfill b_{x,q}=\frac{q^{\left(\genfrac{}{}{0pt}{}{x}{2}\right)}}{{\left[x\right]}_q!}{\left[D_q^{x}g\left(qt\right)\right]}_{t=0}\ge 0,x=0,1,\dots ,0<q<1,or1<q<\infty ,\end{array}$$

and does not involve the parameter θ. Indeed, replacing q by $q^{-1}$ in (5) and (6) and since ${\left[x\right]}_{q^{-1}}!=q^{-\left(\genfrac{}{}{0pt}{}{x}{2}\right)}{\left[x\right]}_q!$ and ${\left[D_{q^{-1}}^{x}g\left(t\right)\right]}_{t=0}={\left[D_q^{x}g\left(qt\right)\right]}_{t=0}$, the equivalent expression is readily deduced.

Remark 2. 

The class of power series q-distributions, for $q\to 1$, reduces to the class of (usual) power series distributions, which was introduced by Noack (1950) [12] and further studied by Khatri (1959) [13] and Patil (1962) [14]. Furthermore, it should be noted that the range of x in (7), as in the case of the power series distributions, need not be the entire set of nonnegative integers; it can be an arbitrary non-null subset of nonnegative integers. Also, note that a truncated version of a power series q-distribution is itself a power series q-distribution in its own right; hence, the properties that hold for a power series q-distribution continue to hold for its truncated forms.

The basic properties of a power series q-distribution are established in the following propositions. Its q-factorial moments are derived first, in terms of the q-derivatives of the series function.

Proposition 1. 

The mth q-factorial moment of the power series q-distribution (7) is given by

$$\begin{array}{c}\hfill E\left({\left[X\right]}_{m,q}\right)=\frac{{\theta }^m}{g\left(\theta \right)}·\frac{d_q^{m}g\left(\theta \right)}{d_q{\theta }^m},m=1,2,\dots .\end{array}$$

(8)

In particular, the q-mean and q-variance are given by

$$\begin{array}{c}\hfill E\left({\left[X\right]}_q\right)=\frac{\theta }{g\left(\theta \right)}·\frac{d_{q}g\left(\theta \right)}{d_q\theta }\end{array}$$

(9)

and

$$\begin{array}{c}\hfill V\left({\left[X\right]}_q\right)=\frac{{\theta }^{2}q}{g\left(\theta \right)}·\frac{d_q^{2}g\left(\theta \right)}{d_q{\theta }^2}-\frac{\theta }{g\left(\theta \right)}·\frac{d_{q}g\left(\theta \right)}{d_q\theta }\left(\frac{\theta }{g\left(\theta \right)}·\frac{d_{q}g\left(\theta \right)}{d_q\theta }-1\right),\end{array}$$

(10)

respectively.

Proof. 

The mth q-factorial moment,

$$E\left({\left[X\right]}_{m,q}\right)=\sum _{x=m}^{\infty }{\left[x\right]}_{m,q}\frac{a_{x,q}{\theta }^x}{g\left(\theta \right)}=\frac{{\theta }^m}{g\left(\theta \right)}\sum _{x=m}^{\infty }{a}_{x,q}{\left[x\right]}_{m,q}{\theta }^{x-m},$$

on using the mth q-derivative of the series function (5),

$$\frac{d_q^{m}g\left(\theta \right)}{d_q{\theta }^m}=\sum _{x=m}^{\infty }{a}_{x,q}{\left[x\right]}_{m,q}{\theta }^{x-m},$$

is readily deduced as (8). In particular, for $m=1$, the q-mean is given by (9). Also, using the expression (Charalambides (2016), p. 43)

$$V\left({\left[X\right]}_q\right)=qE\left({\left[X\right]}_{2,q}\right)-E\left({\left[X\right]}_q\right)\left(E\left({\left[X\right]}_q\right)-1\right),$$

the q-variance is obtained in the form (10). □

The derivation of the $q^{-1}$-factorial moments, $E\left({\left[X\right]}_{m,q^{-1}}\right)$, $m=1,2,\dots$, of several power series q-distributions, in addition to their own interest, are shown to be useful in the study of limiting distributions (Kyriakoussis and Vamvakari (2013) [15] and Charalambides (2016) [2], chapter 4). These moments are given, in terms of the $q^{-1}$-derivatives of the series function, by (8) with q replaced by $q^{-1}$. An alternative expression, in terms of the q-derivatives of the series function, is obtained in the next proposition.

Proposition 2. 

The mth $q^{-1}$-factorial moment of the power series q-distribution (7) is given by

$$\begin{array}{c}\hfill E\left({\left[X\right]}_{m,q^{-1}}\right)=\frac{{\theta }^m{q}^{\left(\genfrac{}{}{0pt}{}{m+1}{2}\right)}}{g\left(\theta \right)}\frac{d_q^{m}g\left(q^{-m}\theta \right)}{d_q{\theta }^m},m=1,2,\dots .\end{array}$$

(11)

In particular, the $q^{-1}$-mean and $q^{-1}$-variance are given by

$$\begin{array}{c}\hfill E\left({\left[X\right]}_{q^{-1}}\right)=\frac{\theta q}{g\left(\theta \right)}·\frac{d_{q}g\left(q^{-1}\theta \right)}{d_q\theta }\end{array}$$

(12)

and

$$\begin{array}{c}\hfill V\left({\left[X\right]}_{q^{-1}}\right)=\frac{{\theta }^2{q}^2}{g\left(\theta \right)}·\frac{d_q^{2}g\left(q^{-2}\theta \right)}{d_q{\theta }^2}-\frac{\theta q}{g\left(\theta \right)}·\frac{d_{q}g\left(q^{-1}\theta \right)}{d_q\theta }\left(\frac{\theta q}{g\left(\theta \right)}·\frac{d_{q}g\left(q^{-1}\theta \right)}{d_q\theta }-1\right),\end{array}$$

(13)

respectively.

Proof. 

The mth $q^{-1}$-factorial moment, since ${\left[x\right]}_{m,q^{-1}}=q^{\left(\genfrac{}{}{0pt}{}{m+1}{2}\right)-mx}{\left[x\right]}_{m,q}$, may be expressed as

$$E\left({\left[X\right]}_{m,q^{-1}}\right)=\sum _{x=m}^{\infty }{\left[x\right]}_{m,q^{-1}}\frac{a_{x,q}{\theta }^x}{g\left(\theta \right)}=\frac{{\theta }^m{q}^{\left(\genfrac{}{}{0pt}{}{m+1}{2}\right)}}{g\left(\theta \right)}\sum _{x=m}^{\infty }{a}_{x,q}{\left[x\right]}_{m,q}{q}^{-mx}{\theta }^{x-m}.$$

Also, the mth q-derivative of the series function $g\left(q^{-m}\theta \right)$, with respect to $\theta$, can be written as

$$\frac{d_q^{m}g\left(q^{-m}\theta \right)}{d_q{\theta }^m}=q^{-m^2}{\left[\frac{d_q^{m}g\left(u\right)}{d_q{u}^m}\right]}_{u=q^{-m}\theta }=\sum _{x=m}^{\infty }{a}_{x,q}{\left[x\right]}_{m,q}{q}^{-mx}{\theta }^{x-m}.$$

Introducing it into the last expression of the mth $q^{-1}$-factorial moment, (11) is obtained. In particular, for $m=1$, the $q^{-1}$-mean is given by (12). Also, using the expression

$$V\left({\left[X\right]}_{q^{-1}}\right)=q^{-1}E\left({\left[X\right]}_{2,q^{-1}}\right)-E\left({\left[X\right]}_{q^{-1}}\right)\left(E\left({\left[X\right]}_{q^{-1}}\right)-1\right),$$

the $q^{-1}$-variance is obtained in the form (13). □

The convolution of power series q-distributions is also a power q-series distribution, according to the following proposition.

Proposition 3. 

(a) The probability generating function $P\left(t\right)=E\left(t^X\right)$ of the power series q-distribution (7) is given, in terms of the series function (5), by

$$\begin{array}{c}\hfill P\left(t\right)=\frac{g\left(\theta t\right)}{g\left(\theta \right)},\left|t\right|<\rho /\theta .\end{array}$$

(14)

(b) Suppose that $X_j$, $j=1,2,\dots ,n$, is a sequence of n independent random variables obeying a power series q-distribution, with series function $g_j\left(\theta \right)$, $j=1,2,\dots ,n$. Then, the sum $S_n={\sum }_{j=1}^n{X}_j$ obeys a power series q-distribution, with series function

$$\begin{array}{c}\hfill g\left(\theta \right)=\prod _{j=1}^n{g}_j\left(\theta \right).\end{array}$$

(15)

Proof. 

(a) The probability generating function $P\left(t\right)={\sum }_{x=0}^{\infty }{p}_x(\theta ;q)t^x$, on using (5) and (7), is readily obtained as (14).

(b) The probability generating function $P_{S_n}\left(t\right)$, of the sum $S_n={\sum }_{j=1}^n{X}_j$, is the product of the generating functions $P_{X_j}\left(t\right)$, $j=1,2,\dots ,n$, of the summands, $P_{S_n}\left(t\right)={\prod }_{j=1}^n{P}_{X_j}\left(t\right)$, and so by (14) is deduced in the form

$$P_{S_n}\left(t\right)=\prod _{j=1}^n\frac{g_j\left(\theta t\right)}{g_j\left(\theta \right)}=\frac{{\prod }_{j=1}^n{g}_j\left(\theta t\right)}{{\prod }_{j=1}^n{g}_j\left(\theta \right)}.$$

Using again (14), the last expression implies (15). □

The second part of Proposition 3 can be directly extended to an infinite series of random variables according to the following corollary.

Corollary 1. 

Suppose that $X_j$, $j=1,2,\dots$, is a sequence of independent random variables obeying a power series q-distribution, with series function $g_j\left(\theta \right)$, $j=1,2,\dots$. Then, the sum $S={\sum }_{j=1}^{\infty }{X}_j$ obeys a power series q-distribution, with series function

$$\begin{array}{c}\hfill g\left(\theta \right)=\prod _{j=1}^{\infty }{g}_j\left(\theta \right),\end{array}$$

(16)

provided ${\prod }_{j=1}^{\infty }{g}_j\left(\theta \right)<\infty$.

3. Particular Power Series q-Distributions

Particular power series q-distributions, which are obtained by specifying the series function, are discussed; their q-factorial moments are deduced and convolution properties are examined.

3.1. q-Poisson Distributions

The q-Poisson distributions, with probability function (1), belong in the family of power series q-distributions, with series function $g\left(\lambda \right)=e_q\left(\lambda \right)=1/E_q(-\lambda )$, where $0<\lambda <1/(1-q)$ and $0<q<1$ (Euler distribution) or $0<\lambda <\infty$ and $1<q<\infty$ (Heine distribution). Indeed, since $D_q{e}_q\left(t\right)=e_q\left(t\right)$ and $e_q\left(0\right)=1$, it follows from (6) that

$$a_{x,q}=\frac{1}{{\left[x\right]}_q!}{\left[D_q^x{e}_q\left(t\right)\right]}_{t=0}=\frac{1}{{\left[x\right]}_q!},x=0,1,\dots ,$$

and the probability function (7) reduces to (1).

The q-factorial moments, by (8) and since $d_q^m{e}_q\left(\lambda \right)/d_q{\lambda }^m=e_q\left(\lambda \right)$, are readily deduced as

$$E\left({\left[X\right]}_{m,q}\right)={\lambda }^m,m=1,2,\dots ,$$

where $0<\lambda <1/(1-q)$ and $0<q<1$ (Euler distribution) or $0<\lambda <\infty$ and $1<q<\infty$ (Heine distribution). In particular, the q-mean is given by

$$E\left({\left[X\right]}_q\right)=\lambda .$$

Furthermore, using (10), the q-variance is obtained as

$$V\left({\left[X\right]}_q\right)=q{\lambda }^2-\lambda (\lambda -1)=\lambda (1+(q-1)\lambda ).$$

The $q^{-1}$-factorial moments, by (11) and since

$$\frac{d_q^m{e}_q\left(q^{-m}\lambda \right)}{d_q{\lambda }^m}=q^{-m^2}{e}_q\left(q^{-m}\lambda \right),$$

are obtained as

$$E\left({\left[X\right]}_{m,q^{-1}}\right)=\frac{{\lambda }^m{q}^{\left(\genfrac{}{}{0pt}{}{m+1}{2}\right)}}{e_q\left(\lambda \right)}·q^{-m^2}{e}_q\left(q^{-m}\lambda \right)=\frac{{\lambda }^m{q}^{-\left(\genfrac{}{}{0pt}{}{m}{2}\right)}{\prod }_{i=1}^{\infty }\left(1-\lambda (1-q)q^{i-1}\right)}{{\prod }_{i=1}^{\infty }\left(1-\lambda (1-q)q^{-m+i-1}\right)},$$

which on using

$$\prod _{i=1}^{\infty }\left(1-\lambda (1-q)q^{-m+i-1}\right)=\prod _{j=1}^m\left(1+\lambda (1-q^{-1})q^{-(j-1)}\right)\prod _{i=1}^{\infty }\left(1-\lambda (1-q)q^{i-1}\right),$$

reduces to

$$E\left({\left[X\right]}_{m,q^{-1}}\right)=\frac{{\lambda }^m{q}^{-\left(\genfrac{}{}{0pt}{}{m}{2}\right)}}{{\prod }_{j=1}^m\left(1+\lambda (1-q^{-1})q^{-(j-1)}\right)},m=1,2,\dots ,$$

where $0<\lambda <1/(1-q)$ and $0<q<1$ (Euler distribution) or $0<\lambda <\infty$ and $1<q<\infty$ (Heine distribution). In particular, the $q^{-1}$-mean is

$$E\left({\left[X\right]}_{q^{-1}}\right)=\frac{\lambda }{1+\lambda (1-q^{-1})}.$$

Also, by (13), the $q^{-1}$-variance is obtained as

$$\begin{array}{cc}\hfill V\left({\left[X\right]}_{q^{-1}}\right)& =\frac{{\lambda }^2{q}^{-2}}{\left(1+\lambda (1-q^{-1})\right)\left(1+\lambda (1-q^{-1})q^{-1})\right)}+\frac{\lambda +{\lambda }^2(1-q^{-1})-{\lambda }^2}{{\left(1+\lambda (1-q^{-1})\right)}^2}\hfill \\ \hfill & =\frac{{\lambda }^2{q}^{-2}+{\lambda }^3{q}^{-2}-{\lambda }^3{q}^{-3}+\lambda -{\lambda }^2{q}^{-1}+{\lambda }^2{q}^{-2}-{\lambda }^2{q}^{-2}-{\lambda }^3{q}^{-2}+{\lambda }^3{q}^{-3}}{{\left(1+\lambda (1-q^{-1})\right)}^2\left(1+\lambda (1-q^{-1})q^{-1})\right)},\hfill \end{array}$$

which reduces to

$$V\left({\left[X\right]}_{q^{-1}}\right)=\frac{\lambda }{{\left(1+\lambda (1-q^{-1})\right)}^2\left(1+\lambda (1-q^{-1})q^{-1})\right)}.$$

A characterization of a q-Poisson distribution through a relation between the first two q-factorial moments is worth mentioning. Clearly,

$$E\left({\left[X\right]}_{2,q}\right)={\left(E\left({\left[X\right]}_q\right)\right)}^2,$$

for $0<\lambda <1/(1-q)$ and $0<q<1$ (Euler distribution) or $0<\lambda <\infty$ and $1<q<\infty$ (Heine distribution). Charalambides and Papadatos (2005) [8] showed that a family of nonnegative integer-valued random variables $X_{\lambda }$, for $0<\lambda <\rho$ and $0<q<1$, with a power series q-distribution, obeys a Euler distribution, if and only if

$$E\left({\left[X_{\lambda }\right]}_{2,q}\right)={\left(E\left({\left[X_{\lambda }\right]}_q\right)\right)}^2,$$

for $0<\lambda <\rho$ and $0<q<1$. Without any change in the proof, the last relation holds true for $0<\lambda <\rho$ and $0<q<1$ or $1<q<\infty$ if and only if the probability function of $X_{\lambda }$ is given by

$$p_x(a\lambda ;q)=E_q(-a\lambda )\frac{{\left(a\lambda \right)}^x}{{\left[x\right]}_q!},x=0,1,\dots ,$$

where $0<a\lambda <1/(1-q)$ and $0<q<1$ or $0<a\lambda <\infty$ and $1<q<\infty$, with $a>0$ an arbitrary constant. The additional characterization provided by this extension may be rephrased as follows. A family of nonnegative integer-valued random variables $X_{\lambda }$, for $0<\lambda <\rho$ and $0<q<1$, with a power series q-distribution, obeys a Heine distribution, if and only if

$$E\left({\left[X_{\lambda }\right]}_{2,q^{-1}}\right)={\left(E\left({\left[X_{\lambda }\right]}_{q^{-1}}\right)\right)}^2,$$

for $0<\lambda <\rho$ and $0<q<1$.

A close examination of the probability generating function of a q-Poisson distribution reveals interesting and useful structural information about the probability distribution. Specifically, from expression (14), with series function $g\left(\lambda \right)=E_q\left(\lambda \right)={\prod }_{i=1}^{\infty }\left(1+\lambda (1-q)q^{i-1}\right)$, and setting $\theta =\lambda (1-q)$, the probability generating function of the Heine distribution is deduced as

$$P\left(t\right)=\frac{E_q\left(\lambda t\right)}{E_q\left(\lambda \right)}=\prod _{i=1}^{\infty }\frac{1+\theta t{q}^{i-1}}{1+\theta q^{j-1}},-\infty <t<\infty ,0<\theta <\infty ,0<q<1,$$

where $P_{X_i}\left(t\right)=\left(1+\theta t{q}^{i-1}\right)/\left(1+\theta q^{j-1}\right)$ is the probability generating function of a Bernoulli distribution. Therefore, according to Corollary 1, the Heine distribution may be expressed as an infinite convolution of independent (and not identically distributed) Bernoulli distributions. This representation of the Heine distribution was first noticed by Benkherouf and Bather (1988).

Also, from (14), with $g\left(\lambda \right)=e_q\left(\lambda \right)={\prod }_{j=1}^{\infty }{\left(1-\lambda (1-q)q^{j-1}\right)}^{-1}$, and setting $\theta =\lambda (1-q)$, the probability generating function of the Euler distribution is obtained as

$$P\left(t\right)=\frac{e_q\left(\lambda t\right)}{e_q\left(\lambda \right)}=\prod _{j=1}^{\infty }\frac{1-\theta q^{j-1}}{1-\theta t{q}^{j-1}},\left|t\right|<1/\theta ,0<\theta <1,0<q<1,$$

where $P_{X_j}\left(t\right)=\left(1-\theta q^{j-1}\right)/(1-\theta t{q}^{j-1})$ is the probability generating function of a geometric distribution. Therefore, according to Corollary 1, the Euler distribution may be expressed as an infinite convolution of independent (and not identically distributed) geometric distributions. It should be noted that this expression of the Euler distribution was derived by Kemp (1992) [16].

3.2. q-Binomial Distribution of the First Kind

The q-binomial distribution of the first kind, with probability function (2), is a power series q-distribution, with series function $g\left(\theta \right)={\prod }_{i=1}^n(1+\theta q^{i-1})$, where $0<\theta <\infty$ and $0<q<1$ or $1<q<\infty$. Indeed, since

$$\begin{array}{cc}\hfill D_{q}g\left(\theta \right)& =\frac{{\prod }_{i=1}^n(1+\theta q^{i-1})-{\prod }_{i=1}^n(1+\theta q^i)}{(1-q)\theta }\hfill \\ \hfill & =\frac{[(1+\theta )-(1+\theta q^n)]{\prod }_{i=1}^{n-1}(1+\theta q^i)}{(1-q)\theta }={\left[n\right]}_q\prod _{i=1}^{n-1}(1+\left(\theta q\right)q^{i-1}),\hfill \end{array}$$

it follows successively that

$$D_q^{x}g\left(\theta \right)={\left[n\right]}_{x,q}{q}^{1+2+\cdots +(x-1)}\prod _{i=1}^{n-x}(1+\left(\theta q^x\right)q^{i-1})={\left[n\right]}_{x,q}{q}^{\left(\genfrac{}{}{0pt}{}{x}{2}\right)}\prod _{i=1}^{n-x}(1+\left(\theta q^x\right)q^{i-1}),$$

for $x=1,2,\dots ,n$, and, by (6), that

$$a_{x,q}=\frac{1}{{\left[x\right]}_q!}{\left[D_q^{x}g\left(t\right)\right]}_{t=0}={\left[\genfrac{}{}{0.0pt}{}{n}{x}\right]}_q{q}^{\left(\genfrac{}{}{0pt}{}{x}{2}\right)},x=0,1,\dots ,n,$$

and the probability function (7) reduces to (2).

The q-factorial moments, by (8) and since

$$\frac{d_q^{m}g\left(\theta \right)}{d_q{\theta }^m}={\left[n\right]}_{m,q}{q}^{\left(\genfrac{}{}{0pt}{}{m}{2}\right)}\prod _{i=1}^{n-m}(1+\left(\theta q^m\right)q^{i-1})={\left[n\right]}_{m,q}{q}^{\left(\genfrac{}{}{0pt}{}{m}{2}\right)}\prod _{i=m+1}^n(1+\theta q^{i-1}),$$

are obtained as

$$E\left({\left[X\right]}_{m,q}\right)=\frac{{\left[n\right]}_{m,q}{\theta }^m{q}^{\left(\genfrac{}{}{0pt}{}{m}{2}\right)}}{{\prod }_{i=1}^m(1+\theta q^{i-1})},m=1,2,\dots ,$$

where $0<\theta <\infty$ and $0<q<1$ or $1<q<\infty$. In particular, the q-mean is

$$E\left({\left[X\right]}_q\right)=\frac{{\left[n\right]}_q\theta }{(1+\theta )}.$$

Also, by (10), the q-variance is obtained as

$$V\left({\left[X\right]}_q\right)=\frac{{\left[n\right]}_q{[n-1]}_q{\theta }^2{q}^2}{(1+\theta )(1+\theta q)}+\frac{{\left[n\right]}_q\theta }{1+\theta }\left(1-\frac{{\left[n\right]}_q\theta }{1+\theta }\right),$$

which, on using the expression $q{[n-1]}_q={\left[n\right]}_q-1$, reduces to

$$V\left({\left[X\right]}_q\right)=\frac{{\left[n\right]}_q\theta }{(1+\theta )(1+\theta q)}\left(1+\frac{{\left[n\right]}_q\theta (q-1)}{1+\theta }\right).$$

The $q^{-1}$-factorial moments, on using (11) with

$$\frac{d_q^{m}g\left(q^{-m}\theta \right)}{d_q{\theta }^m}={\left[n\right]}_{m,q}{q}^{\left(\genfrac{}{}{0pt}{}{m}{2}\right)-m^2}\prod _{i=1}^{n-m}(1+\theta q^{i-1})={\left[n\right]}_{m,q}{q}^{-\left(\genfrac{}{}{0pt}{}{m+1}{2}\right)}\prod _{i=1}^{n-m}(1+\theta q^{i-1}),$$

and since

$$\begin{array}{cc}\hfill \prod _{i=1}^n(1+\theta q^{i-1})& =\prod _{i=1}^{n-m}(1+\theta q^{i-1})\prod _{i=n-m+1}^n(1+\theta q^{i-1})\hfill \\ \hfill & =\prod _{i=1}^{n-m}(1+\theta q^{i-1})\prod _{i=1}^n(1+\theta q^{n-m+i-1}),\hfill \end{array}$$

are obtained as

$$E\left({\left[X\right]}_{m,q^{-1}}\right)=\frac{{\left[n\right]}_{m,q}{\theta }^m}{{\prod }_{i=1}^m(1+\theta q^{n-m+i-1})},m=1,2,\dots ,$$

where $0<\theta <\infty$ and $0<q<1$ or $1<q<\infty$. In particular, the $q^{-1}$-mean is

$$E\left({\left[X\right]}_{q^{-1}}\right)=\frac{{\left[n\right]}_q\theta }{1+\theta q^{n-1}}.$$

Also, by (13) and using the expression ${[n-1]}_q={\left[n\right]}_q-q^{n-1}$, the $q^{-1}$-variance is obtained as

$$\begin{array}{cc}\hfill V\left({\left[X\right]}_{q^{-1}}\right)=& \frac{{\left[n\right]}_q\theta }{1+\theta q^{n-1}}-\frac{{\left[n\right]}_q{\theta }^2{q}^{n-2}}{(1+\theta q^{n-1})(1+\theta q^{n-2})}\hfill \\ \hfill & +\frac{{\left[n\right]}_q^2{\theta }^2{q}^{-1}}{(1+\theta q^{n-1})(1+\theta q^{n-2})}-\frac{{\left[n\right]}_q^2{\theta }^2}{{(1+\theta q^{n-1})}^2},\hfill \end{array}$$

which after some algebra reduces to

$$V\left({\left[X\right]}_{q^{-1}}\right)=\frac{{\left[n\right]}_q\theta }{(1+\theta q^{n-1})(1+\theta q^{n-2})}\left(1+\frac{{\left[n\right]}_q\theta (q^{-1}-1)}{1+\theta q^{n-1}}\right).$$

The probability generating function of the q-binomial distribution of the first kind, on using (14), is deduced as

$$P\left(t\right)=\prod _{i=1}^n\frac{1+\theta t{q}^{i-1}}{1+\theta q^{i-1}},\left|t\right|<\infty ,0<\theta <\infty ,0<q<1\mathrm{or}1<q<\infty ,$$

where $P_{X_i}\left(t\right)=\left(1+\theta t{q}^{i-1}\right)/\left(1+\theta q^{j-1}\right)$ is the probability generating function of a Bernoulli distribution. Therefore, according to Proposition 3(b), the q-binomial distribution of the first kind, may be expressed as a convolution of n independent (and not identically distributed) Bernoulli distributions.

More generally, the q-binomial distribution of the first kind may be expressed as a convolution of n independent q-binomial distributions of the first kind. Specifically, let $X_j$, $j=1,2,\dots ,n$, be a sequence of n independent random variables and assume that $X_j$ follows a q-binomial distribution of the first kind with parameters $r_j$, $\theta q^{s_{j-1}}$, and q, where $s_j={\sum }_{i=1}^j{r}_i$, for $j=1,2,\dots ,n$ and $s_0=0$. Clearly, the probability-generating function of $X_j$ is given by

$$P_{X_j}\left(t\right)=\prod _{i=1}^{r_j}\frac{1+\theta t{q}^{s_{j-1}+i-1}}{1+\theta q^{s_{j-1}+i-1}},\left|t\right|<\infty ,0<\theta <\infty ,0<q<1\mathrm{or}1<q<\infty .$$

Consequently, the probability-generating function of the sum $S_n={\sum }_{j=1}^n{X}_j$, is deduced as

$$P_{S_n}\left(t\right)=\prod _{j=1}^n\prod _{i=1}^{r_j}\frac{1+\theta t{q}^{s_{j-1}+i-1}}{1+\theta q^{s_{j-1}+i-1}}=\prod _{j=1}^n\prod _{i=s_{j-1}+1}^{s_j}\frac{1+\theta t{q}^{i-1}}{1+\theta q^{i-1}},$$

which, for $s_n\equiv m$, simplifies to

$$P_{S_n}\left(t\right)=\prod _{i=1}^m\frac{1+\theta t{q}^{i-1}}{1+\theta q^{i-1}},\left|t\right|<\infty ,0<\theta <\infty ,0<q<1\mathrm{or}1<q<\infty .$$

Therefore, the distribution of $S_n$ is a q-binomial distribution of the first kind with parameters m, $\theta$, and q.

Finally, it is worth noticing that the probability-generating function of the Heine distribution, with parameters $\lambda$ and q,

$$P_X\left(t\right)=\prod _{i=1}^{\infty }\frac{1+\theta t{q}^{i-1}}{1+\theta q^{j-1}},-\infty <t<\infty ,0<\theta <\infty ,0<q<1,$$

where $\theta =\lambda (1-q)$, may be expressed as product, $P_X\left(t\right)=P_{X_n}\left(t\right)P_{Y_n}\left(t\right)$, of the probability generating function of the q-binomial distribution of the first kind, with parameters n, $\theta$, and q,

$$P_{X_n}\left(t\right)=\prod _{i=1}^n\frac{1+\theta t{q}^{i-1}}{1+\theta q^{j-1}},-\infty <t<\infty ,0<\theta <\infty ,0<q<1,$$

and the probability generating function of the Heine distribution, with parameters $\lambda q^n$ and q,

$$P_{Y_n}\left(t\right)=\prod _{i=1}^{\infty }\frac{1+\theta t{q}^{n+i-1}}{1+\theta q^{n+j-1}},-\infty <t<\infty ,0<\theta <\infty ,0<q<1.$$

Therefore, a Heine distribution may be expressed as a convolution of a q-binomial distribution of the first kind and an independent Heine distribution.

3.3. Negative q-Binomial Distribution of the Second Kind

The negative q-binomial distribution of the second kind with probability function (3) is a power series q-distribution, with series function $g\left(\theta \right)={\prod }_{i=1}^n{(1-\theta q^{i-1})}^{-1}$, where $0<\theta <1$ and $0<q<1$. Indeed, since

$$\begin{array}{cc}\hfill D_{q}g\left(\theta \right)& =\frac{{\prod }_{i=1}^n{(1-\theta q^{i-1})}^{-1}-{\prod }_{i=1}^n{(1-\theta q^i)}^{-1}}{(1-q)\theta }\hfill \\ \hfill & =\frac{[(1-\theta q^n)-(1-\theta )]{\prod }_{i=1}^{n+1}(1-\theta q^{i-1})}{(1-q)\theta }={\left[n\right]}_q\prod _{i=1}^{n+1}(1-\theta q^{i-1}),\hfill \end{array}$$

it follows successively that

$$D_q^{x}g\left(\theta \right)={\left[n\right]}_q{[n+1]}_q\cdots {[n+x-1]}_q\prod _{i=1}^{n+x}(1-\theta q^{i-1})={[n+x-1]}_{x,q}\prod _{i=1}^{n+x}(1-\theta q^{i-1}),$$

for $x=1,2,\dots$, and, by (6), that

$$a_{x,q}=\frac{1}{{\left[x\right]}_q!}{\left[D_q^{x}g\left(t\right)\right]}_{t=0}={\left[\genfrac{}{}{0.0pt}{}{n+x-1}{x}\right]}_q,x=0,1,\dots ,$$

and the probability function (7) reduces to (3).

The q-factorial moments, by (8) and since

$$\begin{array}{cc}\hfill D_q^{m}g\left(\theta \right)& ={[n+m-1]}_{m,q}\prod _{i=1}^{n+m}{(1-\theta q^{i-1})}^{-1}\hfill \\ \hfill & ={[n+m-1]}_{m,q}\prod _{i=1}^n{(1-\theta q^{i-1})}^{-1}\prod _{i=1}^m{(1-\theta q^{n+i-1})}^{-1},\hfill \end{array}$$

are obtained as

$$E\left({\left[X\right]}_{m,q}\right)=\frac{{[n+m-1]}_{m,q}{\theta }^m}{{\prod }_{i=1}^m(1-\theta q^{n+i-1})},m=1,2,\dots ,$$

where $0<\theta <1$ and $0<q<1$. In particular, the q-expected value is

$$E\left({\left[X\right]}_q\right)=\frac{{\left[n\right]}_q\theta }{1-\theta q^n}.$$

Also, by (10), the q-variance is obtained as

$$V\left({\left[X\right]}_q\right)=\frac{{\left[n\right]}_q{[n+1]}_q{\theta }^{2}q}{(1-\theta q^n)(1-\theta q^{n+1})}+\frac{{\left[n\right]}_q\theta }{1-\theta q^n}\left(1-\frac{{\left[n\right]}_q\theta }{1-\theta q^n}\right)$$

which, on using the expression ${[n+1]}_q={\left[n\right]}_q+q^n$, reduces to

$$V\left({\left[X\right]}_q\right)=\frac{{\left[n\right]}_q\theta }{(1-\theta q^n)(1-\theta q^{n+1})}\left(1+\frac{{\left[n\right]}_q\theta (q-1)}{1-\theta q^n}\right).$$

The $q^{-1}$-factorial moments, on using (11) with

$$\frac{d_q^{m}g\left(q^{-m}\theta \right)}{d_q{\theta }^m}=q^{-m^2}{[n+m-1]}_{m,q}\prod _{i=1}^{n+m}{(1-\theta q^{-m+i-1})}^{-1},$$

and since

$$\begin{array}{cc}\hfill \prod _{i=1}^{n+m}{(1-\theta q^{-m+i-1})}^{-1}& =\prod _{i=1}^m{(1-\theta q^{-m+i-1})}^{-1}\prod _{i=m+1}^{m+n}{(1-\theta q^{-m+i-1})}^{-1}\hfill \\ \hfill & =\prod _{j=1}^m{(1-\theta q^{-j})}^{-1}\prod _{j=1}^n{(1-\theta q^{j-1})}^{-1},\hfill \end{array}$$

are obtained as

$$E\left({\left[X\right]}_{m,q^{-1}}\right)=\frac{{[n+m-1]}_{m,q}{\theta }^m{q}^{-\left(\genfrac{}{}{0pt}{}{m}{2}\right)}}{{\prod }_{i=1}^m(1-\theta q^{-j})},m=1,2,\dots ,$$

where $0<\theta <1$ and $0<q<1$. In particular, the $q^{-1}$-mean is

$$E\left({\left[X\right]}_{q^{-1}}\right)=\frac{{\left[n\right]}_q\theta }{1-\theta q^{-1}}.$$

Also, by (13), the $q^{-1}$-variance is obtained as

$$V\left({\left[X\right]}_{q^{-1}}\right)=\frac{{\left[n\right]}_q{[n+1]}_q{\theta }^2{q}^{-2}}{(1-\theta q^{-1})(1-\theta q^{-2})}+\frac{{\left[n\right]}_q\theta }{1-\theta q^{-1}}\left(1-\frac{{\left[n\right]}_q\theta }{1-\theta q^{-1}}\right)$$

which, on using the expression ${[n+1]}_q=q{\left[n\right]}_q+1$, reduces to

$$V\left({\left[X\right]}_{q^{-1}}\right)=\frac{{\left[n\right]}_q\theta }{(1-\theta q^{-1})(1-\theta q^{-2})}\left(1+\frac{{\left[n\right]}_q\theta (q^{-1}-1)}{1-\theta q^{-1}}\right).$$

The probability generating function of the negative q-binomial distribution of the second kind, on using (14), is deduced as

$$P\left(t\right)=\prod _{i=1}^n\frac{1-\theta q^{i-1}}{1-\theta t{q}^{i-1}},\left|t\right|<1/\theta ,0<\theta <1,0<q<1,$$

where $P_{X_i}\left(t\right)=(1-\theta q^{i-1})(1-\theta t{q}^{i-1})$ is the probability generating function of a geometric distribution. Therefore, according to Proposition 3(b), the negative q-binomial distribution of the second kind may be expressed as a convolution of n independent (and not identically distributed) geometric distributions.

More generally, the negative q-binomial distribution of the second kind may be expressed as a convolution of n independent negative q-binomial distributions of the second kind. Specifically, let $X_j$, $j=1,2,\dots ,n$, be a sequence of n independent random variables and assume that $X_j$, follows a negative q-binomial distribution of the second kind with parameters $r_j$, $\theta q^{s_{j-1}}$, and q, where $s_j={\sum }_{i=1}^j{r}_i$, for $j=1,2,\dots ,n$ and $s_0=0$. Clearly, the probability-generating function of $X_j$ is given by

$$P_{X_j}\left(t\right)=\prod _{i=1}^{r_j}\frac{1-\theta q^{s_{j-1}+i-1}}{1-\theta t{q}^{s_{j-1}+i-1}},\left|t\right|<1/\theta ,0<\theta <1,0<q<1.$$

Consequently, the probability-generating function of the sum $S_n={\sum }_{j=1}^n{X}_j$, is deduced as

$$P_{S_n}\left(t\right)=\prod _{j=1}^n\prod _{i=1}^{r_j}\frac{1-\theta q^{s_{j-1}+i-1}}{1-\theta t{q}^{s_{j-1}+i-1}}=\prod _{j=1}^n\prod _{i=s_{j-1}+1}^{s_j}\frac{1-\theta q^{i-1}}{1-\theta t{q}^{i-1}},$$

which, for $s_n\equiv m$, simplifies to

$$P_{S_n}\left(t\right)=\prod _{i=1}^m\frac{1-\theta q^{i-1}}{1-\theta t{q}^{i-1}},\left|t\right|<1/\theta ,0<\theta <1,0<q<1.$$

Therefore, the distribution of $S_n$ is a negative q-binomial distribution of the second kind with parameters m, $\theta$, and q.

Finally, it is worth noticing that the probability-generating function of the Euler distribution, with parameters $\lambda$ and q,

$$P_X\left(t\right)=\prod _{i=1}^{\infty }\frac{1-\theta q^{i-1}}{1-\theta t{q}^{j-1}},\left|t\right|<1/\theta ,0<\theta <1,0<q<1,$$

where $\theta =\lambda (1-q)$ may be expressed as product, $P_X\left(t\right)=P_{X_n}\left(t\right)P_{Y_n}\left(t\right)$, of the probability generating function of the negative q-binomial distribution of the second kind, with parameters n, $\theta$, and q,

$$P_{X_n}\left(t\right)=\prod _{i=1}^n\frac{1-\theta q^{i-1}}{1-\theta t{q}^{j-1}},\left|t\right|<1/\theta ,0<\theta <1,0<q<1,$$

and the probability generating function of the Euler distribution, with parameters $\lambda q^n$ and q,

$$P_{Y_n}\left(t\right)=\prod _{i=1}^{\infty }\frac{1-\theta q^{n+i-1}}{1-\theta t{q}^{n+j-1}},\left|t\right|<1/\theta ,0<\theta <1,0<q<1.$$

Therefore, an Euler distribution may be expressed as a convolution of a negative q-binomial distribution of the second kind and an independent Euler distribution.

3.4. q-Logarithmic Distribution

The q-logarithmic distribution, with probability distribution (4), is a power series q-distribution with a series function

$$g\left(\theta \right)=-l_q(1-\theta )=\sum _{j=1}^{\infty }\frac{{\theta }^j}{{\left[j\right]}_q},0<\theta <1,0<q<1.$$

Indeed, taking successively q-derivatives of the series function,

$$D_q^{x}g\left(\theta \right)=\sum _{j=x}^{\infty }{[j-1]}_{x-1,q}{\theta }^{j-x}={[x-1]}_q!\sum _{j=x}^{\infty }{\left[\genfrac{}{}{0.0pt}{}{j-1}{j-x}\right]}_q{\theta }^{j-x},$$

and using the negative q-binomial formula

$$\sum _{k=0}^{\infty }{\left[\genfrac{}{}{0.0pt}{}{x+k-1}{k}\right]}_q{\theta }^k=\prod _{i=1}^x{(1-\theta q^{i-1})}^{-1},$$

we find

$$D_q^{x}g\left(\theta \right)={[x-1]}_q!\prod _{i=1}^x{(1-\theta q^{i-1})}^{-1}$$

and, by (6),

$$a_{x,q}=\frac{1}{{\left[x\right]}_q!}{\left[D_q^{x}g\left(t\right)\right]}_{t=0}=\frac{1}{{\left[x\right]}_q},x=1,2,\dots ,$$

and the probability function (7) reduces to (4).

The q-factorial moments, by (8) and since

$$\frac{d_q^{m}g\left(\theta \right)}{d{\theta }^m}={[m-1]}_q!\prod _{i=1}^m{(1-\theta q^{i-1})}^{-1},$$

are obtained as

$$E\left({\left[X\right]}_{m,q}\right)=\frac{{[-l_q(1-\theta )]}^{-1}{[m-1]}_q!{\theta }^m}{{\prod }_{i=1}^m(1-\theta q^{i-1})},m=1,2,\dots .$$

In particular, the q-mean value is

$$E\left({\left[X\right]}_q\right)=\frac{{[-l_q(1-\theta )]}^{-1}\theta }{1-\theta }.$$

Also, using (11), the q-variance is obtained as

$$V\left({\left[X\right]}_q\right)=\frac{{[-l_q(1-\theta )]}^{-1}\theta }{1-\theta }\left(\frac{1}{1-\theta q}-\frac{{[-l_q(1-\theta )]}^{-1}\theta }{1-\theta }\right).$$

The $q^{-1}$-factorial moments, on using (11) with

$$\frac{d_q^{m}g\left(q^{-m}\theta \right)}{d_q{\theta }^m}={[m-1]}_q!q^{-m^2}\prod _{i=1}^m{(1-\theta q^{-m+i-1})}^{-1},$$

are obtained as

$$E\left({\left[X\right]}_{m,q^{-1}}\right)=\frac{{[-l_q(1-\theta )]}^{-1}{[m-1]}_q!{\theta }^m{q}^{-\left(\genfrac{}{}{0pt}{}{m}{2}\right)}}{{\prod }_{i=1}^m(1-\theta q^{-m+i-1})},m=1,2,\dots .$$

In particular, the $q^{-1}$-mean value is

$$E\left({\left[X\right]}_{q^{-1}}\right)=\frac{{[-l_q(1-\theta )]}^{-1}\theta }{1-\theta q^{-1}}.$$

Also, using (13), the $q^{-1}$-variance is obtained as

$$V\left({\left[X\right]}_{q^{-1}}\right)=\frac{{[-l_q(1-\theta )]}^{-1}\theta }{1-\theta q^{-1}}\left(1+\frac{\theta q^{-1}}{1-\theta q^{-2}}-\frac{{[-l_q(1-\theta )]}^{-1}\theta }{1-\theta }\right).$$

The probability generating function of the q-logarithmic distribution, using (14) is deduced

$$P_X\left(t\right)=\frac{-l_q(1-\theta t)}{-l_q(1-\theta )},\left|t\right|<1/\theta ,0<\theta <1,0<q<1.$$