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

From Kemp [1], we have a family of confluent q-ChuVandermonde 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-ChuVandermonde 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.

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.
Thus an important question is arisen about the asymptotic behaviour for n → ∞ 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 → ∞ of the q-factorial number of order n , where 0 < q < 1 and [t] q = 1−q t 1−q , the q-number t.Analytically we have 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 2π log q −1 x e (log x) 2 2 log q , x > 0 with mean value µ SW = q −1 and standard deviation σ 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.

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 = [X] 1/q , and we then compute the mean value and variance of the random variable Y , say µ q and σ 2 q 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 The q-mean of the family of confluent q-Chu-Vandermonde distributions is given by and since it is written as 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 = [X] 1/q which is given by and Equation ( 9) becomes So, 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 Proof.
The r-th order q-factorial moments of the family of confluent q-Chu-Vandermonde distributions is (c; q) ∞ (c; q) x = (cq r ; q) ∞ (cq r ; q) x−r the sum Equation ( 12) becomes 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 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 where s q (r, k) the q-Stirling numbers of the first kind (see Charalambides [11]).Since x r q = q r(x−r) x r 1/q the sum Equation ( 15) is written as By Equation (11) of the previous proposition 2 and the definition of the q-hypergeometric function Equation (1), Equation ( 16) reduces to Equation (14).

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 = [X] 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.
By using the q-Stirling asymptotic formula (5) we get the following approximation for the p.f. f X (x) From the standardized r.v.Z = [X] 1/q −µq σq with µ q and σ q given in Equation ( 7), we get Using the assumptions b n = o(1) and −c n /b n → ∞ as n → ∞, we have Also, by the previous two equations we get Moreover, by the Equation ( 23) we find and Finally, by the Equation (21) we get and Substituting all the previous approximations Equations ( 22),( 24),( 26),(28) to the p.f. f X (x) we get the approximation As a last step, we need to estimate the products ) and (c n ; q) ∞ /(c n ; q) x = (c n q x ; q) ∞ = ∞ j=1 (1 − c n q x q j−1 ) by integrals.Since the first product is written as and the function has all orders continuous derivatives in [1, ∞), we can apply the Euler-Maclaurin summation formula (see [5], p. 1090) in the sum of the Equation (30).So, where with β k the Bernoulli numbers.Now, expressing the integral appearing in Equation (31) through the dilogarithm function we get where Li 2 (y) = k≥1 y k k 2 the dilogarithm function.The dilogarithm satisfies the Landen's identity Applying the Landen's identity to Equation (33) we obtain Next, we estimate the sum and the quantity R k appearing in Equation (31) that is Equation (18).Moreover, by standardizing the continuous Stieltjes-Wigert distribution Equation ( 6) and then deforming this by the random variable [X] 1/q −µq σq , 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 (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.
Consequently, p qCCV I (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.
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.
where c constant.However, the discrete approximation of the above q-CCV-III distribution for n → ∞, is given by 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, ∞) defined by where the parameter p > 0 is given by p = γ/(t 2 + γ 2 ) with γ 2 = −t(1/ψ(a) + t), where t belongs to the interval with endpoints 0 and −1/ψ(a) and is given by ψ(a) = (q; q) ∞ ∞ j=0 q j (a−q j )(q;q) j with ψ(q + ) = ∞.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 closlelyrelated open problem to study a bilateral family of the considered distributions.

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 qnegative 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.