A class of power series q-distributions Charalambos A. Charalambides Department of Mathematics, University of Athens ccharal@math.uoa.gr 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. The q-Poisson (Heine and Euler), q-binomial, negative q-binomial and q-logarithmic distributions belong in this class. The probability generating functions and q-factorial moments of the power series q-distributions are derived. In particular, the q-mean and the q-variance are deduced. 1 Introduction Benkherouf and Bather[BB88] derived the Heine and Euler distributions, which constitute q-analogs of the Poisson distribution, as feasible priors in a simple Bayesian model for oil exploration. The probability function of the q-Poisson distributions is given by (Charalambides[Cha16, p. 107]) λx px (λ; q) = Eq (−λ) , x = 0, 1, . . . , [x]q ! where 0 < λ < 1/(1 − q) and Q∞ 0 < q < 1 (Euler distribution) or 0 < λ < ∞ and 1 < q < ∞ (Heine i−1 distribution). Q∞ Also, Eq (t) = i=1 (1 + t(1 − q)q ) is a q-exponential function. It should be noted that i−1 −1 eq (t) = i=1 (1 − t(1 − q)q ) is another q-exponential function and that these q-exponential functions are connected by Eq (t)eq (−t) = 1 and Eq−1 (t) = eq (t). Kemp and Kemp [KK91], in their study of the Weldon’s classical dice data, introduced a q-binomial distribu- tion. 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 [KN90] further studied it as stationary distribution of a birth and death process. The probability function of this q-binomial distribution of the first kind is given by x θx q(2)   n px (θ; q) = Qn , x = 0, 1, . . . , n, x q i=1 (1 + θq i−1 ) where 0 < θ < ∞, and 0 < q < 1 or 1 < q < ∞. Charalambides [Cha10] 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 Copyright © by the paper’s authors. Copying permitted for private and academic purposes. In: L. Ferrari, M. Vamvakari (eds.): Proceedings of the GASCom 2018 Workshop, Athens, Greece, 18–20 June 2018, published at http://ceur-ws.org 1 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   n n+x−1 xY px (θ; q) = θ (1 − θq i−1 ), x = 0, 1, . . . , x q i=1 where 0 < θ < 1 and 0 < q < 1. A q-logarithmic distribution was studied by C. D. Kemp[Kem97] as a group size distribution. Its probability function is given by θx px (θ; q) = [−lq (1 − θ)]−1 , x = 1, 2, . . . , [x]q where 0 < θ < 1, 0 < q < 1, and ∞ ∞ ! Y 1 − θq x+i−1 X θj −lq (1 − θ) = lim −1 = x→0 i=1 1 − θq i−1 j=1 [j]q is a q-logarithmic function. The class of power series q-distributions, introduced in section 2, provides a unified approach to the study of these distributions. Its probability generating function and q-factorial moments are derived. Demonstrating this approach, the probability generating function and q-factorial moments of the q-Poisson (Heine and Euler), q-binomial, negative q-binomial, and q-logarithmic distributions are obtained. 2 Power series q-distributions Consider a positive function g(θ) of a positive parameter θ and assume that it is analytic with a q-Taylor expansion ∞ X g(θ) = ax,q θx , 0 < θ < ρ, ρ > 0, (1) x=0 where the coefficient 1 ax,q = [Dx g(t)]t=0 ≥ 0, x = 0, 1, . . . , 0 < q < 1, or 1 < q < ∞, (2) [x]q ! q with Dq = dq /dq t the q-derivative operator, dq g(t) g(t) − g(qt) Dq g(t) = = , dq t (1 − q)t does not involve the parameter θ. Clearly, the function ax,q θx px (θ; q) = , x = 0, 1, . . . , (3) g(θ) with 0 < q < 1 or 1 < q < ∞, and 0 < θ < ρ, satisfies the properties of a probability (mass) function. Definition 2.1. A family of discrete q-distributions px (θ; q), θ ∈ Θ, q ∈ Q, is said to be a class of power series q-distributions, with parameters θ, q and series function g(θ) if it has the representation (3), with series function satisfying condition (1). Remark 2.2. The range of x in (3), as in the case of the (usual) power series distributions), may be reduced. Thus, we may have ax,q > 0 for x ∈ T , with T = {x0 , x0 + 1, . . . , x0 + x1 − 1}, x0 ≥ 0, x1 ≥ 1. Moreover, note that the truncated versions of the a power series q-distribution are also power series q-distributions in their own right. 2 P∞ x The probability generating function P (t) = x=0 px (θ; q)t , on using (1) and (3), is readily deduced as g(θt) P (t) = . (4) g(θ) Clearly, the mth q-derivative, with respect to t, of the probability generating function is ∞ dm q P (t) X = px (θ; q)[x]m,q tx−m . dq tm x=m Thus, the mth q-factorial moment of the power series q-distribution, on using (4), is obtained as  m dq g(θt) θ m dm q g(θ)  1 E([X]m,q ) = · = · , m = 1, 2, . . . . (5) g(θ) dq tm t=1 g(θ) dq θm In particular the q-mean is given by θ dq g(θ) E([X]q ) = · . (6) g(θ) dq θ Also, on using the expression  V ([X]q ) = qE([X]2,q ) − E([X]q ) E([X]q ) − 1 , (7) the q-variance is obtained as qθ2 d2q g(θ)   θ dq g(θ) θ dq g(θ) V ([X]q ) = · − · · − 1 . (8) g(θ) dq θ2 g(θ) dq θ g(θ) dq θ Example 2.3. q-Poisson distributions. These are power series q-distributions, with series function g(λ) = eq (λ) = 1/Eq (−λ), where 0 < λ < 1/(1 − q) and 0 < q < 1 or 0 < λ < ∞ and 1 < q < ∞. Since Dq eq (t) = eq (t) and eq (0) = 1, it follows from (2) that 1 1 ax,q = [Dx eq (t)]t=0 = , x = 0, 1, . . . , [x]q ! q [x]q ! Also, the probability generating function of the q-Poisson distributions, on using (4), is deduced as eq (λt) P (t) = = Eq (−λ)eq (λt). eq (λ) The q-factorial moments, by (5) and since Dqm eq (λ) = eq (λ), are readily deduced as E([X]m,q ) = λm , m = 1, 2, . . . . In particular, the q-mean is given by E([X]q ) = λ. Also, using (7), the q-variance is obtained as V ([X]q ) = qλ2 − λ(λ − 1) = λ(1 + (q − 1)λ). Example Qn 2.4. q-Binomial distribution of the first kind. The series function of this distribution is g(θ) = i−1 i=1 (1 + θq ), where 0 < θ < ∞ and 0 < q < 1 or 1 < q < ∞. Since Qn Qn (1 + θq i−1 ) − i=1 (1 + θq i ) Dq g(θ) = i=1 (1 − q)θ Qn−1 n−1 [(1 + θ) − (1 + θq n )] i=1 (1 + θq i ) Y = = [n]q (1 + (θq)q i−1 ), (1 − q)θ i=1 3 it follows successively that n−x n−x x (1 + (θq x )q i−1 ) = [n]x,q q (2) Y Y Dqx g(θ) = [n]x,q q 1+2+···+(x−1) (1 + (θq x )q i−1 ), i=1 i=1 for x = 1, 2, . . . , n. Thus, by (2),   1 n (x2) ax,q = [Dqx g(t)]t=0 = q , x = 0, 1, . . . , n. [x]q ! x q Also, the probability generating function of the q-binomial distribution of the first kind, on using (4), is deduced as Qn (1 + θtq i−1 ) P (t) = Qi=1 n i−1 ) . i=1 (1 + θq The q-factorial moments, by (5) and since n−m n m m Dqm g(θ) = [n]m,q q ( 2 ) (1 + (θq m )q i−1 ) = [n]m,q q ( 2 ) Y Y (1 + θq i−1 ), i=1 i=m+1 are obtained as m [n]m,q θm q ( 2 ) E([X]m,q ) = Qm i−1 ) , m = 1, 2, . . . . i=1 (1 + θq In particular, the q-mean is [n]q θ E([X]q ) = . (1 + θ) Also, using (7) and, subsequently, the expression q[n − 1]q = [n]q − 1, the q-variance is obtained as [n]q [n − 1]q θ2 q 2   [n]q θ [n]q θ V ([X]q ) = + 1− (1 + θ)(1 + θq) 1+θ 1+θ   [n]q θ [n]q θ(q − 1) = 1+ . (1 + θ)(1 + θq) 1+θ Example 2.5. Negative Qn q-binomial distribution of the second kind. It is a power series q-distribution, with series function g(θ) = i=1 (1 − θq i−1 )−1 , where 0 < θ < 1 and 0 < q < 1. Since Qn Qn (1 − θq i−1 )−1 − i=1 (1 − θq i )−1 Dq g(θ) = i=1 (1 − q)θ Qn+1 n+1 [(1 − θq n ) − (1 − θ)] i=1 (1 − θq i−1 ) Y = = [n]q (1 − θq i−1 ), (1 − q)θ i=1 it follows successively that n+x Y n+x Y Dqx g(θ) = [n]q [n + 1]q · · · [n + x − 1]q (1 − θq i−1 ) = [n + x − 1]x,q (1 − θq i−1 ), i=1 i=1 for x = 1, 2, . . . . Thus, by (2),   1 x n+x−1 ax,q = [D g(t)]t=0 = , x = 0, 1, . . . . [x]q ! q x q Also, the probability generating function of the negative q-binomial distribution of the second kind, on using (4), is deduced as Qn (1 − θtq i−1 )−1 P (t) = Qi=1 n i−1 )−1 . i=1 (1 − θq 4 The q-factorial moments, by (5) and since n+m Y Dqm g(θ) = [n + m − 1]m,q (1 − θq i−1 )−1 i=1 n Y m Y = [n + m − 1]m,q (1 − θq i−1 )−1 (1 − θq n+i−1 )−1 , i=1 i=1 are obtained as m Y E([X]m,q ) = [n + m − 1]m,q θm (1 − θq n+i−1 )−1 , m = 1, 2, . . . . i=1 In particular, the q-expected value is [n]q θ E([X]q ) = . 1 − θq n Also, using (7) and, subsequently, the expression [n + 1]q = [n]q + q n , the q-variance is successively obtained as [n]q [n + 1]q θ2 q   [n]q θ [n]q θ V ([X]q ) = + 1 − (1 − θq n )(1 − θq n+1 ) 1 − θq n 1 − θq n   [n]q θ [n]q θ(q − 1) = 1 + . (1 − θq n )(1 − θq n+1 ) 1 − θq n Example 2.6. q-Logarithmic distribution. The series function of this distribution is ∞ X θj g(θ) = −lq (1 − θ) = , 0 < θ < 1, 0 < q < 1. j=1 [j]q Taking successively its q-derivatives, ∞ ∞   X X j−1 Dqx g(θ) = [j − 1]x−1,q θj−x = [x − 1]q ! θj−x , j=x j=x j−x q and using the negative q-binomial formula ∞   x X x+k−1 k Y θ = (1 − θq i−1 )−1 , k q i=1 k=0 we find x Y Dqx g(θ) = [x − 1]q ! (1 − θq i−1 )−1 . i=1 Thus, by (2), 1 1 ax,q = [Dx g(t)]t=0 = , x = 1, 2, . . . . [x]q ! q [x]q Also, the probability generating function of the q-logarithmic distribution, on using (4), is deduced as −lq (1 − θt) P (t) = . −lq (1 − θ) The q-factorial moments, by (5) and since m Y Dqm g(θ) = [m − 1]q ! (1 − θq i−1 )−1 , i=1 are obtained as [−lq (1 − θ)]−1 [m − 1]q !θm E([X]m,q ) = Qm i−1 ) , m = 1, 2, . . . . i=1 (1 − θq 5 In particular, the q-mean value is [−lq (1 − θ)]−1 θ E([X]q ) = . 1−θ Also, using (7), the q-variance is obtained as [−lq (1 − θ)]−1 θ2 q [−lq (1 − θ)]−1 θ [−lq (1 − θ)]−1 θ   V ([X]q ) = + 1− (1 − θ)(1 − θq) 1−θ 1−θ −1 −1   [−lq (1 − θ)] θ 1 [−lq (1 − θ)] θ = − . 1−θ 1 − θq 1−θ References [BB88] L. Benkherouf, J. A. Bather. Oil exploration: sequential decisions in the face of uncertainty.Journal of Applied Probability, 25:529-543, 1988. [Cha10] Ch. A. Charalambides The q-Bernstein basis as a q-binomial distribution. Journal of Statististical Planning and Inference, 140:2184-2190, 2010. [Cha16] Ch. A. Charalambides. Discrete q-Distributions. John Wiley & Sons, Hoboken, New Jersey, 2016. [Kem97] C. D. Kemp. A q-logarithmic distribution. In Balakrishnan, N. (Ed.) Advances in Combinatorial Methods and Applications to Probability and Statistics, Birkhäuser, Boston, MA, pp. 465-470, 1997. [KK91] A. Kemp, C D. Kemp. Weldon’s dice data revisited. American Statistician, 45:216-222, 1991. [KN90] A. Kemp, J. Newton. Certain state-dependent processes for dichotomized parasite population. Journal of Applied Probability, 27:251-258, 1990. 6