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]) p x (λ; q) = E q (−λ) λ x [x] q ! , x = 0, 1, . . . , where 0 < λ < 1/(1 − q) and 0 < q < 1 (Euler distribution) or 0 < λ < ∞ and 1 < q < ∞ (Heine distribution). Also, E q (t) = ∞ i=1 (1 + t(1 − q)q i−1 ) is a q-exponential function. It should be noted that e q (t) = ∞ i=1 (1 − t(1 − q)q i−1 ) −1 is another q-exponential function and that these q-exponential functions are connected by E q (t)e q (−t) = 1 and E q −1 (t) = e q (t).

Kemp and Kemp [KK91], in their study of the 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 [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

p x (θ; q) = n x q θ x q ( x 2 )

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

p x (θ; q) = n + x − 1 x q θ x n i=1

(1 − θq i−1 ), x = 0, 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

p x (θ; q) = [−l q (1 − θ)] −1 θ x [x] q , x = 1, 2, . . . ,

where 0 < θ < 1, 0 < q < 1, and

−l q (1 − θ) = lim x→0 ∞ i=1 1 − θq x+i−1 1 − θq i−1 − 1 = ∞ j=1 θ j [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.

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

g(θ) = ∞ x=0 a x,q θ x , 0 < θ < ρ, ρ > 0,(1)

where the coefficient

a x,q = 1 [x] q ! [D x q g(t)] t=0 ≥ 0, x = 0, 1, . . . , 0 < q < 1, or 1 < q < ∞,(2)

with D q = d q /d q t the q-derivative operator,

D q g(t) = d q g(t) d q t = g(t) − g(qt) (1 − q)t ,

does not involve the parameter θ. Clearly, the function

p x (θ; q) = a x,q θ x g(θ) , x = 0, 1, . . . ,(3)

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 p x (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 a x,q > 0 for x ∈ T , with

T = {x 0 , x 0 + 1, . . . , x 0 + x 1 − 1}, x 0 ≥ 0, x 1 ≥ 1.

Moreover, note that the truncated versions of the a power series q-distribution are also power series q-distributions in their own right.

The probability generating function P (t) = ∞ x=0 p x (θ; q)t x , on using (1) and (3), is readily deduced as

P (t) = g(θt) g(θ) .(4)

Clearly, the mth q-derivative, with respect to t, of the probability generating function is

d m q P (t) d q t m = ∞ x=m p x (θ; q)[x] m,q t x−m .

Thus, the mth q-factorial moment of the power series q-distribution, on using (4), is obtained as

E([X] m,q ) = 1 g(θ) • d m q g(θt) d q t m t=1 = θ m g(θ) • d m q g(θ) d q θ m , m = 1, 2, . . . .(5)

In particular the q-mean is given by

E([X] q ) = θ g(θ) • d q g(θ) d q θ .(6)

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

V ([X] q ) = qθ 2 g(θ) • d 2 q g(θ) d q θ 2 − θ g(θ) • d q g(θ) d q θ θ g(θ) • d q g(θ) d q θ − 1 . (8)

Example 2.3. q-Poisson distributions. These are power series q-distributions, with series function g(λ) = e q (λ) = 1/E q (−λ), where 0 < λ < 1/(1 − q) and 0 < q < 1 or 0 < λ < ∞ and 1 < q < ∞. Since D q e q (t) = e q (t) and e q (0) = 1, it follows from (2) that

a x,q = 1 [x] q ! [D x q e q (t)] t=0 = 1 [x] q !

, x = 0, 1, . . . , Also, the probability generating function of the q-Poisson distributions, on using (4), is deduced as

P (t) =

e q (λt) e q (λ) = E q (−λ)e q (λt).

The q-factorial moments, by (5) and since D m q e q (λ) = e q (λ), are readily deduced as

E([X] m,q ) = λ m , 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 ) = qλ 2 − λ(λ − 1) = λ(1 + (q − 1)λ).

Example 2.4. q-Binomial distribution of the first kind. The series function of this distribution is g(θ) = n i=1 (1 + θq i−1 ), where 0 < θ < ∞ and 0 < q < 1 or 1 < q < ∞. Since

D q g(θ) = n i=1 (1 + θq i−1 ) − n i=1 (1 + θq i ) (1 − q)θ = [(1 + θ) − (1 + θq n )] n−1 i=1 (1 + θq i ) (1 − q)θ = [n] q n−1 i=1

(1 + (θq)q i−1 ), it follows successively that

D x q g(θ) = [n] x,q q 1+2+•••+(x−1) n−x i=1 (1 + (θq x )q i−1 ) = [n] x,q q ( x 2 ) n−x i=1

(1 + (θq x )q i−1 ), for x = 1, 2, . . . , n. Thus, by (2),

a x,q = 1 [x] q ! [D x q g(t)] t=0 = n x q q ( x 2 ) , x = 0, 1, . . . , n.

Also, the probability generating function of the q-binomial distribution of the first kind, on using (4), is deduced as

P (t) = n i=1 (1 + θtq i−1 ) n i=1 (1 + θq i−1 )

.

The q-factorial moments, by (5) and since

D m q g(θ) = [n] m,q q ( m 2 ) n−m i=1 (1 + (θq m )q i−1 ) = [n] m,q q ( m 2 ) n i=m+1

(1 + θq i−1 ), are obtained as

E([X] m,q ) = [n] m,q θ m q ( m 2 ) m i=1 (1 + θq i−1 )

, m = 1, 2, . . . .

In particular, the q-mean is

E([X] q ) = [n] q θ (1 + θ) .

Also, using (7) and, subsequently, the expression q[n − 1] q = [n] q − 1, the q-variance is obtained as

V ([X] q ) = [n] q [n − 1] q θ 2 q 2 (1 + θ)(1 + θq) + [n] q θ 1 + θ 1 − [n] q θ 1 + θ = [n] q θ (1 + θ)(1 + θq) 1 + [n] q θ(q − 1) 1 + θ .

Example 2.5. Negative q-binomial distribution of the second kind. It is a power series q-distribution, with series function g(θ) = n i=1 (1 − θq i−1 ) −1 , where 0 < θ < 1 and 0 < q < 1. Since

D q g(θ) = n i=1 (1 − θq i−1 ) −1 − n i=1 (1 − θq i ) −1 (1 − q)θ = [(1 − θq n ) − (1 − θ)] n+1 i=1 (1 − θq i−1 ) (1 − q)θ = [n] q n+1 i=1

(1 − θq i−1 ), it follows successively that

D x q g(θ) = [n] q [n + 1] q • • • [n + x − 1] q n+x i=1 (1 − θq i−1 ) = [n + x − 1] x,q n+x i=1

(1 − θq i−1 ), for x = 1, 2, . . . . Thus, by (2),

a x,q = 1 [x] q ! [D x q g(t)] t=0 = n + x − 1 x q , x = 0, 1, . . . .

Also, the probability generating function of the negative q-binomial distribution of the second kind, on using (4), is deduced as

P (t) = n i=1 (1 − θtq i−1 ) −1 n i=1 (1 − θq i−1 ) −1 .

The q-factorial moments, by (5) and since

D m q g(θ) = [n + m − 1] m,q n+m i=1 (1 − θq i−1 ) −1 = [n + m − 1] m,q n i=1 (1 − θq i−1 ) −1 m i=1 (1 − θq n+i−1 ) −1 , are obtained as E([X] m,q ) = [n + m − 1] m,q θ m m i=1

(1 − θq n+i−1 ) −1 , m = 1, 2, . . . .

In particular, the q-expected value is

E([X] q ) = [n] q θ 1 − θq n .

Also, using (7) and, subsequently, the expression [n + 1] q = [n] q + q n , the q-variance is successively obtained as

V ([X] q ) = [n] q [n + 1] q θ 2 q (1 − θq n )(1 − θq n+1 ) + [n] q θ 1 − θq n 1 − [n] q θ 1 − θq n = [n] q θ (1 − θq n )(1 − θq n+1 ) 1 + [n] q θ(q − 1) 1 − θq n .

Example 2.6. q-Logarithmic distribution. The series function of this distribution is

g(θ) = −l q (1 − θ) = ∞ j=1 θ j [j] q , 0 < θ < 1, 0 < q < 1.

Taking successively its q-derivatives,

D x q g(θ) = ∞ j=x [j − 1] x−1,q θ j−x = [x − 1] q ! ∞ j=x j − 1 j − x q θ j−x ,

and using the negative q-binomial formula ∞ k=0

x + k − 1 k

q θ k = x i=1

(1 − θq i−1 ) −1 , we find

D x q g(θ) = [x − 1] q ! x i=1

(1 − θq i−1 ) −1 .

Thus, by (2), a x,q = 1 [x] q ! [D x q g(t)] t=0 = 1 [x] q , x = 1, 2, . . . . Also, the probability generating function of the q-logarithmic distribution, on using (4), is deduced as

P (t) = −l q (1 − θt) −l q (1 − θ) .

The q-factorial moments, by (5) and since

D m q g(θ) = [m − 1] q ! m i=1

(1 − θq i−1 ) −1 , are obtained as

E([X] m,q ) = [−l q (1 − θ)] −1 [m − 1] q !θ m m i=1 (1 − θq i−1 )

, m = 1, 2, . . . .