A class of power series q-distributions, generated by considering a qTaylor 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.
Introduction x = 0; 1; : : : ; px( ; q) =
xq(x2) n x q i=1(1 + qi 1) ;
Qn x = 0; 1; : : : ; n; where 0 < < 1, and 0 < q < 1 or 1 < q < 1.
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 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 and assume that it is analytic with a q-Taylor 1 X ax;q x; x=0 The probability generating function P (t) = P1
x=0 px( ; q)tx, on using (1) and (3), is readily deduced as Clearly, the mth q-derivative, with respect to t, of the probability generating function is Thus, the mth q-factorial moment of the power series q-distribution, on using (4), is obtained as Example 2.3. q-Poisson distributions. These are power series q-distributions, with series function g( ) = eq( ) = 1=Eq( ), where 0 < < 1=(1 q) and 0 < q < 1 or 0 < < 1 and 1 < q < 1. Since Dqeq(t) = eq(t) and eq(0) = 1, it follows from (2) that In particular, the q-mean is given by Also, using (7), the q-variance is obtained as eq( t) eq( ) P (t) = = Eq(
)eq( t): it follows successively that for x = 1; 2; : : : ; n. Thus, by (2),
n x x n x Dqxg( ) = [n]x;qq1+2+ +(x 1) Y (1 + ( qx)qi 1) = [n]x;qq 2 ( ) Y (1 + ( qx)qi 1); i=1
i=1 x q Also, using (7) and, subsequently, the expression q[n 1, the q-variance is obtained as Example 2.5. Negative q-binomial distribution of the second kind. It is a power series q-distribution, with series function g( ) = Qin=1(1 qi 1) 1, where 0 < < 1 and 0 < q < 1. Since it follows successively that for x = 1; 2; : : : . Thus, by (2),
Dqxg( ) = [n]q[n + 1]q [n + x qi 1) = [n + x 1]x;q
qi 1); n+x Y (1 i=1 1 + [n]1q (q qn 1 qn
[n]q 1 1) : qn Example 2.6. q-Logarithmic distribution. The series function of this distribution is Taking successively its q-derivatives, and using the negative q-binomial formula ) = X1 j j=1 [j]q
;
t)) : E([X]m;q) = [ lq(1
Qim=1(1 )] 1[m
1]q! m qi 1) ; In particular, the q-mean value is : Also, using (7), the q-variance is obtained as = : )] 1 [BB88]