Gaussian q-distribution

Definition

Let q be a real number in the interval 0, 1). The [probability density function of the Gaussian q-distribution is given by

:s_q(x) = \begin{cases} 0 & \text{if } x \nu. \end{cases}

where

:\nu = \nu(q) = \frac{1}{\sqrt{1-q}} ,

: c(q)=2(1-q)^{1/2}\sum_{m=0}^\infty \frac{(-1)^m q^{m(m+1)}}{(1-q^{2m+1})(1-q^2)_{q^2}^m} .

The q-analogue [t]q of the real number t is given by

: [t]_q=\frac{q^t-1}{q-1}.

The q-analogue of the exponential function is the q-exponential, E, which is given by

: E_q^{x}=\sum_{j=0}^{\infty}q^{j(j-1)/2}\frac{x^{j}}{[j]!}

where the q-analogue of the factorial is the q-factorial, [n]q!, which is in turn given by

: [n]_q!=[n]_q[n-1]_q\cdots [2]_q ,

for an integer n 2 and [1]q! = [0]q! = 1.

The cumulative distribution function of the Gaussian q-distribution is given by

: G_q(x) = \begin{cases} 0 & \text{if } x \displaystyle \frac{1}{c(q)}\int_{-\nu}^{x} E_{q^2}^{-q^2 t^2/[2]} , d_qt & \text{if } -\nu \leq x \leq \nu \[12pt] 1 & \text{if } x\nu \end{cases}

where the integration symbol denotes the Jackson integral.

The function G**q is given explicitly by

: G_q(x)= \begin{cases} 0 & \text{if } x \displaystyle \frac{1}{2} + \frac{1-q}{c(q)} \sum_{n=0}^\infty \frac{q^{n(n+1)}(q-1)^n}{(1-q^{2n+1})(1-q^2)_{q^2}^{n}}x^{2n+1} & \text{if } -\nu \leq x \leq \nu \ 1 & \text{if}\ x \nu \end{cases}

where

: (a+b)q^n=\prod{i=0}^{n-1}(a+q^ib) .

Moments

The moments of the Gaussian q-distribution are given by

: \frac{1}{c(q)}\int_{-\nu}^\nu E_{q^2}^{-q^2 x^2/[2]} , x^{2n} , d_qx =[2n-1]!! ,

: \frac{1}{c(q)}\int_{-\nu}^\nu E_{q^{2}}^{-q^2 x^2/[2]} , x^{2n+1} , d_qx=0 ,

where the symbol [2n − 1]!! is the q-analogue of the double factorial given by

: [2n-1][2n-3]\cdots[1]= [2n-1]!!. ,

References

• Exton, H. (1983), q-Hypergeometric Functions and Applications, New York: Halstead Press, Chichester: Ellis Horwood, 1983, , ,