Gaussian q-distribution

In

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

${\displaystyle s_{q}(x)={\begin{cases}0&{\text{if }}x<-\nu \\{\frac {1}{c(q)}}E_{q^{2}}^{\frac {-q^{2}x^{2}}{[2]_{q}}}&{\text{if }}-\nu \leq x\leq \nu \\0&{\mbox{if }}x>\nu .\end{cases}}}$

where

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

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

The q-analogue [t]_q of the real number ${\displaystyle t}$ is given by

${\displaystyle [t]_{q}={\frac {q^{t}-1}{q-1}}.}$

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

${\displaystyle E_{q}^{x}=\sum _{j=0}^{\infty }q^{j(j-1)/2}{\frac {x^{j}}{[j]!}}}$

where the q-analogue of the

${\displaystyle [n]_{q}!=[n]_{q}[n-1]_{q}\cdots [2]_{q}\,}$

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

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

${\displaystyle G_{q}(x)={\begin{cases}0&{\text{if }}x<-\nu \\[12pt]\displaystyle {\frac {1}{c(q)}}\int _{-\nu }^{x}E_{q^{2}}^{-q^{2}t^{2}/[2]}\,d_{q}t&{\text{if }}-\nu \leq x\leq \nu \\[12pt]1&{\text{if }}x>\nu \end{cases}}}$

where the integration symbol denotes the Jackson integral.

The function G_q is given explicitly by

${\displaystyle G_{q}(x)={\begin{cases}0&{\text{if }}x<-\nu ,\\\displaystyle {\frac {1}{2}}+{\frac {1-q}{c(q)}}\sum _{n=0}^{\infty }{\frac {q^{n(n+1)}(q-1)^{n}}{(1-q^{2n+1})(1-q^{2})_{q^{2}}^{n}}}x^{2n+1}&{\text{if }}-\nu \leq x\leq \nu \\1&{\text{if}}\ x>\nu \end{cases}}}$

where

${\displaystyle (a+b)_{q}^{n}=\prod _{i=0}^{n-1}(a+q^{i}b).}$

Moments

The moments of the Gaussian q-distribution are given by

${\displaystyle {\frac {1}{c(q)}}\int _{-\nu }^{\nu }E_{q^{2}}^{-q^{2}x^{2}/[2]}\,x^{2n}\,d_{q}x=[2n-1]!!,}$

${\displaystyle {\frac {1}{c(q)}}\int _{-\nu }^{\nu }E_{q^{2}}^{-q^{2}x^{2}/[2]}\,x^{2n+1}\,d_{q}x=0,}$

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

${\displaystyle [2n-1][2n-3]\cdots [1]=[2n-1]!!.\,}$

See also

• Q-Gaussian process

References

• Díaz, R.; Pariguan, E. (2009). "On the Gaussian q-distribution".
  S2CID 115175228
  .
• Diaz, R.; Teruel, C. (2005). "q,k-Generalized Gamma and Beta Functions" (PDF).
  S2CID 73643153
  .
• van Leeuwen, H.; Maassen, H. (1995). "A q deformation of the Gauss distribution" (PDF).
  S2CID 13934946
  .
• Exton, H. (1983), q-Hypergeometric Functions and Applications, New York: Halstead Press, Chichester: Ellis Horwood, 1983,
  ISBN 978-0470274538