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Gaussian q-distribution
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<h2 id="definition">Definition</h2> <p>Let <i>q</i> be a <a href="/facts/Real_number/R02gw5Pb">real number</a> in the interval [0, 1). The <a href="/facts/Probability_density_function/zvfybna4">probability density function</a> of the Gaussian <i>q</i>-distribution is given by </p> s q ( x ) = { 0 if x < − ν 1 c ( q ) E q 2 − q 2 x 2 [ 2 ] q if − ν ≤ x ≤ ν 0 if x > ν . {\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}}} <p>where </p> ν = ν ( q ) = 1 1 − q , {\displaystyle \nu =\nu (q)={\frac {1}{\sqrt {1-q}}},} c ( q ) = 2 ( 1 − q ) 1 / 2 ∑ m = 0 ∞ ( − 1 ) m q m ( m + 1 ) ( 1 − q 2 m + 1 ) ( 1 − q 2 ) q 2 m . {\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}}}.} <p>The <i>q</i>-analogue [<i>t</i>]<i>q</i> of the real number t {\displaystyle t} is given by </p> [ t ] q = q t − 1 q − 1 . {\displaystyle [t]_{q}={\frac {q^{t}-1}{q-1}}.} <p>The <i>q</i>-analogue of the <a href="/facts/Exponential_function/ewlYxvR2">exponential function</a> is the <a href="/facts/Q-exponential/t9gJxytF">q-exponential</a>, <i>E</i><i>x</i><i>q</i>, which is given by </p> E q x = ∑ j = 0 ∞ q j ( j − 1 ) / 2 x j [ j ] ! {\displaystyle E_{q}^{x}=\sum _{j=0}^{\infty }q^{j(j-1)/2}{\frac {x^{j}}{[j]!}}} <p>where the <i>q</i>-analogue of the <a href="/facts/Factorial/kkfy1Bwe">factorial</a> is the <a href="/facts/Q-factorial/keASLfpe">q-factorial</a>, [<i>n</i>]<i>q</i>!, which is in turn given by </p> [ n ] q ! = [ n ] q [ n − 1 ] q ⋯ [ 2 ] q {\displaystyle [n]_{q}!=[n]_{q}[n-1]_{q}\cdots [2]_{q}\,} <p>for an integer <i>n</i> > 2 and [1]<i>q</i>! = [0]<i>q</i>! = 1. </p> <p>The <a href="/facts/Cumulative_distribution_function/WaKU8tp4">cumulative distribution function</a> of the Gaussian <i>q</i>-distribution is given by </p> G q ( x ) = { 0 if x < − ν 1 c ( q ) ∫ − ν x E q 2 − q 2 t 2 / [ 2 ] d q t if − ν ≤ x ≤ ν 1 if x > ν {\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}}} <p>where the <a href="/facts/Integral/V7lyV997">integration</a> symbol denotes the <a href="/facts/Jackson_integral/1zDdQoFQ">Jackson integral</a>. </p><p>The function <i>G</i><i>q</i> is given explicitly by </p> G q ( x ) = { 0 if x < − ν , 1 2 + 1 − q c ( q ) ∑ n = 0 ∞ q n ( n + 1 ) ( q − 1 ) n ( 1 − q 2 n + 1 ) ( 1 − q 2 ) q 2 n x 2 n + 1 if − ν ≤ x ≤ ν 1 if x > ν {\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}}} <p>where </p> ( a + b ) q n = ∏ i = 0 n − 1 ( a + q i b ) . {\displaystyle (a+b)_{q}^{n}=\prod _{i=0}^{n-1}(a+q^{i}b).} <h2 id="moments">Moments</h2> <p>The <a href="/facts/Moment_(mathematics)/GL7vJ1nb">moments</a> of the Gaussian <i>q</i>-distribution are given by </p> 1 c ( q ) ∫ − ν ν E q 2 − q 2 x 2 / [ 2 ] x 2 n d q x = [ 2 n − 1 ] ! ! , {\displaystyle {\frac {1}{c(q)}}\int _{-\nu }^{\nu }E_{q^{2}}^{-q^{2}x^{2}/[2]}\,x^{2n}\,d_{q}x=[2n-1]!!,} 1 c ( q ) ∫ − ν ν E q 2 − q 2 x 2 / [ 2 ] x 2 n + 1 d q x = 0 , {\displaystyle {\frac {1}{c(q)}}\int _{-\nu }^{\nu }E_{q^{2}}^{-q^{2}x^{2}/[2]}\,x^{2n+1}\,d_{q}x=0,} <p>where the symbol [2<i>n</i> − 1]!! is the <i>q</i>-analogue of the <a href="/facts/Double_factorial/VSU8AgzB">double factorial</a> given by </p> [ 2 n − 1 ] [ 2 n − 3 ] ⋯ [ 1 ] = [ 2 n − 1 ] ! ! . {\displaystyle [2n-1][2n-3]\cdots [1]=[2n-1]!!.\,} <h2 id="see-also">See also</h2> <ul><li><a href="/facts/Q-Gaussian_process/I5kGyLUY">Q-Gaussian process</a></li></ul> <ul><li>Díaz, R.; Pariguan, E. (2009). "On the Gaussian q-distribution". <i><a href="/facts/Journal_of_Mathematical_Analysis_and_Applications/IeXhWStY">Journal of Mathematical Analysis and Applications</a></i>. 358: 1–9. <a href="/facts/ArXiv_(identifier)/H6EtgnBe">arXiv</a>:<a href="https://arxiv.org/abs/0807.1918">0807.1918</a>. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1016%2Fj.jmaa.2009.04.046">10.1016/j.jmaa.2009.04.046</a>. <a href="/facts/S2CID_(identifier)/ldJsHa2Y">S2CID</a> <a href="https://api.semanticscholar.org/CorpusID:115175228">115175228</a>.</li> <li>Diaz, R.; Teruel, C. (2005). <a href="http://staff.www.ltu.se/~norbert/home_journal/electronic/121art6.pdf">"q,k-Generalized Gamma and Beta Functions"</a> (PDF). <i><a href="/facts/Journal_of_Nonlinear_Mathematical_Physics/EwPCUm6g">Journal of Nonlinear Mathematical Physics</a></i>. 12 (1): 118–134. <a href="/facts/ArXiv_(identifier)/H6EtgnBe">arXiv</a>:<a href="https://arxiv.org/abs/math/0405402">math/0405402</a>. <a href="/facts/Bibcode_(identifier)/9HtdQSGB">Bibcode</a>:<a href="https://ui.adsabs.harvard.edu/abs/2005JNMP...12..118D">2005JNMP...12..118D</a>. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.2991%2Fjnmp.2005.12.1.10">10.2991/jnmp.2005.12.1.10</a>. <a href="/facts/S2CID_(identifier)/ldJsHa2Y">S2CID</a> <a href="https://api.semanticscholar.org/CorpusID:73643153">73643153</a>.</li> <li>van Leeuwen, H.; Maassen, H. (1995). <a href="http://www.math.ru.nl/~maassen/preps/qGauss.pdf">"A <i>q</i> deformation of the Gauss distribution"</a> (PDF). <i><a href="/facts/Journal_of_Mathematical_Physics/hfGFBt5M">Journal of Mathematical Physics</a></i>. 36 (9): 4743. <a href="/facts/Bibcode_(identifier)/9HtdQSGB">Bibcode</a>:<a href="https://ui.adsabs.harvard.edu/abs/1995JMP....36.4743V">1995JMP....36.4743V</a>. <a href="/facts/CiteSeerX_(identifier)/SceDmd3c">CiteSeerX</a> <a href="https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.24.6957">10.1.1.24.6957</a>. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1063%2F1.530917">10.1063/1.530917</a>. <a href="/facts/Hdl_(identifier)/rdebSxmC">hdl</a>:<a href="https://hdl.handle.net/2066%2F141604">2066/141604</a>. <a href="/facts/S2CID_(identifier)/ldJsHa2Y">S2CID</a> <a href="https://api.semanticscholar.org/CorpusID:13934946">13934946</a>.</li> <li>Exton, H. (1983), <i>q-Hypergeometric Functions and Applications</i>, New York: Halstead Press, Chichester: Ellis Horwood, 1983, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0853124914, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0470274530, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0470274538</li></ul>