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q-gamma function
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<h2 id="transformation-properties">Transformation properties</h2> <p>The q {\displaystyle q} -gamma function satisfies the q-analog of the Gauss multiplication formula (Gasper & Rahman (2004)): Γ q ( n x ) Γ r ( 1 / n ) Γ r ( 2 / n ) ⋯ Γ r ( ( n − 1 ) / n ) = ( 1 − q n 1 − q ) n x − 1 Γ r ( x ) Γ r ( x + 1 / n ) ⋯ Γ r ( x + ( n − 1 ) / n ) , r = q n . {\displaystyle \Gamma _{q}(nx)\Gamma _{r}(1/n)\Gamma _{r}(2/n)\cdots \Gamma _{r}((n-1)/n)=\left({\frac {1-q^{n}}{1-q}}\right)^{nx-1}\Gamma _{r}(x)\Gamma _{r}(x+1/n)\cdots \Gamma _{r}(x+(n-1)/n),\ r=q^{n}.} </p> <h3>Integral representation</h3> <p>The q {\displaystyle q} -gamma function has the following integral representation (<a href="/facts/Mourad_E._H._Ismail/rKHL37ak">Ismail</a> (1981)): 1 Γ q ( z ) = sin ( π z ) π ∫ 0 ∞ t − z d t ( − t ( 1 − q ) ; q ) ∞ . {\displaystyle {\frac {1}{\Gamma _{q}(z)}}={\frac {\sin(\pi z)}{\pi }}\int _{0}^{\infty }{\frac {t^{-z}\mathrm {d} t}{(-t(1-q);q)_{\infty }}}.} </p> <h3>Stirling formula</h3> <p>Moak obtained the following q-analogue of the Stirling formula (see Moak (1984)): log Γ q ( x ) ∼ ( x − 1 / 2 ) log [ x ] q + L i 2 ( 1 − q x ) log q + C q ^ + 1 2 H ( q − 1 ) log q + ∑ k = 1 ∞ B 2 k ( 2 k ) ! ( log q ^ q ^ x − 1 ) 2 k − 1 q ^ x p 2 k − 3 ( q ^ x ) , x → ∞ , {\displaystyle \log \Gamma _{q}(x)\sim (x-1/2)\log[x]_{q}+{\frac {\mathrm {Li} _{2}(1-q^{x})}{\log q}}+C_{\hat {q}}+{\frac {1}{2}}H(q-1)\log q+\sum _{k=1}^{\infty }{\frac {B_{2k}}{(2k)!}}\left({\frac {\log {\hat {q}}}{{\hat {q}}^{x}-1}}\right)^{2k-1}{\hat {q}}^{x}p_{2k-3}({\hat {q}}^{x}),\ x\to \infty ,} q ^ = { q i f 0 < q ≤ 1 1 / q i f q ≥ 1 } , {\displaystyle {\hat {q}}=\left\{{\begin{aligned}q\quad \mathrm {if} \ &0<q\leq 1\\1/q\quad \mathrm {if} \ &q\geq 1\end{aligned}}\right\},} C q = 1 2 log ( 2 π ) + 1 2 log ( q − 1 log q ) − 1 24 log q + log ∑ m = − ∞ ∞ ( r m ( 6 m + 1 ) − r ( 3 m + 1 ) ( 2 m + 1 ) ) , {\displaystyle C_{q}={\frac {1}{2}}\log(2\pi )+{\frac {1}{2}}\log \left({\frac {q-1}{\log q}}\right)-{\frac {1}{24}}\log q+\log \sum _{m=-\infty }^{\infty }\left(r^{m(6m+1)}-r^{(3m+1)(2m+1)}\right),} where r = exp ( 4 π 2 / log q ) {\displaystyle r=\exp(4\pi ^{2}/\log q)} , H {\displaystyle H} denotes the <a href="/facts/Heaviside_step_function/bgsUtxgP">Heaviside step function</a>, B k {\displaystyle B_{k}} stands for the <a href="/facts/Bernoulli_number/RH0mblhs">Bernoulli number</a>, L i 2 ( z ) {\displaystyle \mathrm {Li} _{2}(z)} is the dilogarithm, and p k {\displaystyle p_{k}} is a polynomial of degree k {\displaystyle k} satisfying p k ( z ) = z ( 1 − z ) p k − 1 ′ ( z ) + ( k z + 1 ) p k − 1 ( z ) , p 0 = p − 1 = 1 , k = 1 , 2 , ⋯ . {\displaystyle p_{k}(z)=z(1-z)p'_{k-1}(z)+(kz+1)p_{k-1}(z),p_{0}=p_{-1}=1,k=1,2,\cdots .} </p> <h2 id="raabe-type-formulas">Raabe-type formulas</h2> <p>Due to I. Mező, the q-analogue of the <a href="/facts/Gamma_function/SjGQcnyw">Raabe formula</a> exists, at least if we use the q {\displaystyle q} -gamma function when | q | > 1 {\displaystyle |q|>1} . With this restriction, ∫ 0 1 log Γ q ( x ) d x = ζ ( 2 ) log q + log q − 1 q 6 + log ( q − 1 ; q − 1 ) ∞ ( q > 1 ) . {\displaystyle \int _{0}^{1}\log \Gamma _{q}(x)dx={\frac {\zeta (2)}{\log q}}+\log {\sqrt {\frac {q-1}{\sqrt[{6}]{q}}}}+\log(q^{-1};q^{-1})_{\infty }\quad (q>1).} El Bachraoui considered the case 0 < q < 1 {\displaystyle 0<q<1} and proved that ∫ 0 1 log Γ q ( x ) d x = 1 2 log ( 1 − q ) − ζ ( 2 ) log q + log ( q ; q ) ∞ ( 0 < q < 1 ) . {\displaystyle \int _{0}^{1}\log \Gamma _{q}(x)dx={\frac {1}{2}}\log(1-q)-{\frac {\zeta (2)}{\log q}}+\log(q;q)_{\infty }\quad (0<q<1).} </p> <h2 id="special-values">Special values</h2> <p>The following special values are known.<a class="footnote-ref" id="fnref:1" href="#fn:1"><sup>1</sup></a> Γ e − π ( 1 2 ) = e − 7 π / 16 e π − 1 1 + 2 4 2 15 / 16 π 3 / 4 Γ ( 1 4 ) , {\displaystyle \Gamma _{e^{-\pi }}\left({\frac {1}{2}}\right)={\frac {e^{-7\pi /16}{\sqrt {e^{\pi }-1}}{\sqrt[{4}]{1+{\sqrt {2}}}}}{2^{15/16}\pi ^{3/4}}}\,\Gamma \left({\frac {1}{4}}\right),} Γ e − 2 π ( 1 2 ) = e − 7 π / 8 e 2 π − 1 2 9 / 8 π 3 / 4 Γ ( 1 4 ) , {\displaystyle \Gamma _{e^{-2\pi }}\left({\frac {1}{2}}\right)={\frac {e^{-7\pi /8}{\sqrt {e^{2\pi }-1}}}{2^{9/8}\pi ^{3/4}}}\,\Gamma \left({\frac {1}{4}}\right),} Γ e − 4 π ( 1 2 ) = e − 7 π / 4 e 4 π − 1 2 7 / 4 π 3 / 4 Γ ( 1 4 ) , {\displaystyle \Gamma _{e^{-4\pi }}\left({\frac {1}{2}}\right)={\frac {e^{-7\pi /4}{\sqrt {e^{4\pi }-1}}}{2^{7/4}\pi ^{3/4}}}\,\Gamma \left({\frac {1}{4}}\right),} Γ e − 8 π ( 1 2 ) = e − 7 π / 2 e 8 π − 1 2 9 / 4 π 3 / 4 1 + 2 Γ ( 1 4 ) . {\displaystyle \Gamma _{e^{-8\pi }}\left({\frac {1}{2}}\right)={\frac {e^{-7\pi /2}{\sqrt {e^{8\pi }-1}}}{2^{9/4}\pi ^{3/4}{\sqrt {1+{\sqrt {2}}}}}}\,\Gamma \left({\frac {1}{4}}\right).} These are the analogues of the classical formula Γ ( 1 2 ) = π {\displaystyle \Gamma \left({\frac {1}{2}}\right)={\sqrt {\pi }}} . </p><p>Moreover, the following analogues of the familiar identity Γ ( 1 4 ) Γ ( 3 4 ) = 2 π {\displaystyle \Gamma \left({\frac {1}{4}}\right)\Gamma \left({\frac {3}{4}}\right)={\sqrt {2}}\pi } hold true: Γ e − 2 π ( 1 4 ) Γ e − 2 π ( 3 4 ) = e − 29 π / 16 ( e 2 π − 1 ) 1 + 2 4 2 33 / 16 π 3 / 2 Γ ( 1 4 ) 2 , {\displaystyle \Gamma _{e^{-2\pi }}\left({\frac {1}{4}}\right)\Gamma _{e^{-2\pi }}\left({\frac {3}{4}}\right)={\frac {e^{-29\pi /16}\left(e^{2\pi }-1\right){\sqrt[{4}]{1+{\sqrt {2}}}}}{2^{33/16}\pi ^{3/2}}}\,\Gamma \left({\frac {1}{4}}\right)^{2},} Γ e − 4 π ( 1 4 ) Γ e − 4 π ( 3 4 ) = e − 29 π / 8 ( e 4 π − 1 ) 2 23 / 8 π 3 / 2 Γ ( 1 4 ) 2 , {\displaystyle \Gamma _{e^{-4\pi }}\left({\frac {1}{4}}\right)\Gamma _{e^{-4\pi }}\left({\frac {3}{4}}\right)={\frac {e^{-29\pi /8}\left(e^{4\pi }-1\right)}{2^{23/8}\pi ^{3/2}}}\,\Gamma \left({\frac {1}{4}}\right)^{2},} Γ e − 8 π ( 1 4 ) Γ e − 8 π ( 3 4 ) = e − 29 π / 4 ( e 8 π − 1 ) 16 π 3 / 2 1 + 2 Γ ( 1 4 ) 2 . {\displaystyle \Gamma _{e^{-8\pi }}\left({\frac {1}{4}}\right)\Gamma _{e^{-8\pi }}\left({\frac {3}{4}}\right)={\frac {e^{-29\pi /4}\left(e^{8\pi }-1\right)}{16\pi ^{3/2}{\sqrt {1+{\sqrt {2}}}}}}\,\Gamma \left({\frac {1}{4}}\right)^{2}.} </p> <h2 id="matrix-version">Matrix version</h2> <p>Let A {\displaystyle A} be a complex square matrix and <a href="/facts/Positive-definite_matrix/Hbr6AuPS">positive-definite matrix</a>. Then a q {\displaystyle q} -gamma matrix function can be defined by q {\displaystyle q} -integral:<a class="footnote-ref" id="fnref:2" href="#fn:2"><sup>2</sup></a> Γ q ( A ) := ∫ 0 1 1 − q t A − I E q ( − q t ) d q t {\displaystyle \Gamma _{q}(A):=\int _{0}^{\frac {1}{1-q}}t^{A-I}E_{q}(-qt)\mathrm {d} _{q}t} where E q {\displaystyle E_{q}} is the <a href="/facts/Q-exponential/t9gJxytF">q-exponential</a> function. </p> <h2 id="other-q-gamma-functions">Other <i>q</i>-gamma functions</h2> <p>For other q {\displaystyle q} -gamma functions, see Yamasaki 2006.<a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a> </p> <h2 id="numerical-computation">Numerical computation</h2> <p>An iterative algorithm to compute the q-gamma function was proposed by Gabutti and Allasia.<a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a> </p> <h2 id="further-reading">Further reading</h2> <ul><li>Zhang, Ruiming (2007), "On asymptotics of <i>q</i>-gamma functions", <i>Journal of Mathematical Analysis and Applications</i>, 339 (2): 1313–1321, <a href="/facts/ArXiv_(identifier)/H6EtgnBe">arXiv</a>:<a href="https://arxiv.org/abs/0705.2802">0705.2802</a>, <a href="/facts/Bibcode_(identifier)/9HtdQSGB">Bibcode</a>:<a href="https://ui.adsabs.harvard.edu/abs/2008JMAA..339.1313Z">2008JMAA..339.1313Z</a>, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1016%2Fj.jmaa.2007.08.006">10.1016/j.jmaa.2007.08.006</a>, <a href="/facts/S2CID_(identifier)/ldJsHa2Y">S2CID</a> <a href="https://api.semanticscholar.org/CorpusID:115163047">115163047</a></li> <li>Zhang, Ruiming (2010), "On asymptotics of Γ<i>q</i>(<i>z</i>) as <i>q</i> approaching 1", <a href="/facts/ArXiv_(identifier)/H6EtgnBe">arXiv</a>:<a href="https://arxiv.org/abs/1011.0720">1011.0720</a> [<a href="https://arxiv.org/archive/math.CA">math.CA</a>]</li> <li>Ismail, Mourad E. H.; Muldoon, Martin E. (1994), "Inequalities and monotonicity properties for gamma and <i>q</i>-gamma functions", in Zahar, R. V. M. (ed.), <i>Approximation and computation a festschrift in honor of Walter Gautschi: Proceedings of the Purdue conference, December 2-5, 1993</i>, vol. 119, Boston: Birkhäuser Verlag, pp. 309–323, <a href="/facts/ArXiv_(identifier)/H6EtgnBe">arXiv</a>:<a href="https://arxiv.org/abs/1301.1749">1301.1749</a>, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1007%2F978-1-4684-7415-2_19">10.1007/978-1-4684-7415-2_19</a>, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-1-4684-7415-2, <a href="/facts/S2CID_(identifier)/ldJsHa2Y">S2CID</a> <a href="https://api.semanticscholar.org/CorpusID:118563435">118563435</a></li></ul> <ul><li>Jackson, F. H. (1905), "The Basic Gamma-Function and the Elliptic Functions", <i>Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character</i>, 76 (508), The Royal Society: 127–144, <a href="/facts/Bibcode_(identifier)/9HtdQSGB">Bibcode</a>:<a href="https://ui.adsabs.harvard.edu/abs/1905RSPSA..76..127J">1905RSPSA..76..127J</a>, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1098%2Frspa.1905.0011">10.1098/rspa.1905.0011</a>, <a href="/facts/ISSN_(identifier)/DPAflDvU">ISSN</a> <a href="https://search.worldcat.org/issn/0950-1207">0950-1207</a>, <a href="/facts/JSTOR_(identifier)/YTeVmaJ7">JSTOR</a> <a href="https://www.jstor.org/stable/92601">92601</a></li> <li>Gasper, George; Rahman, Mizan (2004), <i>Basic hypergeometric series</i>, Encyclopedia of Mathematics and its Applications, vol. 96 (2nd ed.), <a href="/facts/Cambridge_University_Press/fgEBSSRq">Cambridge University Press</a>, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-521-83357-8, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=2128719">2128719</a></li> <li>Ismail, Mourad (1981), "The Basic Bessel Functions and Polynomials", <i>SIAM Journal on Mathematical Analysis</i>, 12 (3): 454–468, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1137%2F0512038">10.1137/0512038</a></li> <li>Moak, Daniel S. (1984), "The Q-analogue of Stirling's formula", <i>Rocky Mountain J. Math.</i>, 14 (2): 403–414, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1216%2FRMJ-1984-14-2-403">10.1216/RMJ-1984-14-2-403</a></li> <li>Mező, István (2012), "A q-Raabe formula and an integral of the fourth Jacobi theta function", <i>Journal of Number Theory</i>, 133 (2): 692–704, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1016%2Fj.jnt.2012.08.025">10.1016/j.jnt.2012.08.025</a>, <a href="/facts/Hdl_(identifier)/rdebSxmC">hdl</a>:<a href="https://hdl.handle.net/2437%2F166217">2437/166217</a></li> <li>El Bachraoui, Mohamed (2017), "Short proofs for q-Raabe formula and integrals for Jacobi theta functions", <i>Journal of Number Theory</i>, 173 (2): 614–620, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1016%2Fj.jnt.2016.09.028">10.1016/j.jnt.2016.09.028</a></li> <li>Askey, Richard (1978), "The q-gamma and q-beta functions.", <i>Applicable Analysis</i>, 8 (2): 125–141, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1080%2F00036817808839221">10.1080/00036817808839221</a></li> <li>Andrews, George E. (1986), <i>q-Series: Their development and application in analysis, number theory, combinatorics, physics, and computer algebra.</i>, Regional Conference Series in Mathematics, vol. 66, <a href="/facts/American_Mathematical_Society/fr5gotCt">American Mathematical Society</a></li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>Mező, István (2011), "Several special values of Jacobi theta functions", arXiv:1106.1042 [math.NT] <a href="/wiki/ArXiv_(identifier)" target="_blank">/wiki/ArXiv_(identifier)</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>Salem, Ahmed (June 2012). "On a q-gamma and a q-beta matrix functions". Linear and Multilinear Algebra. 60 (6): 683–696. doi:10.1080/03081087.2011.627562. S2CID 123011613. <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> <li id="fn:3"><p>Yamasaki, Yoshinori (December 2006). "On q-Analogues of the Barnes Multiple Zeta Functions". Tokyo Journal of Mathematics. 29 (2): 413–427. arXiv:math/0412067. doi:10.3836/tjm/1170348176. MR 2284981. S2CID 14082358. Zbl 1192.11060. <a href="/wiki/ArXiv_(identifier)" target="_blank">/wiki/ArXiv_(identifier)</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li> <li id="fn:4"><p>Gabutti, Bruno; Allasia, Giampietro (17 September 2008). "Evaluation of q-gamma function and q-analogues by iterative algorithms". Numerical Algorithms. 49 (1–4): 159–168. Bibcode:2008NuAlg..49..159G. doi:10.1007/s11075-008-9196-5. S2CID 6314057. <a href="/wiki/Bibcode_(identifier)" target="_blank">/wiki/Bibcode_(identifier)</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li> </ol>