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q-gamma function

Q-analog of the gamma function

In q-analog theory, the q {\displaystyle q} -gamma function, or basic gamma function, is a generalization of the ordinary gamma function closely related to the double gamma function. It was introduced by Jackson (1905). It is given by Γ q ( x ) = ( 1 − q ) 1 − x ∏ n = 0 ∞ 1 − q n + 1 1 − q n + x = ( 1 − q ) 1 − x ( q ; q ) ∞ ( q x ; q ) ∞ {\displaystyle \Gamma _{q}(x)=(1-q)^{1-x}\prod _{n=0}^{\infty }{\frac {1-q^{n+1}}{1-q^{n+x}}}=(1-q)^{1-x}\,{\frac {(q;q)_{\infty }}{(q^{x};q)_{\infty }}}} when | q | < 1 {\displaystyle |q|<1} , and Γ q ( x ) = ( q − 1 ; q − 1 ) ∞ ( q − x ; q − 1 ) ∞ ( q − 1 ) 1 − x q ( x 2 ) {\displaystyle \Gamma _{q}(x)={\frac {(q^{-1};q^{-1})_{\infty }}{(q^{-x};q^{-1})_{\infty }}}(q-1)^{1-x}q^{\binom {x}{2}}} if | q | > 1 {\displaystyle |q|>1} . Here ( ⋅ ; ⋅ ) ∞ {\displaystyle (\cdot ;\cdot )_{\infty }} is the infinite q {\displaystyle q} -Pochhammer symbol. The q {\displaystyle q} -gamma function satisfies the functional equation Γ q ( x + 1 ) = 1 − q x 1 − q Γ q ( x ) = [ x ] q Γ q ( x ) {\displaystyle \Gamma _{q}(x+1)={\frac {1-q^{x}}{1-q}}\Gamma _{q}(x)=[x]_{q}\Gamma _{q}(x)} In addition, the q {\displaystyle q} -gamma function satisfies the q-analog of the Bohr–Mollerup theorem, which was found by Richard Askey (Askey (1978)).

For non-negative integers n {\displaystyle n} , Γ q ( n ) = [ n − 1 ] q ! {\displaystyle \Gamma _{q}(n)=[n-1]_{q}!} where [ ⋅ ] q {\displaystyle [\cdot ]_{q}} is the q {\displaystyle q} -factorial function. Thus the q {\displaystyle q} -gamma function can be considered as an extension of the q {\displaystyle q} -factorial function to the real numbers.

The relation to the ordinary gamma function is made explicit in the limit lim q → 1 ± Γ q ( x ) = Γ ( x ) . {\displaystyle \lim _{q\to 1\pm }\Gamma _{q}(x)=\Gamma (x).} There is a simple proof of this limit by Gosper. See the appendix of (Andrews (1986)).

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Transformation properties

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}.}

Integral representation

The q {\displaystyle q} -gamma function has the following integral representation (Ismail (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 }}}.}

Stirling formula

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 Heaviside step function, B k {\displaystyle B_{k}} stands for the Bernoulli number, 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 .}

Raabe-type formulas

Due to I. Mező, the q-analogue of the Raabe formula 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).}

Special values

The following special values are known.¹ 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 }}} .

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}.}

Matrix version

Let A {\displaystyle A} be a complex square matrix and positive-definite matrix. Then a q {\displaystyle q} -gamma matrix function can be defined by q {\displaystyle q} -integral:² 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 q-exponential function.

Other q-gamma functions

For other q {\displaystyle q} -gamma functions, see Yamasaki 2006.³

Numerical computation

An iterative algorithm to compute the q-gamma function was proposed by Gabutti and Allasia.⁴

References

Mező, István (2011), "Several special values of Jacobi theta functions", arXiv:1106.1042 [math.NT] /wiki/ArXiv_(identifier) ↩
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. /wiki/Doi_(identifier) ↩
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. /wiki/ArXiv_(identifier) ↩
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. /wiki/Bibcode_(identifier) ↩

Transformation properties

Integral representation

Stirling formula

Raabe-type formulas

Special values

Matrix version

Other q-gamma functions

Numerical computation

Further reading

References