q-gamma function

In q-analog theory, the $q$ -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 $Γ_{q} (x) = (1 - q)^{1 - x} \prod_{n = 0}^{\infty} \frac{1 - q^{n + 1}}{1 - q^{n + x}} = (1 - q)^{1 - x} \frac{(q; q)_{\infty}}{(q^{x}; q)_{\infty}}$ when $| q | < 1$ , and $Γ_{q} (x) = \frac{(q^{- 1}; q^{- 1})_{\infty}}{(q^{- x}; q^{- 1})_{\infty}} (q - 1)^{1 - x} q^{(\binom{x}{2})}$ if $| q | > 1$ . Here $(\cdot; \cdot)_{\infty}$ is the infinite $q$ -Pochhammer symbol. The $q$ -gamma function satisfies the functional equation $Γ_{q} (x + 1) = \frac{1 - q^{x}}{1 - q} Γ_{q} (x) = [x]_{q} Γ_{q} (x)$ In addition, the $q$ -gamma function satisfies the q-analog of the Bohr–Mollerup theorem, which was found by Richard Askey (Askey (1978)).

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

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

Transformation properties

The $q$ -gamma function satisfies the q-analog of the Gauss multiplication formula (Gasper & Rahman (2004)): $Γ_{q} (n x) Γ_{r} (1 / n) Γ_{r} (2 / n) \dots Γ_{r} ((n - 1) / n) = {(\frac{1 - q^{n}}{1 - q})}^{n x - 1} Γ_{r} (x) Γ_{r} (x + 1 / n) \dots Γ_{r} (x + (n - 1) / n), r = q^{n} .$

Integral representation

The $q$ -gamma function has the following integral representation (Ismail (1981)): $\frac{1}{Γ_{q} (z)} = \frac{\sin (π z)}{π} \int_{0}^{\infty} \frac{t^{- z} d t}{(- t (1 - q); q)_{\infty}} .$

Stirling formula

Moak obtained the following q-analogue of the Stirling formula (see Moak (1984)): $\log Γ_{q} (x) \sim (x - 1 / 2) \log [x]_{q} + \frac{{L i}_{2} (1 - q^{x})}{\log q} + C_{\hat{q}} + \frac{1}{2} H (q - 1) \log q + \sum_{k = 1}^{\infty} \frac{B_{2 k}}{(2 k)!} {(\frac{\log \hat{q}}{{\hat{q}}^{x} - 1})}^{2 k - 1} {\hat{q}}^{x} p_{2 k - 3} ({\hat{q}}^{x}), x \to \infty,$ $\hat{q} = {\begin{matrix} q i f & 0 < q \leq 1 \\ 1 / q i f & q \geq 1 \end{matrix}},$ $C_{q} = \frac{1}{2} \log (2 π) + \frac{1}{2} \log (\frac{q - 1}{\log q}) - \frac{1}{24} \log q + \log \sum_{m = - \infty}^{\infty} (r^{m (6 m + 1)} - r^{(3 m + 1) (2 m + 1)}),$ where $r = \exp (4 π^{2} / \log q)$ , $H$ denotes the Heaviside step function, $B_{k}$ stands for the Bernoulli number, ${L i}_{2} (z)$ is the dilogarithm, and $p_{k}$ is a polynomial of degree $k$ satisfying $p_{k} (z) = z (1 - z) p'_{k - 1} (z) + (k z + 1) p_{k - 1} (z), p_{0} = p_{- 1} = 1, k = 1, 2, \dots .$

Raabe-type formulas

Due to I. Mező, the q-analogue of the Raabe formula exists, at least if we use the $q$ -gamma function when $| q | > 1$ . With this restriction, $\int_{0}^{1} \log Γ_{q} (x) d x = \frac{ζ (2)}{\log q} + \log \sqrt{\frac{q - 1}{\sqrt[6]{q}}} + \log (q^{- 1}; q^{- 1})_{\infty} (q > 1) .$ El Bachraoui considered the case $0 < q < 1$ and proved that $\int_{0}^{1} \log Γ_{q} (x) d x = \frac{1}{2} \log (1 - q) - \frac{ζ (2)}{\log q} + \log (q; q)_{\infty} (0 < q < 1) .$

Special values

The following special values are known.^[1] $Γ_{e^{- π}} (\frac{1}{2}) = \frac{e^{- 7 π / 16} \sqrt{e^{π} - 1} \sqrt[4]{1 + \sqrt{2}}}{2^{15 / 16} π^{3 / 4}} Γ (\frac{1}{4}),$ $Γ_{e^{- 2 π}} (\frac{1}{2}) = \frac{e^{- 7 π / 8} \sqrt{e^{2 π} - 1}}{2^{9 / 8} π^{3 / 4}} Γ (\frac{1}{4}),$ $Γ_{e^{- 4 π}} (\frac{1}{2}) = \frac{e^{- 7 π / 4} \sqrt{e^{4 π} - 1}}{2^{7 / 4} π^{3 / 4}} Γ (\frac{1}{4}),$ $Γ_{e^{- 8 π}} (\frac{1}{2}) = \frac{e^{- 7 π / 2} \sqrt{e^{8 π} - 1}}{2^{9 / 4} π^{3 / 4} \sqrt{1 + \sqrt{2}}} Γ (\frac{1}{4}) .$ These are the analogues of the classical formula $Γ (\frac{1}{2}) = \sqrt{π}$ .

Moreover, the following analogues of the familiar identity $Γ (\frac{1}{4}) Γ (\frac{3}{4}) = \sqrt{2} π$ hold true: $Γ_{e^{- 2 π}} (\frac{1}{4}) Γ_{e^{- 2 π}} (\frac{3}{4}) = \frac{e^{- 29 π / 16} (e^{2 π} - 1) \sqrt[4]{1 + \sqrt{2}}}{2^{33 / 16} π^{3 / 2}} Γ {(\frac{1}{4})}^{2},$ $Γ_{e^{- 4 π}} (\frac{1}{4}) Γ_{e^{- 4 π}} (\frac{3}{4}) = \frac{e^{- 29 π / 8} (e^{4 π} - 1)}{2^{23 / 8} π^{3 / 2}} Γ {(\frac{1}{4})}^{2},$ $Γ_{e^{- 8 π}} (\frac{1}{4}) Γ_{e^{- 8 π}} (\frac{3}{4}) = \frac{e^{- 29 π / 4} (e^{8 π} - 1)}{16 π^{3 / 2} \sqrt{1 + \sqrt{2}}} Γ {(\frac{1}{4})}^{2} .$

Matrix version

Let $A$ be a complex square matrix and positive-definite matrix. Then a $q$ -gamma matrix function can be defined by $q$ -integral:^[2] $Γ_{q} (A) := \int_{0}^{\frac{1}{1 - q}} t^{A - I} E_{q} (- q t) d_{q} t$ where $E_{q}$ is the q-exponential function.

Other q-gamma functions

For other $q$ -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]
↑ 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.
↑ 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.
↑ 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.

Jackson, F. H. (1905), "The Basic Gamma-Function and the Elliptic Functions", Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character, 76 (508), The Royal Society: 127–144, Bibcode:1905RSPSA..76..127J, doi:10.1098/rspa.1905.0011, ISSN 0950-1207, JSTOR 92601
Gasper, George; Rahman, Mizan (2004), Basic hypergeometric series, Encyclopedia of Mathematics and its Applications, vol. 96 (2nd ed.), Cambridge University Press, ISBN 978-0-521-83357-8, MR 2128719
Ismail, Mourad (1981), "The Basic Bessel Functions and Polynomials", SIAM Journal on Mathematical Analysis, 12 (3): 454–468, doi:10.1137/0512038
Moak, Daniel S. (1984), "The Q-analogue of Stirling's formula", Rocky Mountain J. Math., 14 (2): 403–414, doi:10.1216/RMJ-1984-14-2-403
Mező, István (2012), "A q-Raabe formula and an integral of the fourth Jacobi theta function", Journal of Number Theory, 133 (2): 692–704, doi:10.1016/j.jnt.2012.08.025, hdl:2437/166217
El Bachraoui, Mohamed (2017), "Short proofs for q-Raabe formula and integrals for Jacobi theta functions", Journal of Number Theory, 173 (2): 614–620, arXiv:1609.06895, doi:10.1016/j.jnt.2016.09.028
Askey, Richard (1978), "The q-gamma and q-beta functions.", Applicable Analysis, 8 (2): 125–141, doi:10.1080/00036817808839221
Andrews, George E. (1986), q-Series: Their development and application in analysis, number theory, combinatorics, physics, and computer algebra., Regional Conference Series in Mathematics, vol. 66, American Mathematical Society

[1] Mező, István (2011), "Several special values of Jacobi theta functions", arXiv:1106.1042 [math.NT]

[2] 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.

[3] 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.

[4] 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.

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