Q-gamma function

Short description: Function in q-analog theory

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 Γq(x)=(1−q)1−x∏n=0∞1−qn+11−qn+x=(1−q)1−x(q;q)∞(qx;q)∞ when |q|<1, and Γq(x)=(q−1;q−1)∞(q−x;q−1)∞(q−1)1−xq(x2) if |q|>1. Here (⋅;⋅)∞ is the infinite q-Pochhammer symbol. The q-gamma function satisfies the functional equation Γq(x+1)=1−qx1−qΓq(x)=[x]qΓq(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, Γq(n)=[n−1]q! where [⋅]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 limq->1±Γq(x)=Γ(x). There is a simple proof of this limit by Gosper. See the appendix of (Andrews (1986)).

## Contents

* 1 Transformation properties
* 1.1 Integral representation
* 1.2 Stirling formula
* 2 Raabe-type formulas
* 3 Special values
* 4 Matrix version
* 5 Other q-gamma functions
* 6 Numerical computation
* 7 Further reading
* 8 References

## Transformation properties

The q-gamma function satisfies the q-analog of the Gauss multiplication formula ((Gasper Rahman)): Γq(nx)Γr(1/n)Γr(2/n)⋯Γr((n−1)/n)=(1−qn1−q)nx−1Γr(x)Γr(x+1/n)⋯Γr(x+(n−1)/n), r=qn.

### Integral representation

The q-gamma function has the following integral representation (Ismail (1981)): 1Γq(z)=sin⁡(πz)π∫0∞t−zdt(−t(1−q);q)∞.

### Stirling formula

Moak obtained the following q-analogue of the Stirling formula (see (Moak 1984)): log⁡Γq(x)∼(x−1/2)log⁡[x]q+Li2(1−qx)log⁡q+Cq^+12H(q−1)log⁡q+∑k=1∞B2k(2k)!(log⁡q^q^x−1)2k−1q^xp2k−3(q^x), x->∞, q^={qif 0<q≤11/qif q≥1}, Cq=12log⁡(2π)+12log⁡(q−1log⁡q)−124log⁡q+log⁡∑m=−∞∞(rm(6m+1)−r(3m+1)(2m+1)), where r=exp⁡(4π2/log⁡q), H denotes the Heaviside step function, Bk stands for the Bernoulli number, Li2(z) is the dilogarithm, and pk is a polynomial of degree k satisfying pk(z)=z(1−z)p'k−1(z)+(kz+1)pk−1(z),p0=p−1=1,k=1,2,⋯.

## 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, ∫01log⁡Γq(x)dx=ζ(2)log⁡q+log⁡q−1q6+log⁡(q−1;q−1)∞(q>1). El Bachraoui considered the case 0<q<1 and proved that ∫01log⁡Γq(x)dx=12log⁡(1−q)−ζ(2)log⁡q+log⁡(q;q)∞(0<q<1).

## Special values

The following special values are known.[1] Γe−π(12)=e−7π/16eπ−11+24215/16π3/4Γ(14), Γe−2π(12)=e−7π/8e2π−129/8π3/4Γ(14), Γe−4π(12)=e−7π/4e4π−127/4π3/4Γ(14), Γe−8π(12)=e−7π/2e8π−129/4π3/41+2Γ(14). These are the analogues of the classical formula Γ(12)=π.

Moreover, the following analogues of the familiar identity Γ(14)Γ(34)=2π hold true: Γe−2π(14)Γe−2π(34)=e−29π/16(e2π−1)1+24233/16π3/2Γ(14)2, Γe−4π(14)Γe−4π(34)=e−29π/8(e4π−1)223/8π3/2Γ(14)2, Γe−8π(14)Γe−8π(34)=e−29π/4(e8π−1)16π3/21+2Γ(14)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] Γq(A):=∫011−qtA−IEq(−qt)dqt where Eq 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]

## Further reading

* Zhang, Ruiming (2007), "On asymptotics of q-gamma functions", Journal of Mathematical Analysis and Applications 339 (2): 1313–1321, doi:10.1016/j.jmaa.2007.08.006, Bibcode: 2008JMAA..339.1313Z
* Zhang, Ruiming (2010), "On asymptotics of Γq(z) as q approaching 1", arXiv:1011.0720 [math.CA]
* Ismail, Mourad E. H.; Muldoon, Martin E. (1994). "Inequalities and monotonicity properties for gamma and q-gamma functions". in Zahar, R. V. M.. Approximation and computation a festschrift in honor of Walter Gautschi: Proceedings of the Purdue conference, December 2-5, 1993. 119. Boston: Birkhäuser Verlag. pp. 309–323. doi:10.1007/978-1-4684-7415-2_19. ISBN 978-1-4684-7415-2.

## References

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.
3. ↑ Yamasaki, Yoshinori (December 2006). "On q-Analogues of the Barnes Multiple Zeta Functions". Tokyo Journal of Mathematics 29 (2): 413–427. doi:10.3836/tjm/1170348176.
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. doi:10.1007/s11075-008-9196-5. Bibcode: 2008NuAlg..49..159G.

* 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 (The Royal Society) 76 (508): 127–144, doi:10.1098/rspa.1905.0011, ISSN 0950-1207, Bibcode: 1905RSPSA..76..127J
* Gasper, George; Rahman, Mizan (2004), Basic hypergeometric series, Encyclopedia of Mathematics and its Applications, 96 (2nd ed.), Cambridge University Press, ISBN 978-0-521-83357-8
* 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
* El Bachraoui, Mohamed (2017), "Short proofs for q-Raabe formula and integrals for Jacobi theta functions", Journal of Number Theory 173 (2): 614–620, 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, 66, American Mathematical Society

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