q-gamma function

In q-analog theory, the -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

when , and

if . Here is the infinite q-Pochhammer symbol. The -gamma function satisfies the functional equation

In addition, the -gamma function satisfies the q-analog of the Bohr–Mollerup theorem, which was found by Richard Askey (Askey (1978)).
For non-negative integers n,

where is the q-factorial function. Thus the -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

There is a simple proof of this limit by Gosper. See the appendix of (Andrews (1986)).

Transformation properties

The -gamma function satisfies the q-analog of the Gauss multiplication formula (Gasper & Rahman (2004)):

Integral representation

The -gamma function has the following integral representation (Ismail (1981)):

Stirling formula

Moak obtained the following q-analogue of the Stirling formula (see Moak (1984)):

where , denotes the Heaviside step function, stands for the Bernoulli number, is the dilogarithm, and is a polynomial of degree satisfying

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 . With this restriction

El Bachraoui considered the case and proved that

Special values

The following special values are known.[1]

These are the analogues of the classical formula .

Moreover, the following analogues of the familiar identity hold true:

Matrix Version

Let be a complex square matrix and Positive-definite matrix. Then a q-gamma matrix function can be defined by q-integral:[2]

where 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, arXiv:0705.2802, Bibcode:2008JMAA..339.1313Z, doi:10.1016/j.jmaa.2007.08.006, S2CID 115163047
  • 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. (ed.), Approximation and computation a festschrift in honor of Walter Gautschi: Proceedings of the Purdue conference, December 2-5, 1993, vol. 119, Boston: Birkhäuser Verlag, pp. 309–323, arXiv:1301.1749, doi:10.1007/978-1-4684-7415-2_19, ISBN 978-1-4684-7415-2, S2CID 118563435

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. 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.
  • 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, 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
  • 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, vol. 66, American Mathematical Society
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