q-gamma function

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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[edit]

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

Integral Representation[edit]

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

Stirling Formula[edit]

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

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

Raabe-type formulas[edit]

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[edit]

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:

References[edit]

  • 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, JSTOR 92601 
  • 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, MR 2128719 
  • Ismail, Mourad (1981), "The Basic Bessel Functions and Polynomials", SIAM Journal on Mathematical Analysis, 12 (3): 454–468 
  • 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 
  • 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 
Notes