In
q-analog
In mathematics, a ''q''-analog of a theorem, identity or expression is a generalization involving a new parameter ''q'' that returns the original theorem, identity or expression in the limit as . Typically, mathematicians are interested in ''q' ...
theory, the
-gamma function, or basic gamma function, is a generalization of the ordinary
gamma function
In mathematics, the gamma function (represented by , the capital letter gamma from the Greek alphabet) is one commonly used extension of the factorial function to complex numbers. The gamma function is defined for all complex numbers except ...
closely related to the
double gamma function
In mathematics, the multiple gamma function \Gamma_N is a generalization of the Euler gamma function and the Barnes G-function. The double gamma function was studied by . At the end of this paper he mentioned the existence of multiple gamma funct ...
. It was introduced by . It is given by
when
, and
if
. Here
is the infinite
q-Pochhammer symbol
In mathematical area of combinatorics, the ''q''-Pochhammer symbol, also called the ''q''-shifted factorial, is the product
(a;q)_n = \prod_^ (1-aq^k)=(1-a)(1-aq)(1-aq^2)\cdots(1-aq^),
with (a;q)_0 = 1.
It is a ''q''-analog of the Pochhammer symb ...
. The
-gamma function satisfies the functional equation
In addition, the
-gamma function satisfies the q-analog of the
Bohr–Mollerup theorem
In mathematical analysis, the Bohr–Mollerup theorem is a theorem proved by the Danish mathematicians Harald Bohr and Johannes Mollerup. The theorem characterizes the gamma function, defined for by
:\Gamma(x)=\int_0^\infty t^ e^\,dt
as the '' ...
, which was found by
Richard Askey
Richard Allen Askey (4 June 1933 – 9 October 2019) was an American mathematician, known for his expertise in the area of special functions. The Askey–Wilson polynomials (introduced by him in 1984 together with James A. Wilson) are on the t ...
().
For non-negative integers ''n'',
where
is the
q-factorial
In mathematical area of combinatorics, the ''q''-Pochhammer symbol, also called the ''q''-shifted factorial, is the product
(a;q)_n = \prod_^ (1-aq^k)=(1-a)(1-aq)(1-aq^2)\cdots(1-aq^),
with (a;q)_0 = 1.
It is a ''q''-analog of the Pochhammer sym ...
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 ().
Transformation properties
The
-gamma function satisfies the q-analog of the Gauss multiplication formula ():
Integral representation
The
-gamma function has the following integral representation ():
Stirling formula
Moak obtained the following q-analogue of the Stirling formula (see ):
where
,
denotes the
Heaviside step function
The Heaviside step function, or the unit step function, usually denoted by or (but sometimes , or ), is a step function, named after Oliver Heaviside (1850–1925), the value of which is zero for negative arguments and one for positive argume ...
,
stands for the
Bernoulli number
In mathematics, the Bernoulli numbers are a sequence of rational numbers which occur frequently in analysis. The Bernoulli numbers appear in (and can be defined by) the Taylor series expansions of the tangent and hyperbolic tangent functions, ...
,
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