The q-gamma function is defined by
This is a q-analogue of the well-known gamma function since we have
Note that the q-gamma function satisfies the functional equation
which is a q-extension of the well-known functional equation
for the ordinary gamma function. For nonintegral values of z this ordinary gamma function also satisfies the relation
which can be used to show that
This limit can be used to show that the orthogonality relation (3.27.2) for the Stieltjes-Wigert polynomials follows from the orthogonality relation (3.21.2) for the q-Laguerre polynomials.
The q-binomial coefficient is defined by
where n denotes a nonnegative integer.
This definition can be generalized in the following way. For arbitrary
complex 
we have
Or more general for all complex 
and 
we have
For instance this implies that
Note that
For integer values of the parameter 
we have
and when the parameter 
is an integer too we may write
This latter formula can be used to show that
This can be used to write the generating functions (1.8.46) and (1.8.52) for the Chebyshev polynomials of the first and the second kind in the following form :
and
respectively.
Finally we remark that
