Q-Pochhammer symbol
In mathematics, in the area of combinatorics, a q-Pochhammer symbol, also called a q-shifted factorial, is a q-analog of the Pochhammer symbol. It is defined as
with
by definition. The q-Pochhammer symbol is a major building block in the construction of q-analogs; for instance, in the theory of basic hypergeometric series, it plays the role that the ordinary Pochhammer symbol plays in the theory of generalized hypergeometric series.
Unlike the ordinary Pochhammer symbol, the q-Pochhammer symbol can be extended to an infinite product:
This is an analytic function of q in the interior of the unit disk, and can also be considered as a formal power series in q. The special case
is known as Euler's function, and is important in combinatorics, number theory, and the theory of modular forms.
The finite product can be expressed in terms of the infinite product:
which extends the definition to negative integers n. Thus, for nonnegative n, one has
and
The q-Pochhammer symbol is the subject of a number of q-series identities, particularly the infinite series expansions
and
which are both special cases of the q-binomial theorem:
The q-Pochhammer symbol is closely related to the enumerative combinatorics of partitions. The coefficient of in
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