In the mathematical field of combinatorics, the q-Pochhammer symbol, also called the q-shifted factorial, is the product
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.
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.
which extends the definition to negative integers n. Thus, for nonnegative n, one has
The q-Pochhammer symbol is the subject of a number of q-series identities, particularly the infinite series expansions
which are both special cases of the q-binomial theorem:
Fridrikh Karpelevich found the following identity (see Olshanetsky and Rogov (1995) for the proof):
The q-Pochhammer symbol is closely related to the enumerative combinatorics of partitions.
Since, by conjugation of partitions, this is the same as the number of partitions of m into parts of size at most n, by identification of generating series we obtain the identity
is the number of partitions of m into n or n-1 distinct parts.
This gives a weight-preserving bijection between the set of partitions into n or n − 1 distinct parts and the set of pairs consisting of a triangular partition having n − 1 parts and a partition with at most n parts.
By identifying generating series, this leads to the identity
The q-binomial theorem itself can also be handled by a slightly more involved combinatorial argument of a similar flavor (see also the expansions given in the next subsection).
Since identities involving q-Pochhammer symbols so frequently involve products of many symbols, the standard convention is to write a product as a single symbol of multiple arguments:
A q-series is a series in which the coefficients are functions of q, typically expressions of
[2] Early results are due to Euler, Gauss, and Cauchy.
The systematic study begins with Eduard Heine (1843).
[3] The q-analog of n, also known as the q-bracket or q-number of n, is defined to be
From this one can define the q-analog of the factorial, the q-factorial, as
counts permutations of an n-element set S. Equivalently, it counts the number of sequences of nested sets
[4] By comparison, when q is a prime power and V is an n-dimensional vector space over the field with q elements, the q-analogue
[4] The preceding considerations suggest that one can regard a sequence of nested sets as a flag over a conjectural field with one element.
A product of negative integer q-brackets can be expressed in terms of the q-factorial as
where it is easy to see that the triangle of these coefficients is symmetric in the sense that for all
One can also see from the previous recurrence relations that the next variants of the
-binomial theorem are expanded in terms of these coefficients as follows:[5]
The coefficient above counts the number of flags
of subspaces in an n-dimensional vector space over the field with q elements such that
This converges to the usual gamma function as q approaches 1 from inside the unit disc.
for non-negative integer values of n. Alternatively, this may be taken as an extension of the q-factorial function to the real number system.