q-Pochhammer symbol

In mathematics, in the area of combinatorics, a q-Pochhammer symbol, also called a q-shifted factorial, is a q-analog[further explanation needed] of the Pochhammer symbol. It is defined as

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.

which extends the definition to negative integers n. Thus, for nonnegative n, one has

which is useful for some of the generating functions of partition functions.

The q-Pochhammer symbol is the subject of a number of q-series identities, particularly the infinite series expansions

Fridrikh Karpelevich found the following identity (see Olshanetsky and Rogov (1995) for the proof):

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:

By removing a triangular partition with n − 1 parts from such a partition, we are left with an arbitrary partition with at most n 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:

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

A product of negative integer q-brackets can be expressed in terms of the q-factorial as

From the q-factorials, one can move on to define the q-binomial coefficients, also known as the Gaussian binomial coefficients, as

One also obtains a q-analog of the gamma function, called the q-gamma function, and defined as

This converges to the usual gamma function as q approaches 1 from inside the unit disc. Note that

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.