Q-Pochhammer symbol

In mathematical area of combinatorics, the q-Pochhammer symbol, also called the q-shifted factorial, is the product with It is a q-analog of the Pochhammer symbol , in the sense that 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 Alternatively, 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 and which are both special cases of the q-binomial theorem: Fridrikh Karpelevich found the following identity (see for the proof): The q-Pochhammer symbol is closely related to the enumerative combinatorics of partitions. The coefficient of in is the number of partitions of m into at most n parts. 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 as in the above section. We also have that the coefficient of in is the number of partitions of m into n or n-1 distinct parts. 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.

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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 q → 1. Typically, mathematicians are interested in q-analogs that arise naturally, rather than in arbitrarily contriving q-analogs of known results. The earliest q-analog studied in detail is the basic hypergeometric series, which was introduced in the 19th century. q-analogs are most frequently studied in the mathematical fields of combinatorics and special functions.

Basic hypergeometric series

In mathematics, basic hypergeometric series, or q-hypergeometric series, are q-analogue generalizations of generalized hypergeometric series, and are in turn generalized by elliptic hypergeometric series. A series xn is called hypergeometric if the ratio of successive terms xn+1/xn is a rational function of n. If the ratio of successive terms is a rational function of qn, then the series is called a basic hypergeometric series. The number q is called the base. The basic hypergeometric series was first considered by .

Euler function

In mathematics, the Euler function is given by Named after Leonhard Euler, it is a model example of a q-series and provides the prototypical example of a relation between combinatorics and complex analysis. The coefficient in the formal power series expansion for gives the number of partitions of k. That is, where is the partition function. The Euler identity, also known as the Pentagonal number theorem, is is a pentagonal number.

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