In mathematics, in the area of combinatorics, a qPochhammer symbol, also called a qshifted factorial, is a qanalog of the common Pochhammer symbol. It is defined as \[ (a;q)_n = \prod_{k=0}^{n1} (1aq^k)=(1a)(1aq)(1aq^2)\cdots(1aq^{n1}) \] with (a;q)_{0} = 1 by definition. The qPochhammer symbol is a major building block in the construction of qanalogs; 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 qPochhammer symbol can be extended to an infinite product: \[ (a;q)_\infty = \prod_{k=0}^{\infty} (1aq^k). \] 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 \[ \phi(q) = (q;q)_\infty=\prod_{k=1}^\infty (1q^k) \] is known as Euler's function, and is important in combinatorics, number theory, and the theory of modular forms. A qseries is a series in which the coefficients are functions of q, typically depending on q via qPochhammer symbols. Identities The finite product can be expressed in terms of the infinite product: \[ (a;q)_n = \frac{(a;q)_\infty} {(aq^n;q)_\infty}, \] which extends the definition to negative integers n. Thus, for nonnegative n, one has \[ (a;q)_{n} = \frac{1}{(aq^{n};q)_n}=\prod_{k=1}^n \frac{1}{(1a/q^k)} \] and \[ (a;q)_{n} = \frac{(q/a)^n q^{n(n1)/2}} {(q/a;q)_n}. \] The qPochhammer symbol is the subject of a number of qseries identities, particularly the infinite series expansions \[ (x;q)_\infty = \sum_{n=0}^\infty \frac{(1)^n q^{n(n1)/2}}{(q;q)_n} x^n \] and \[ \frac{1}{(x;q)_\infty}=\sum_{n=0}^\infty \frac{x^n}{(q;q)_n}, \] which are both special cases of the qbinomial theorem: \[ \frac{(ax;q)_\infty}{(x;q)_\infty} = \sum_{n=0}^\infty \frac{(a;q)_n}{(q;q)_n} x^n. \] Combinatorial interpretation The qPochhammer symbol is closely related to the enumerative combinatorics of partitions. The coefficient of qman in \[ (a;q)_\infty^{1} = \prod_{k=0}^{\infty} (1aq^k)^{1} \] 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: \[ (a;q)_\infty^{1} = \sum_{k=0}^\infty \left(\prod_{j=1}^k \frac{1}{1q^j} \right) a^k = \sum_{k=0}^\infty \frac{a^k}{(q;q)_k} \] as in the above section. We also have that the coefficient of qman in \[ (a;q)_\infty = \prod_{k=0}^{\infty} (1+aq^k) \] is the number of partitions of m into n or n1 distinct parts. By removing a triangular partition with n1 parts from such a partition, we are left with an arbitrary partition with at most n parts. This gives a weightpreserving bijection between the set of partitions into n or n1 distinct parts and the set of pairs consisting of a triangular partition having n1 parts and a partition with at most n parts. By identifying generating series, this leads to the identity: \[ (a;q)_\infty = \prod_{k=0}^\infty (1+aq^k) = \sum_{k=0}^\infty \left(q^{k\choose 2} \prod_{j=1}^k \frac{1}{1q^j}\right) a^k = \sum_{k=0}^\infty \frac{q^{k\choose 2}}{(q;q)_k} a^k \] also described in the above section. The qbinomial theorem itself can also be handled by a slightly more involved combinatorial argument of a similar flavour. Since identities involving qPochhammer symbols so frequently involve products of many symbols, the standard convention is to write a product as a single symbol of multiple arguments: \[ (a_1,a_2,\ldots,a_m;q)_n = (a_1;q)_n (a_2;q)_n \ldots (a_m;q)_n. \] Relationship to other qfunctions Noticing that \[ \lim_{q\rightarrow 1}\frac{1q^n}{1q}=n, \] we define the qanalog of n, also known as the qbracket or qnumber of n to be \[ [n]_q=\frac{1q^n}{1q}. \] From this one can define the qanalog of the factorial, the qfactorial, as \[ \big[n]_q! =\prod_{k=1}^n [k]_q Again, one recovers the usual factorial by taking the limit as q approaches 1. This can be interpreted as the number of flags in an ndimensional vector space over the field with q elements, and taking the limit as q goes to 1 yields the interpretation of an ordering on a set as a flag in a vector space over the field with one element. A product of negative integer qbrackets can be expressed in terms of the qfactorial as: \[ \prod_{k=1}^n [k]_q = \frac{(1)^n\,[n]_q!}{q^{n(n+1)/2}} \] From the qfactorials, one can move on to define the qbinomial coefficients, also known as Gaussian coefficients, Gaussian polynomials, or Gaussian binomial coefficients: \[ \begin{bmatrix} n\\ k \end{bmatrix}_q = \frac{[n]_q!}{[nk]_q! [k]_q!}. \] One can check that \[ \begin{bmatrix} n+1\\ k \end{bmatrix}_q = \begin{bmatrix} n\\ k \end{bmatrix}_q + q^{nk+1} \begin{bmatrix} n\\ k1 \end{bmatrix}_q. \] One also obtains a qanalog of the Gamma function, called the qgamma function, and defined as \[ \Gamma_q(x)=\frac{(1q)^{1x} (q;q)_\infty}{(q^x;q)_\infty} \] This converges to the usual Gamma function as q approaches 1 from inside the unit disc.. Note that \[\Gamma_q(x+1)=[x]_q\Gamma_q(x)\, \] for any x and \[ \Gamma_q(n+1)=[n]_q!\frac{}{}. \] for nonnegative integer values of n. Alternatively, this may be taken as an extension of the qfactorial function to the real number system. Basic hypergeometric series References George Gasper and Mizan Rahman, Basic Hypergeometric Series, 2nd Edition, (2004), Encyclopedia of Mathematics and Its Applications, 96, Cambridge University Press, Cambridge. ISBN 0521833574. Retrieved from "http://en.wikipedia.org/"

