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| {{lowercase|title=''q''-Pochhammer symbol}}
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| In [[mathematics]], in the area of [[combinatorics]], a '''''q''-Pochhammer symbol''', also called a '''''q''-shifted factorial''', is a [[q-analog|''q''-analog]] of the common [[Pochhammer symbol]]. It is defined as
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| :<math>(a;q)_n = \prod_{k=0}^{n-1} (1-aq^k)=(1-a)(1-aq)(1-aq^2)\cdots(1-aq^{n-1})</math>
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| with
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| :<math>(a;q)_0 = 1</math>
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| 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]].
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| Unlike the ordinary Pochhammer symbol, the ''q''-Pochhammer symbol can be extended to an infinite product:
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| :<math>(a;q)_\infty = \prod_{k=0}^{\infty} (1-aq^k).</math>
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| 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
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| :<math>\phi(q) = (q;q)_\infty=\prod_{k=1}^\infty (1-q^k)</math>
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| is known as [[Euler's function]], and is important in [[combinatorics]], [[number theory]], and the theory of [[modular forms]].
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| A ''q''-series is a [[Series (mathematics)|series]] in which the coefficients are functions of ''q'', typically depending on ''q'' via ''q''-Pochhammer symbols.
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| ==Identities==
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| The finite product can be expressed in terms of the infinite product:
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| :<math>(a;q)_n = \frac{(a;q)_\infty} {(aq^n;q)_\infty}, </math>
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| which extends the definition to negative integers ''n''. Thus, for nonnegative ''n'', one has
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| :<math>(a;q)_{-n} = \frac{1}{(aq^{-n};q)_n}=\prod_{k=1}^n \frac{1}{(1-a/q^k)}</math>
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| and
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| :<math>(a;q)_{-n} = \frac{(-q/a)^n q^{n(n-1)/2}} {(q/a;q)_n}.</math>
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| The ''q''-Pochhammer symbol is the subject of a number of ''q''-series identities, particularly the infinite series expansions
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| :<math>(x;q)_\infty = \sum_{n=0}^\infty \frac{(-1)^n q^{n(n-1)/2}}{(q;q)_n} x^n</math>
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| and
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| :<math>\frac{1}{(x;q)_\infty}=\sum_{n=0}^\infty \frac{x^n}{(q;q)_n}</math>,
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| which are both special cases of the [[q-binomial theorem]]:
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| :<math>\frac{(ax;q)_\infty}{(x;q)_\infty} = \sum_{n=0}^\infty \frac{(a;q)_n}{(q;q)_n} x^n.</math>
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| ==Combinatorial interpretation==
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| The ''q''-Pochhammer symbol is closely related to the enumerative combinatorics of partitions. The coefficient of <math>q^m a^n</math> in
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| :<math>(a;q)_\infty^{-1} = \prod_{k=0}^{\infty} (1-aq^k)^{-1}</math>
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| is the number of partitions of ''m'' into at most ''n'' parts.
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| 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:
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| :<math>(a;q)_\infty^{-1} = \sum_{k=0}^\infty \left(\prod_{j=1}^k \frac{1}{1-q^j} \right) a^k
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| = \sum_{k=0}^\infty \frac{a^k}{(q;q)_k}</math>
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| as in the above section.
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| We also have that the coefficient of <math>q^m a^n</math> in
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| :<math>(-a;q)_\infty = \prod_{k=0}^{\infty} (1+aq^k)</math>
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| is the number of partitions of ''m'' into ''n'' or ''n''-1 distinct parts.
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| 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:
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| :<math>(-a;q)_\infty = \prod_{k=0}^\infty (1+aq^k)
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| = \sum_{k=0}^\infty \left(q^{k\choose 2} \prod_{j=1}^k \frac{1}{1-q^j}\right) a^k
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| = \sum_{k=0}^\infty \frac{q^{k\choose 2}}{(q;q)_k} a^k</math>
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| also described in the above section.
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| The ''q''-binomial theorem itself can also be handled by a slightly more involved combinatorial argument of a similar flavour.
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| == Multiple arguments convention ==
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| 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:
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| :<math>(a_1,a_2,\ldots,a_m;q)_n = (a_1;q)_n (a_2;q)_n \ldots (a_m;q)_n.</math>
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| == Relationship to other ''q''-functions ==
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| Noticing that
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| :<math>\lim_{q\rightarrow 1}\frac{1-q^n}{1-q}=n,</math>
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| we define the ''q''-analog of ''n'', also known as the '''''q''-bracket''' or '''''q''-number''' of ''n'' to be
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| :<math>[n]_q=\frac{1-q^n}{1-q}.</math> | |
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| From this one can define the ''q''-analog of the [[factorial]], the '''''q''-factorial''', as
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| :{| | |
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| |<math>\big[n]_q!</math>
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| |<math>=\prod_{k=1}^n [k]_q</math>
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| |-
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| ||<math>= [1]_q [2]_q \cdots [n-1]_q [n]_q</math>
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| |-
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| |<math>=\frac{1-q}{1-q} \frac{1-q^2}{1-q} \cdots \frac{1-q^{n-1}}{1-q} \frac{1-q^n}{1-q}</math>
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| |-
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| |<math>=1(1+q)\cdots (1+q+\cdots + q^{n-2}) (1+q+\cdots + q^{n-1})</math>
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| |-
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| |<math>=\frac{(q;q)_n}{(1-q)^n}.</math>
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| |}
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| Again, one recovers the usual factorial by taking the limit as ''q'' approaches 1. This can be interpreted as the number of [[flag (linear algebra)|flags]] in an ''n''-dimensional 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]].
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| A product of negative integer ''q''-brackets can be expressed in terms of the ''q''-factorial as:
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| :<math>\prod_{k=1}^n [-k]_q = \frac{(-1)^n\,[n]_q!}{q^{n(n+1)/2}}</math> | |
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| From the ''q''-factorials, one can move on to define the '''''q''-binomial coefficients''', also known as '''Gaussian coefficients''', '''Gaussian polynomials''', or '''[[Gaussian binomial coefficient]]s''':
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| :<math>
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| \begin{bmatrix}
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| n\\
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| k
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| \end{bmatrix}_q
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| =
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| \frac{[n]_q!}{[n-k]_q! [k]_q!}.
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| </math>
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| One can check that
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| :<math>
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| \begin{bmatrix}
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| n+1\\
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| k
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| \end{bmatrix}_q
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| =
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| \begin{bmatrix}
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| n\\
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| k
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| \end{bmatrix}_q
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| +
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| q^{n-k+1}
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| \begin{bmatrix}
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| n\\
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| k-1
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| \end{bmatrix}_q.
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| </math>
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| One also obtains a ''q''-analog of the [[Gamma function]], called the '''[[q-gamma function]]''', and defined as
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| :<math>\Gamma_q(x)=\frac{(1-q)^{1-x} (q;q)_\infty}{(q^x;q)_\infty}</math>
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| This converges to the usual Gamma function as ''q'' approaches 1 from inside the unit disc. Note that
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| :<math>\Gamma_q(x+1)=[x]_q\Gamma_q(x)\,</math>
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| for any ''x'' and
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| :<math>\Gamma_q(n+1)=[n]_q!\frac{}{}.</math>
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| 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.
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| ==See also==
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| * [[Basic hypergeometric series]]
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| * [[Pochhammer symbol]]
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| * [[Q-derivative]]
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| * [[Q-theta function]]
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| * [[Elliptic gamma function]]
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| * [[Theta function|Jacobi theta function]]
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| ==References==
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| * George Gasper and [[Mizan Rahman]], ''Basic Hypergeometric Series, 2nd Edition'', (2004), Encyclopedia of Mathematics and Its Applications, '''96''', Cambridge University Press, Cambridge. ISBN 0-521-83357-4.
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| * Roelof Koekoek and Rene F. Swarttouw, ''[http://fa.its.tudelft.nl/~koekoek/askey/ The Askey scheme of orthogonal polynomials and its q-analogues]'', section 0.2.
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| * Exton, H. (1983), ''q-Hypergeometric Functions and Applications'', New York: Halstead Press, Chichester: Ellis Horwood, 1983, ISBN 0853124914, ISBN 0470274530, ISBN 978-0470274538
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| ==External links==
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| * {{MathWorld|urlname=q-Analog|title=''q''-Analog}}
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| * {{MathWorld|urlname=q-Bracket|title=''q''-Bracket}}
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| * {{MathWorld|urlname=q-Factorial|title=''q''-Factorial}}
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| * {{MathWorld|urlname=q-BinomialCoefficient|title=''q''-Binomial Coefficient}}
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| [[Category:Number theory]]
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| [[Category:Q-analogs]]
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Alyson is what my spouse loves to call me but I don't like when individuals use my complete name. I am really fond of handwriting but I can't make it my occupation truly. She functions as a journey agent but soon she'll be on her own. Mississippi is exactly where her house is but her spouse wants them to move.
Feel free to visit my blog :: online psychics - visit here -