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| In [[mathematics]], Heine's '''basic hypergeometric series''', or '''hypergeometric q-series''', are [[q-analog]] generalizations of [[generalized hypergeometric series]], and are in turn generalized by [[elliptic hypergeometric series]].
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| A series ''x''<sub>''n''</sub> is called hypergeometric if the ratio of successive terms ''x''<sub>''n''+1</sub>/''x''<sub>''n''</sub> is a [[rational function]] of ''n''. If the ratio of successive terms is a rational function of ''q''<sup>''n''</sup>, then the series is called a basic hypergeometric series. The number ''q'' is called the base.
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| The basic hypergeometric series <sub>2</sub>φ<sub>1</sub>(''q''<sup>α</sup>,''q''<sup>β</sup>;''q''<sup>γ</sup>;''q'',''x'') was first considered by {{harvs|txt|authorlink=Eduard Heine|first=Eduard|last= Heine|year=1846}}. It becomes the hypergeometric series ''F''(α,β;γ;''x'') in the limit when the base ''q'' is 1.
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| ==Definition==
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| There are two forms of basic hypergeometric series, the '''unilateral basic hypergeometric series''' φ, and the more general '''bilateral basic geometric series''' ψ.
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| The '''unilateral basic hypergeometric series''' is defined as
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| :<math>\;_{j}\phi_k \left[\begin{matrix}
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| a_1 & a_2 & \ldots & a_{j} \\
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| b_1 & b_2 & \ldots & b_k \end{matrix}
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| ; q,z \right] = \sum_{n=0}^\infty
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| \frac {(a_1, a_2, \ldots, a_{j};q)_n} {(b_1, b_2, \ldots, b_k,q;q)_n} \left((-1)^nq^{n\choose 2}\right)^{1+k-j}z^n</math>
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| where
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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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| and where
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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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| is the [[q-shifted factorial]].
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| The most important special case is when ''j'' = ''k''+1, when it becomes
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| :<math>\;_{k+1}\phi_k \left[\begin{matrix}
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| a_1 & a_2 & \ldots & a_{k}&a_{k+1} \\
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| b_1 & b_2 & \ldots & b_{k} \end{matrix}
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| ; q,z \right] = \sum_{n=0}^\infty
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| \frac {(a_1, a_2, \ldots, a_{k+1};q)_n} {(b_1, b_2, \ldots, b_k,q;q)_n} z^n.</math>
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| This series is called balanced if ''a''<sub>1</sub>''...''a''<sub>''k''+1</sub>'' = ''b''<sub>1</sub>...''b''<sub>''k''</sub>''q''.
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| This series is called well poised if ''a''<sub>1</sub>''q'' = ''a''<sub>2</sub>''b''<sub>1</sub> = ... = ''a''<sub>k+1</sub>''b''<sub>''k''</sub>, and very well poised if in addition ''a''<sub>2</sub> = −''a''<sub>3</sub> = ''qa''<sub>1</sub><sup>1/2</sup>.
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| The '''bilateral basic hypergeometric series''', corresponding to the [[bilateral hypergeometric series]], is defined as
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| :<math>\;_j\psi_k \left[\begin{matrix}
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| a_1 & a_2 & \ldots & a_j \\
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| b_1 & b_2 & \ldots & b_k \end{matrix}
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| ; q,z \right] = \sum_{n=-\infty}^\infty
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| \frac {(a_1, a_2, \ldots, a_j;q)_n} {(b_1, b_2, \ldots, b_k;q)_n} \left((-1)^nq^{n\choose 2}\right)^{k-j}z^n.</math>
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| The most important special case is when ''j'' = ''k'', when it becomes
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| :<math>\;_k\psi_k \left[\begin{matrix}
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| a_1 & a_2 & \ldots & a_k \\
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| b_1 & b_2 & \ldots & b_k \end{matrix}
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| ; q,z \right] = \sum_{n=-\infty}^\infty
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| \frac {(a_1, a_2, \ldots, a_k;q)_n} {(b_1, b_2, \ldots, b_k;q)_n} z^n.</math>
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| The unilateral series can be obtained as a special case of the bilateral one by setting one of the ''b'' variables equal to ''q'', at least when none of the ''a'' variables is a power of ''q''., as all the terms with ''n''<0 then vanish.
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| ==Simple series==
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| Some simple series expressions include
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| :<math>\frac{z}{1-q} \;_{2}\phi_1 \left[\begin{matrix}
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| q \; q \\
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| q^2 \end{matrix}\; ; q,z \right] =
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| \frac{z}{1-q}
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| + \frac{z^2}{1-q^2}
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| + \frac{z^3}{1-q^3}
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| + \ldots </math>
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| and
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| :<math>\frac{z}{1-q^{1/2}} \;_{2}\phi_1 \left[\begin{matrix}
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| q \; q^{1/2} \\
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| q^{3/2} \end{matrix}\; ; q,z \right] =
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| \frac{z}{1-q^{1/2}}
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| + \frac{z^2}{1-q^{3/2}}
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| + \frac{z^3}{1-q^{5/2}}
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| + \ldots </math>
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| and
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| :<math>\;_{2}\phi_1 \left[\begin{matrix}
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| q \; -1 \\
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| -q \end{matrix}\; ; q,z \right] = 1+
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| \frac{2z}{1+q}
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| + \frac{2z^2}{1+q^2}
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| + \frac{2z^3}{1+q^3}
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| + \ldots. </math>
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| ==The ''q''-binomial theorem==
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| The ''q''-binomial theorem (first published in 1811 by [[Heinrich August Rothe]])<ref>{{citation
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| | last = Bressoud | first = D. M.
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| | doi = 10.1017/S0305004100058114
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| | issue = 2
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| | journal = Mathematical Proceedings of the Cambridge Philosophical Society
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| | mr = 600238
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| | pages = 211–223
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| | title = Some identities for terminating ''q''-series
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| | volume = 89
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| | year = 1981}}.</ref><ref>{{citation
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| | last = Benaoum | first = H. B.
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| | arxiv = math-ph/9812011
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| | doi = 10.1088/0305-4470/31/46/001
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| | issue = 46
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| | journal = Journal of Physics A: Mathematical and General
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| | pages = L751–L754
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| | title = ''h''-analogue of Newton's binomial formula
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| | volume = 31}}.</ref> states that
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| :<math>\;_{1}\phi_0 (a;q,z) =\frac{(az;q)_\infty}{(z;q)_\infty}= \prod_{n=0}^\infty
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| \frac {1-aq^n z}{1-q^n z}</math>
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| which follows by repeatedly applying the identity
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| :<math>\;_{1}\phi_0 (a;q,z) =
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| \frac {1-az}{1-z} \;_{1}\phi_0 (a;q,qz).</math>
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| The special case of ''a'' = 0 is closely related to the [[q-exponential]].
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| ==Ramanujan's identity==
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| [[Ramanujan]] gave the identity
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| :<math>\;_1\psi_1 \left[\begin{matrix} a \\ b \end{matrix} ; q,z \right]
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| = \sum_{n=-\infty}^\infty \frac {(a;q)_n} {(b;q)_n} z^n
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| = \frac {(b/a,q,q/az,az;q)_\infty }
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| {(b,b/az,q/a,z;q)_\infty} </math>
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| valid for |''q''| < 1 and |''b''/''a''| < |''z''| < 1. Similar identities for <math>\;_6\psi_6</math> have been given by Bailey. Such identities can be understood to be generalizations of the [[Jacobi triple product]] theorem, which can be written using q-series as
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| :<math>\sum_{n=-\infty}^\infty q^{n(n+1)/2}z^n =
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| (q;q)_\infty \; (-1/z;q)_\infty \; (-zq;q)_\infty.</math>
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| [[Ken Ono]] gives a related [[formal power series]]
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| :<math>A(z;q) \stackrel{\rm{def}}{=} \frac{1}{1+z} \sum_{n=0}^\infty
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| \frac{(z;q)_n}{(-zq;q)_n}z^n =
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| \sum_{n=0}^\infty (-1)^n z^{2n} q^{n^2}.</math>
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| ==Watson's contour integral==
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| As an analogue of the [[Barnes integral]] for the hypergeometric series, Watson showed that
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| :<math>
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| {}_2\phi_1(a,b;c;q,z) = \frac{-1}{2\pi i}\frac{(a,b;q)_\infty}{(q,c;q)_\infty}
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| \int_{-i\infty}^{i\infty}\frac{(qq^s,cq^s;q)_\infty}{(aq^s,bq^s;q)_\infty}\frac{\pi(-z)^s}{\sin \pi s}ds
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| </math>
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| where the poles of <math>(aq^s,bq^s;q)_\infty</math> lie to the left of the contour and the remaining poles lie to the right. There is a similar contour integral for <sub> ''r''+1</sub>φ<sub>''r''</sub>. This contour integral gives an analytic continuation of the basic hypergeometric function in ''z''.
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| ==Notes==
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| {{reflist}}
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| ==References==
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| *{{dlmf|id=17|first=G. E.|last=Andrews|title=q-Hypergeometric and Related Functions}}
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| * W.N. Bailey, ''Generalized Hypergeometric Series'', (1935) Cambridge Tracts in Mathematics and Mathematical Physics, No.32, Cambridge University Press, Cambridge.
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| * William Y. C. Chen and Amy Fu, ''[http://cfc.nankai.edu.cn/publications/04-accepted/Chen-Fu-04A/semi.pdf Semi-Finite Forms of Bilateral Basic Hypergeometric Series]'' (2004)
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| * Gwynneth H. Coogan and [[Ken Ono]], ''[http://www.math.wisc.edu/~ono/reprints/067.pdf A q-series identity and the Arithmetic of Hurwitz Zeta Functions]'', (2003) Proceedings of the [[American Mathematical Society]] '''131''', pp. 719–724
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| * Sylvie Corteel and Jeremy Lovejoy, ''[http://www.labri.fr/Perso/~lovejoy/1psi1.pdf Frobenius Partitions and the Combinatorics of Ramanujan's <math>\,_1\psi_1</math> Summation]''
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| *{{Citation | last1=Fine | first1=Nathan J. | title=Basic hypergeometric series and applications | url=http://www.ams.org/bookstore?fn=20&arg1=survseries&ikey=SURV-27 | publisher=[[American Mathematical Society]] | location=Providence, R.I. | series=Mathematical Surveys and Monographs | isbn=978-0-8218-1524-3 | mr=956465 | year=1988 | volume=27}}
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| *{{Citation | last1=Gasper | first1=George | last2=Rahman | first2=Mizan | title=Basic hypergeometric series | publisher=[[Cambridge University Press]] | edition=2nd | series=Encyclopedia of Mathematics and its Applications | isbn=978-0-521-83357-8 | doi=10.2277/0521833574 | mr=2128719 | year=2004 | volume=96}}
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| *{{citation|first=Eduard |last=Heine|year=1846|journal= Journal für die reine und angewandte Mathematik|pages=210–212|volume=32|title=Über die Reihe <math>1+\frac{(q^\alpha-1)(q^\beta-1)}{(q-1)(q^\gamma-1)}x + \frac{(q^\alpha-1)(q^{\alpha+1}-1)(q^\beta-1)(q^{\beta+1}-1)}{(q-1)(q^2-1)(q^\gamma-1)(q^{\gamma+1}-1)}x^2+\cdots</math>|url=http://resolver.sub.uni-goettingen.de/purl?GDZPPN002145391}}
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| * [[Eduard Heine]], ''Theorie der Kugelfunctionen'', (1878) ''1'', pp 97–125.
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| * Eduard Heine, ''Handbuch die Kugelfunctionen. Theorie und Anwendung'' (1898) Springer, Berlin.
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| [[Category:Q-analogs]]
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| [[Category:Hypergeometric functions]]
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