Basic hypergeometric series
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In mathematics, the basic hypergeometric series, also sometimes called the hypergeometric q-series, are q-analog generalizations of ordinary hypergeometric series. Two basic series are commonly defined, the unilateral basic hypergeometric series, and the bilateral basic geometric series.
The naming is in analogy to an ordinary hypergeometric series. An ordinary series {xn} is termed an ordinary hypergeometric series if the ratio of successive terms xn + 1 / xn is a rational function of n. But if the ratio of successive terms is a rational function of qn, then the series is called a basic hypergeometric series.
The basic hypergeometric series was first considered by Eduard Heine in the 19th century, as a way of capturing the common features of the Jacobi theta functions and elliptic functions.
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[edit] Definition
The unilateral basic hypergeometric series is defined as
![\;_{j}\phi_k \left[\begin{matrix}
a_1 & a_2 & \ldots & a_{j} \\
b_1 & b_2 & \ldots & b_k \end{matrix}
; q,z \right] = \sum_{n=0}^\infty
\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](http://upload.wikimedia.org/math/2/c/3/2c3c423d7b1b9f3997f092b9a15925e9.png)
where
is the q-shifted factorial. The most important special case is when j = k+1, when it becomes
![\;_{k+1}\phi_k \left[\begin{matrix}
a_1 & a_2 & \ldots & a_{k+1} \\
b_1 & b_2 & \ldots & b_k \end{matrix}
; q,z \right] = \sum_{n=0}^\infty
\frac {(a_1, a_2, \ldots, a_{k+1};q)_n} {(b_1, b_2, \ldots, b_k,q;q)_n} z^n.](http://upload.wikimedia.org/math/0/3/b/03b7a747193073fef313388cac2d9bac.png)
The bilateral basic hypergeometric series, corresponding to the bilateral hypergeometric series, is defined as
![\;_j\psi_k \left[\begin{matrix}
a_1 & a_2 & \ldots & a_j \\
b_1 & b_2 & \ldots & b_k \end{matrix}
; q,z \right] = \sum_{n=-\infty}^\infty
\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.](http://upload.wikimedia.org/math/0/a/0/0a0998915eaf9112cd573f4ea7c80043.png)
The most important special case is when j = k, when it becomes
![\;_k\psi_k \left[\begin{matrix}
a_1 & a_2 & \ldots & a_k \\
b_1 & b_2 & \ldots & b_k \end{matrix}
; q,z \right] = \sum_{n=-\infty}^\infty
\frac {(a_1, a_2, \ldots, a_k;q)_n} {(b_1, b_2, \ldots, b_k;q)_n} z^n.](http://upload.wikimedia.org/math/2/f/5/2f52a9ddbb6b83ac7b1f6aa873bdce01.png)
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.
[edit] Simple series
Some simple series expressions include
![\frac{z}{1-q} \;_{2}\phi_1 \left[\begin{matrix}
q \; q \\
q^2 \end{matrix}\; ; q,z \right] =
\frac{z}{1-q}
+ \frac{z^2}{1-q^2}
+ \frac{z^3}{1-q^3}
+ \ldots](http://upload.wikimedia.org/math/7/e/8/7e8df00949af12cfad161caaacbe12f5.png)
and
![\frac{z}{1-q^{1/2}} \;_{2}\phi_1 \left[\begin{matrix}
q \; q^{1/2} \\
q^{3/2} \end{matrix}\; ; q,z \right] =
\frac{z}{1-q^{1/2}}
+ \frac{z^2}{1-q^{3/2}}
+ \frac{z^3}{1-q^{5/2}}
+ \ldots](http://upload.wikimedia.org/math/1/c/f/1cfa9f8cfcd4b2965a0594635e0d9e50.png)
and
![\;_{2}\phi_1 \left[\begin{matrix}
q \; -1 \\
-q \end{matrix}\; ; q,z \right] = 1+
\frac{2z}{1+q}
+ \frac{2z^2}{1+q^2}
+ \frac{2z^3}{1+q^3}
+ \ldots.](http://upload.wikimedia.org/math/5/2/3/523611058a40eedf1dd82fdae8c7ddc1.png)
[edit] Simple identities
Some simple identities include
and
The special case of a = 0 is closely related to the q-exponential.
[edit] Ramanujan's identity
Ramanujan gave the identity
![\;_1\psi_1 \left[\begin{matrix} a \\ b \end{matrix} ; q,z \right]
= \sum_{n=-\infty}^\infty \frac {(a;q)_n} {(b;q)_n}
= \frac {(b/a;q)_\infty\; (q;q)_\infty\; (q/az;q)_\infty\; (az;q)_\infty }
{(b;q)_\infty\; (b/az;q)_\infty\; (q/a;q)_\infty\; (z;q)_\infty}](http://upload.wikimedia.org/math/c/6/d/c6dc552d91f344ab01fcc6e65d171b73.png)
valid for | q | < 1 and | b / a | < | z | < 1. Similar identities for 
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
Ken Ono gives a related formal power series
[edit] References
- Eduard Heine, Theorie der Kugelfunctionen, (1878) 1, pp 97-125.
- Eduard Heine, Handbuch die Kugelfunctionen. Theorie und Anwendung (1898) Springer, Berlin.
- W.N. Bailey, Generalized Hypergeometric Series, (1935) Cambridge Tracts in Mathematics and Mathematical Physics, No.32, Cambridge University Press, Cambridge.
- Gasper, George; Rahman, Mizan (2004), Basic hypergeometric series, Encyclopedia of Mathematics and its Applications, 96 (2nd ed.), Cambridge University Press, MR2128719, ISBN 978-0-521-83357-8
- William Y. C. Chen and Amy Fu, Semi-Finite Forms of Bilateral Basic Hypergeometric Series (2004)
- Sylvie Corteel and Jeremy Lovejoy, Frobenius Partitions and the Combinatorics of Ramanujan's
Summation, (undated) - Gwynneth H. Coogan and Ken Ono, A q-series identity and the Arithmetic of Hurwitz Zeta Functions, (2003) Proceedings of the American Mathematical Society 131, pp. 719-724
