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 ${x n}$ is termed an ordinary hypergeometric series if the ratio of successive terms $x n + 1 / x n$ is a rational function of n. But if the ratio of successive terms is a rational function of $q n$ , 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.

1 Definition
2 Simple series
3 Simple identities
4 Ramanujan's identity
5 References

[edit] Definition

The unilateral basic hypergeometric series in k variables is defined as

$\;_{k+1}\phi_k \left[\begin{matrix} a_1 & a_2 & \ldots & a_{k+1} \\ b_1 & b_2 & \ldots & b_k \end{matrix} ; q,z \right] = \sum_{n=0}^\infty \frac {(a_1, a_2, \ldots, a_{k+1};q)_n} {(b_1, b_2, \ldots, b_k,q;q)_n} z^n$

where

$(a_1,a_2,\ldots,a_m;q)_n = (a_1;q)_n (a_2;q)_n \ldots (a_m;q)_n$

is the q-shifted factorial.

The bilateral basic hypergeometric series in k variables is defined as

$\;_k\psi_k \left[\begin{matrix} a_1 & a_2 & \ldots & a_k \\ b_1 & b_2 & \ldots & b_k \end{matrix} ; q,z \right] = \sum_{n=-\infty}^\infty \frac {(a_1, a_2, \ldots, a_k;q)_n} {(b_1, b_2, \ldots, b_k;q)_n} z^n.$

[edit] Simple series

Some simple series expressions include

$\frac{z}{1-q} \;_{2}\phi_1 \left[\begin{matrix} q \; q \\ q^2 \end{matrix}\; ; q,z \right] = \frac{z}{1-q} + \frac{z^2}{1-q^2} + \frac{z^3}{1-q^3} + \ldots$

and

$\frac{z}{1-q^{1/2}} \;_{2}\phi_1 \left[\begin{matrix} q \; q^{1/2} \\ q^{3/2} \end{matrix}\; ; q,z \right] = \frac{z}{1-q^{1/2}} + \frac{z^2}{1-q^{3/2}} + \frac{z^3}{1-q^{5/2}} + \ldots$

and

$\;_{2}\phi_1 \left[\begin{matrix} q \; -1 \\ -q \end{matrix}\; ; q,z \right] = 1+ \frac{2z}{1+q} + \frac{2z^2}{1+q^2} + \frac{2z^3}{1+q^3} + \ldots$

[edit] Simple identities

Some simple identities include

$\;_{1}\phi_0 (a;q,z) = \prod_{n=0}^\infty \frac {1-aq^n z}{1-q^n z}$

and

$\;_{1}\phi_0 (a;q,z) = \frac {1-az}{1-z} \;_{1}\phi_0 (a;q,qz)$

The special case of $a = 0$ is closely related to the q-exponential.

[edit] Ramanujan's identity

Ramanujan gave the identity

$\;_1\psi_1 \left[\begin{matrix} a \\ b \end{matrix} ; q,z \right] = \sum_{n=-\infty}^\infty \frac {(a;q)_n} {(b;q)_n} = \frac {(b/a;q)_\infty\; (q;q)_\infty\; (q/az;q)_\infty\; (az;q)_\infty } {(b;q)_\infty\; (b/az;q)_\infty\; (q/a;q)_\infty\; (z;q)_\infty}$

valid for $| q | < 1$ and $| b / a | < | z | < 1$ . Similar identities for $\;_6\psi_6$ 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

$\sum_{n=-\infty}^\infty q^{n(n+1)/2}z^n = (q;q)_\infty \; (-1/z;q)_\infty \; (-zq;q)_\infty.$

Ken Ono gives a related formal power series

$A(z;q) = \frac{1}{1+z} \sum_{n=0}^\infty \frac{(z;q)_n}{(-zq;q)_n}z^n = \sum_{n=0}^\infty (-1)^n z^{2n} q^{n^2}.$

[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.
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.
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 $\,_1\psi_1$ 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