Basic hypergeometric series

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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. A series xn is called hypergeometric if the ratio of successive terms xn+1/xn is a rational function of n. If the ratio of successive terms is a rational function of qn, then the series is called a basic hypergeometric series. The number q is called the base.

The basic hypergeometric series 2φ1(qα,qβ;qγ;q,x) was first considered by Eduard Heine (1846). It becomes the hypergeometric series F(α,β;γ;x) in the limit when the base q is 1.

Definition[edit]

There are two forms of basic hypergeometric series, the unilateral basic hypergeometric series φ, and the more general bilateral basic geometric series ψ. 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

where

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

and where

(a;q)_n = \prod_{k=0}^{n-1} (1-aq^k)=(1-a)(1-aq)(1-aq^2)\cdots(1-aq^{n-1}).

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}&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.

This series is called balanced if a1...ak+1 = b1...bkq. This series is called well poised if a1q = a2b1 = ... = ak+1bk, and very well poised if in addition a2 = −a3 = qa11/2.

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.

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.

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.

Simple series[edit]

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.

The q-binomial theorem[edit]

The q-binomial theorem (first published in 1811 by Heinrich August Rothe)[1][2] states that

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

which follows by repeatedly applying the identity

\;_{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.

Ramanujan's identity[edit]

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} z^n
= \frac {(b/a,q,q/az,az;q)_\infty }
{(b,b/az,q/a,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) \stackrel{\rm{def}}{=} \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}.

Watson's contour integral[edit]

As an analogue of the Barnes integral for the hypergeometric series, Watson showed that


{}_2\phi_1(a,b;c;q,z) = \frac{-1}{2\pi i}\frac{(a,b;q)_\infty}{(q,c;q)_\infty}
\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

where the poles of (aq^s,bq^s;q)_\infty lie to the left of the contour and the remaining poles lie to the right. There is a similar contour integral for r+1φr. This contour integral gives an analytic continuation of the basic hypergeometric function in z.

Notes[edit]

  1. ^ Bressoud, D. M. (1981), "Some identities for terminating q-series", Mathematical Proceedings of the Cambridge Philosophical Society 89 (2): 211–223, doi:10.1017/S0305004100058114, MR 600238 .
  2. ^ Benaoum, H. B., "h-analogue of Newton's binomial formula", Journal of Physics A: Mathematical and General 31 (46): L751–L754, arXiv:math-ph/9812011, doi:10.1088/0305-4470/31/46/001 .

References[edit]