In mathematics, basic hypergeometric series, or q-hypergeometric series, are q-analogue 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
{}2\phi
\alpha | |
1(q |
,q\beta;q\gamma;q,x)
F(\alpha,\beta;\gamma;x)
q=1
There are two forms of basic hypergeometric series, the unilateral basic hypergeometric series φ, and the more general bilateral basic hypergeometric series ψ.The unilateral basic hypergeometric series is defined as
j\phik\left[\begin{matrix}a1&a2&\ldots&aj\ b1&b2&\ldots&bk\end{matrix};q,z\right]=
infty | |
\sum | |
n=0 |
(a1,a2,\ldots,aj;q)n | |
(b1,b2,\ldots,bk,q;q)n |
\left((-1)nqn\choose\right)1+k-jzn
(a1,a2,\ldots,am;q)n=(a1;q)n(a2;q)n\ldots(am;q)n
(a;q)n=
n-1 | |
\prod | |
k=0 |
(1-aqk)=(1-a)(1-aq)(1-aq2) … (1-aqn-1)
k+1\phik\left[\begin{matrix}a1&a2&\ldots&ak&ak+1\ b1&b2&\ldots&bk\end{matrix};q,z\right]=
infty | |
\sum | |
n=0 |
(a1,a2,\ldots,ak+1;q)n | |
(b1,b2,\ldots,bk,q;q)n |
zn.
\limq\to j\phik\left[\begin{matrix}
a1 | |
q |
&
a2 | |
q |
&\ldots&
aj | |
q |
b1 | |
\ q |
&
b2 | |
q |
&\ldots&
bk | |
q |
\end{matrix};q,(q-1)1+k-jz\right]= jFk\left[\begin{matrix}a1&a2&\ldots&aj\ b1&b2&\ldots&bk\end{matrix};z\right]
j\psik\left[\begin{matrix}a1&a2&\ldots&aj\ b1&b2&\ldots&bk\end{matrix};q,z\right]=
infty | |
\sum | |
n=-infty |
(a1,a2,\ldots,aj;q)n | |
(b1,b2,\ldots,bk;q)n |
\left((-1)nqn\choose\right)k-jzn.
The most important special case is when j = k, when it becomes
k\psik\left[\begin{matrix}a1&a2&\ldots&ak\ b1&b2&\ldots&bk\end{matrix};q,z\right]=
infty | |
\sum | |
n=-infty |
(a1,a2,\ldots,ak;q)n | |
(b1,b2,\ldots,bk;q)n |
zn.
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.
Some simple series expressions include
z | |
1-q |
2\phi1\left[\begin{matrix}q q\ q2\end{matrix} ;q,z\right]=
z | |
1-q |
+
z2 | |
1-q2 |
+
z3 | |
1-q3 |
+\ldots
and
z | |
1-q1/2 |
2\phi1\left[\begin{matrix}q q1/2\ q3/2\end{matrix} ;q,z\right]=
z | |
1-q1/2 |
+
z2 | |
1-q3/2 |
+
z3 | |
1-q5/2 |
+\ldots
and
2\phi1\left[\begin{matrix}q -1\ -q\end{matrix} ;q,z\right]=1+
2z | |
1+q |
+
2z2 | |
1+q2 |
+
2z3 | |
1+q3 |
+\ldots.
The q-binomial theorem (first published in 1811 by Heinrich August Rothe)[1] [2] states that
1\phi0(a;q,z)=
(az;q)infty | |
(z;q)infty |
=
infty | |
\prod | |
n=0 |
1-aqnz | |
1-qnz |
which follows by repeatedly applying the identity
1\phi0(a;q,z)=
1-az | |
1-z |
1\phi0(a;q,qz).
The special case of a = 0 is closely related to the q-exponential.
Cauchy binomial theorem is a special case of the q-binomial theorem.[3]
N | |
\sum | |
n=0 |
ynqn(n+1)/2\begin{bmatrix}N\\n\end{bmatrix}q=\prod
N | |
k=1 |
\left(1+yqk\right) (|q|<1)
Srinivasa Ramanujan gave the identity
1\psi1\left[\begin{matrix}a\ b\end{matrix};q,z\right]=
infty | |
\sum | |
n=-infty |
(a;q)n | |
(b;q)n |
zn =
(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\psi6
infty | |
\sum | |
n=-infty |
qn(n+1)/2zn=(q;q)infty (-1/z;q)infty (-zq;q)infty.
Ken Ono gives a related formal power series[4]
A(z;q)\stackrel{\rm{def}}{=}
1 | |
1+z |
infty | |
\sum | |
n=0 |
(z;q)n | |
(-zq;q)n |
zn=
infty | |
\sum | |
n=0 |
(-1)nz2n
n2 | |
q |
.
As an analogue of the Barnes integral for the hypergeometric series, Watson showed that
{}2\phi1(a,b;c;q,z)=
-1 | |
2\pii |
(a,b;q)infty | |
(q,c;q)infty |
iinfty | |
\int | |
-iinfty |
| |||||||||||||
|
\pi(-z)s | |
\sin\pis |
ds
(aqs,bq
s;q) | |
infty |
The basic hypergeometric matrix function can be defined as follows:
{}2\phi1(A,B;C;q,z):=
| ||||
\sum | ||||
n=0 |
zn, (A;q)0:=1, (A;q)n:=\prod
n-1 | |
k=0 |
(1-Aqk).
1\psi1