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 x_n is called hypergeometric if the ratio of successive terms x_n+1/x_n 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 ${\displaystyle {}_{2}\phi _{1}(q^{\alpha },q^{\beta };q^{\gamma };q,x)}$ was first considered by Eduard Heine (1846). It becomes the hypergeometric series ${\displaystyle F(\alpha ,\beta ;\gamma ;x)}$ in the limit when base ${\displaystyle q=1}$.

Definition

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

${\displaystyle \;_{j}\phi _{k}\left[{\begin{matrix}a_{1}&a_{2}&\ldots &a_{j}\\b_{1}&b_{2}&\ldots &b_{k}\end{matrix));q,z\right]=\sum _{n=0}^{\infty }{\frac {(a_{1},a_{2},\ldots ,a_{j};q)_{n)){(b_{1},b_{2},\ldots ,b_{k},q;q)_{n))}\left((-1)^{n}q^{n \choose 2}\right)^{1+k-j}z^{n))$

where

${\displaystyle (a_{1},a_{2},\ldots ,a_{m};q)_{n}=(a_{1};q)_{n}(a_{2};q)_{n}\ldots (a_{m};q)_{n))$

and

${\displaystyle (a;q)_{n}=\prod _{k=0}^{n-1}(1-aq^{k})=(1-a)(1-aq)(1-aq^{2})\cdots (1-aq^{n-1})}$

is the q-shifted factorial. The most important special case is when j = k + 1, when it becomes

${\displaystyle \;_{k+1}\phi _{k}\left[{\begin{matrix}a_{1}&a_{2}&\ldots &a_{k}&a_{k+1}\\b_{1}&b_{2}&\ldots &b_{k}\end{matrix));q,z\right]=\sum _{n=0}^{\infty }{\frac {(a_{1},a_{2},\ldots ,a_{k+1};q)_{n)){(b_{1},b_{2},\ldots ,b_{k},q;q)_{n))}z^{n}.}$

This series is called balanced if a₁ ... a_{k + 1} = b₁ ...b_kq. This series is called well poised if a₁q = a₂b₁ = ... = a_{k + 1}b_k, and very well poised if in addition a₂ = −a₃ = qa₁^1/2. The unilateral basic hypergeometric series is a q-analog of the hypergeometric series since

${\displaystyle \lim _{q\to 1}\;_{j}\phi _{k}\left[{\begin{matrix}q^{a_{1))&q^{a_{2))&\ldots &q^{a_{j))\\q^{b_{1))&q^{b_{2))&\ldots &q^{b_{k))\end{matrix));q,(q-1)^{1+k-j}z\right]=\;_{j}F_{k}\left[{\begin{matrix}a_{1}&a_{2}&\ldots &a_{j}\\b_{1}&b_{2}&\ldots &b_{k}\end{matrix));z\right]}$

holds (Koekoek & Swarttouw (1996)).
The bilateral basic hypergeometric series, corresponding to the bilateral hypergeometric series, is defined as

${\displaystyle \;_{j}\psi _{k}\left[{\begin{matrix}a_{1}&a_{2}&\ldots &a_{j}\\b_{1}&b_{2}&\ldots &b_{k}\end{matrix));q,z\right]=\sum _{n=-\infty }^{\infty }{\frac {(a_{1},a_{2},\ldots ,a_{j};q)_{n)){(b_{1},b_{2},\ldots ,b_{k};q)_{n))}\left((-1)^{n}q^{n \choose 2}\right)^{k-j}z^{n}.}$

The most important special case is when j = k, when it becomes

${\displaystyle \;_{k}\psi _{k}\left[{\begin{matrix}a_{1}&a_{2}&\ldots &a_{k}\\b_{1}&b_{2}&\ldots &b_{k}\end{matrix));q,z\right]=\sum _{n=-\infty }^{\infty }{\frac {(a_{1},a_{2},\ldots ,a_{k};q)_{n)){(b_{1},b_{2},\ldots ,b_{k};q)_{n))}z^{n}.}$

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

Some simple series expressions include

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

and

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

and

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

The q-binomial theorem

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

${\displaystyle \;_{1}\phi _{0}(a;q,z)={\frac {(az;q)_{\infty )){(z;q)_{\infty ))}=\prod _{n=0}^{\infty }{\frac {1-aq^{n}z}{1-q^{n}z))}$

which follows by repeatedly applying the identity

${\displaystyle \;_{1}\phi _{0}(a;q,z)={\frac {1-az}{1-z))\;_{1}\phi _{0}(a;q,qz).}$

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

Cauchy binomial theorem

Cauchy binomial theorem is a special case of the q-binomial theorem.^[3]

${\displaystyle \sum _{n=0}^{N}y^{n}q^{n(n+1)/2}{\begin{bmatrix}N\\n\end{bmatrix))_{q}=\prod _{k=1}^{N}\left(1+yq^{k}\right)\qquad (|q|<1)}$

Ramanujan's identity

Srinivasa Ramanujan gave the identity

${\displaystyle \;_{1}\psi _{1}\left[{\begin{matrix}a\\b\end{matrix));q,z\right]=\sum _{n=-\infty }^{\infty }{\frac {(a;q)_{n)){(b;q)_{n))}z^{n}={\frac {(b/a,q,q/az,az;q)_{\infty )){(b,b/az,q/a,z;q)_{\infty ))))$

valid for |q| < 1 and |b/a| < |z| < 1. Similar identities for ${\displaystyle \;_{6}\psi _{6))$ 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

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

Ken Ono gives a related formal power series^[4]

${\displaystyle A(z;q){\stackrel {\rm {def)){=)){\frac {1}{1+z))\sum _{n=0}^{\infty }{\frac {(z;q)_{n)){(-zq;q)_{n))}z^{n}=\sum _{n=0}^{\infty }(-1)^{n}z^{2n}q^{n^{2)).}$

Watson's contour integral

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

${\displaystyle {}_{2}\phi _{1}(a,b;c;q,z)={\frac {-1}{2\pi i)){\frac {(a,b;q)_{\infty )){(q,c;q)_{\infty ))}\int _{-i\infty }^{i\infty }{\frac {(qq^{s},cq^{s};q)_{\infty )){(aq^{s},bq^{s};q)_{\infty ))}{\frac {\pi (-z)^{s)){\sin \pi s))ds}$

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

Matrix version

The basic hypergeometric matrix function can be defined as follows:

${\displaystyle {}_{2}\phi _{1}(A,B;C;q,z):=\sum _{n=0}^{\infty }{\frac {(A;q)_{n}(B;q)_{n)){(C;q)_{n}(q;q)_{n))}z^{n},\quad (A;q)_{0}:=1,\quad (A;q)_{n}:=\prod _{k=0}^{n-1}(1-Aq^{k}).}$

The ratio test shows that this matrix function is absolutely convergent.^[5]