Basic hypergeometric series

In mathematics, Heine's basic hypergeometric series, or hypergeometric q-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 ₂φ₁(q^α,q^β;q^γ;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

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

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

where

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

and

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

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 ... 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 bilateral basic hypergeometric series, corresponding to the bilateral hypergeometric series, is defined as

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

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

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

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

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

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+1φ_r. This contour integral gives an analytic continuation of the basic hypergeometric function in z.

Notes

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

• Andrews, G. E. (2010), "q-Hypergeometric and Related Functions", in Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W., NIST Handbook of Mathematical Functions, Cambridge University Press, ISBN 978-0521192255, MR 2723248
• W.N. Bailey, Generalized Hypergeometric Series, (1935) Cambridge Tracts in Mathematics and Mathematical Physics, No.32, Cambridge University Press, Cambridge.
• William Y. C. Chen and Amy Fu, Semi-Finite Forms of Bilateral Basic Hypergeometric Series (2004)
• 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
• Sylvie Corteel and Jeremy Lovejoy, Frobenius Partitions and the Combinatorics of Ramanujan's $\,_{1}\psi _{1}$ Summation
• Fine, Nathan J. (1988), Basic hypergeometric series and applications, Mathematical Surveys and Monographs, 27, Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-1524-3, MR 956465
• Gasper, George; Rahman, Mizan (2004), Basic hypergeometric series, Encyclopedia of Mathematics and its Applications, 96 (2nd ed.), Cambridge University Press, doi:10.2277/0521833574, ISBN 978-0-521-83357-8, MR 2128719
• Heine, Eduard (1846), "Über die Reihe $1+{\frac {(q^{\alpha }-1)(q^{\beta }-1)}{(q-1)(q^{\gamma }-1)}}x+{\frac {(q^{\alpha }-1)(q^{{\alpha +1}}-1)(q^{\beta }-1)(q^{{\beta +1}}-1)}{(q-1)(q^{2}-1)(q^{\gamma }-1)(q^{{\gamma +1}}-1)}}x^{2}+\cdots$", Journal für die reine und angewandte Mathematik, 32: 210–212
• Eduard Heine, Theorie der Kugelfunctionen, (1878) 1, pp 97–125.
• Eduard Heine, Handbuch die Kugelfunctionen. Theorie und Anwendung (1898) Springer, Berlin.