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Heine Hypergeometric Series


\begin{displaymath}
{}_r\phi_s\left[{\matrix{\alpha_1, \alpha_2, \ldots, \alpha_...
...alpha_r;q)_n\over(q;q)_n(\beta_1;q)_n\cdots(\beta_s;q)_n} z^n,
\end{displaymath} (1)

where
$\displaystyle (a;q)_n$ $\textstyle =$ $\displaystyle (1-a)(1-aq)(1-aq^2)\cdots(1-aq^{n-1}),$ (2)
$\displaystyle (a;q)_0$ $\textstyle =$ $\displaystyle 1.$ (3)

In particular,
\begin{displaymath}
{}_2\psi_1(a,b;c;q,z)=\sum_{n=0}^\infty {(a;q)_n(b;q)_n z^n\over (q;q)_n(c;q)_n}
\end{displaymath} (4)

(Andrews 1986, p. 10). Heine proved the transformation formula
\begin{displaymath}
{}_2\phi_1(a,b;c;q,z)={(b;q)_\infty(az;q)_\infty\over (c;q)_\infty(z;q)_\infty} {}_2\phi_1(c/b,a;az;q,b),
\end{displaymath} (5)

and Rogers (1893) obtained the formulas


\begin{displaymath}
{}_2\phi_1(a,b;c;q,z)={(c/b;q)_\infty(bz;q)_\infty\over(z;q)_\infty(c;q)_\infty} {}_2\phi_1(b,abz/c;bz;q,c/b)
\end{displaymath} (6)


\begin{displaymath}
{}_2\phi_1(a,b,c;q,z)={(abz/c;q)_\infty(z;q)_\infty} {}_2\phi_1(c/a,c/b;c;q,abz/c)
\end{displaymath} (7)

(Andrews 1986, pp. 10-11).

See also q-Series


References

Andrews, G. E. $q$-Series: Their Development and Application in Analysis, Number Theory, Combinatorics, Physics, and Computer Algebra. Providence, RI: Amer. Math. Soc., p. 10, 1986.

Heine, E. ``Über die Reihe $1+{(q^\alpha-1)(q^\beta-1)\over(q-1)(q^\gamma-1)}x$
$+{(q^\alpha-1)(q^{\alpha+1}-1)(q^\beta-1)(q^{\beta+1}-1)\over
(q-1)(q^2-1)(q^\gamma-1)(q^{\gamma+1}-1)}x^2+\ldots$.'' J. reine angew. Math. 32, 210-212, 1846.

Heine, E. ``Untersuchungen über die Reihe $1+{(1-q^\alpha)(1-q^\beta)\over(1-q)(1-q^\gamma)}\cdot x+{(1-q^\alpha)(1-q^{\al...
...(1-q^{\beta+1})\over
(1-q)(1-q^2)(1-q^\gamma)(1-q^{\gamma+1})}\cdot x^2+\ldots$.'' J. reine angew. Math. 34, 285-328, 1847.

Heine, E. Theorie der Kugelfunctionen und der verwandten Functionen, Vol. 1. Berlin: Reimer, 1878.

Rogers, L. J. ``On a Three-Fold Symmetry in the Elements of Heine's Series.'' Proc. London Math. Soc. 24, 171-179, 1893.




© 1996-9 Eric W. Weisstein
1999-05-25