Heine Hypergeometric Series

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

where

$$(a;q)_n = (1-a)(1-aq)(1-aq^2)\cdots(1-aq^{n-1}),$$ (2)

$$(a;q)_0 = 1.$$ (3)

In particular,

$${}_2\psi_1(a,b;c;q,z)=\sum_{n=0}^\infty {(a;q)_n(b;q)_n z^n\over (q;q)_n(c;q)_n}$$ (4)

(Andrews 1986, p. 10). Heine proved the transformation formula

$${}_2\phi_1(a,b;c;q,z)={(b;q)_\infty(az;q)_\infty\over (c;q)_\infty(z;q)_\infty} {}_2\phi_1(c/b,a;az;q,b),$$ (5)

and Rogers (1893) obtained the formulas

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

$${}_2\phi_1(a,b,c;q,z)={(abz/c;q)_\infty(z;q)_\infty} {}_2\phi_1(c/a,c/b;c;q,abz/c)$$ (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.