Baileyによる8φ7の3項変換公式

前回の記事で示した $q$ 積分を用いてBaileyによる $_{8} ϕ_{7}$ の3項変換公式を示す. 今回も
$\begin{array}{r} W (a; b, c, d, e, f; x) =_{8} ϕ_{7} [\begin{array}{c} a, \sqrt{a} q, - \sqrt{a} q, b, c, d, e, f \\ \sqrt{a}, - \sqrt{a}, a q / b, a q / c, a q / d, a q / e, a q / f \end{array}; x] \end{array}$
という記法を用いる.

Baileyの

_{8} ϕ_{7}

3項変換公式

$\begin{aligned} W (a; b, c, d, e, f; \frac{a^{2} q^{2}}{b c d e f}) \\ = \frac{(a q, a q / d e, a q / d f, a q / e f, e q / c, f q / c, b / a, b e f / a; q)_{\infty}}{(a q / d, a q / e, a q / f, a q / d e f, q / c, e f q / c, b e / a, b f / a; q)_{\infty}} W (e f / c; a q / b c, a q / c d, e f / a, e, f; \frac{b d}{a}) \\ + \frac{(a q, b q / a, b q / c, b q / d, b q / e, b q / f, d, e, f, a q / b c, b d e f / a^{2} q, a^{2} q^{2} / b d e f; q)_{\infty}}{(a q / b, a q / c, a q / d, a q / e, a q / f, b d / a, b e / a, b f / a, d e f / a q, a q^{2} / d e f, q / c, b^{2} q / a; q)_{\infty}} \\ \cdot W (b^{2} / a; b, b c / a, b d / a, b e / a, b f / a; \frac{a^{2} q^{2}}{b c d e f}) \end{aligned}$

Non-terminating $q$ -Whippleの変換公式の $q$ 積分表示より,
$\begin{aligned} W (a; b, c, d, e, f; \frac{a^{2} q^{2}}{b c d e f}) \\ = \frac{1}{d e f - a q} \frac{(a q, a q / b c, d, e, f, a q / d e, a q / d f, a q / e f; q)_{\infty}}{(a q / b, a q / c, a q / d, a q / e, a q / f, a q^{2} / d e f, d e f / a, q; q)_{\infty}} \int_{a q}^{d e f} \frac{(a t q / b d e f, a t q / c d e f, t q / d e f, t / a; q)_{\infty}}{(t / d e, t / d f, t / e f, a t q / b c d e f; q)_{\infty}} d_{q} t \\ = \frac{1}{d e f - a q} \frac{(a q, a q / b c, d, e, f, a q / d e, a q / d f, a q / e f; q)_{\infty}}{(a q / b, a q / c, a q / d, a q / e, a q / f, a q^{2} / d e f, d e f / a, q; q)_{\infty}} \int_{a q}^{b d e f / a} \frac{(a t q / b d e f, a t q / c d e f, t q / d e f, t / a; q)_{\infty}}{(t / d e, t / d f, t / e f, a t q / b c d e f; q)_{\infty}} d_{q} t \\ + \frac{1}{d e f - a q} \frac{(a q, a q / b c, d, e, f, a q / d e, a q / d f, a q / e f; q)_{\infty}}{(a q / b, a q / c, a q / d, a q / e, a q / f, a q^{2} / d e f, d e f / a, q; q)_{\infty}} \int_{b d e f / a}^{d e f} \frac{(a t q / b d e f, a t q / c d e f, t q / d e f, t / a; q)_{\infty}}{(t / d e, t / d f, t / e f, a t q / b c d e f; q)_{\infty}} d_{q} t \end{aligned}$
であり, 前回の記事で示した $q$ 積分
$\begin{aligned} \int_{a}^{b} \frac{(t q / a, t q / b, c t, d t; q)_{\infty}}{(e t, f t, g t, h t; q)_{\infty}} d_{q} t \\ = (b - a) \frac{(b c, b d, c d / e h, c d / f h, c d / g h, a q / b, b q / a, q; q)_{\infty}}{(b c d / h, b e, b f, b g, b h, a e, a f, a g; q)_{\infty}} W (b c d / h q; c / h, d / h, b e, b f, b g; a h) (c d = a b e f g h) \end{aligned}$
を用いると, $h$ を $a q / b c d e f$ に選んで,
$\begin{aligned} \int_{a q}^{b d e f / a} \frac{(a t q / b d e f, a t q / c d e f, t q / d e f, t / a; q)_{\infty}}{(t / d e, t / d f, t / e f, a t q / b c d e f; q)_{\infty}} d_{q} t \\ = (\frac{b d e f}{a} - a q) \frac{(b q / a, b q / c, b q / d, b q / e, b q / f, a^{2} q^{2} / b d e f, b d e f / a^{2}, q; q)_{\infty}}{(b^{2} q / a, b d / a, b e / a, b f / a, q / c, a q / d e, a q / d f, a q / e f; q)_{\infty}} W (b^{2} / a; b, b c / a, b d / a, b e / a, b f / a; \frac{a^{2} q^{2}}{b c d e f}) \end{aligned}$
となる. $h$ を $1 / e f$ に選んで,
$\begin{aligned} \int_{b d e f / a}^{d e f} \frac{(a t q / b d e f, a t q / c d e f, t q / d e f, t / a; q)_{\infty}}{(t / d e, t / d f, t / e f, a t q / b c d e f; q)_{\infty}} d_{q} t \\ = (d e f - \frac{b d e f}{a}) \frac{(a q / c, d e f / a, e q / c, f q / c, b e f / a, b q / a, a q / b, q; q)_{\infty}}{(e f q / c, d, e, f, a q / b c, b e / a, b f / a, q / c; q)_{\infty}} W (e f / c; e f / a, a q / c d, a q / b c, e, f; \frac{b d}{a}) \end{aligned}$
だから,
$\begin{aligned} W (a; b, c, d, e, f; \frac{a^{2} q^{2}}{b c d e f}) \\ = \frac{1}{d e f - a q} \frac{(a q, a q / b c, d, e, f, a q / d e, a q / d f, a q / e f; q)_{\infty}}{(a q / b, a q / c, a q / d, a q / e, a q / f, a q^{2} / d e f, d e f / a, q; q)_{\infty}} \int_{a q}^{b d e f / a} \frac{(a t q / b d e f, a t q / c d e f, t q / d e f, t / a; q)_{\infty}}{(t / d e, t / d f, t / e f, a t q / b c d e f; q)_{\infty}} d_{q} t \\ + \frac{1}{d e f - a q} \frac{(a q, a q / b c, d, e, f, a q / d e, a q / d f, a q / e f; q)_{\infty}}{(a q / b, a q / c, a q / d, a q / e, a q / f, a q^{2} / d e f, d e f / a, q; q)_{\infty}} \int_{b d e f / a}^{d e f} \frac{(a t q / b d e f, a t q / c d e f, t q / d e f, t / a; q)_{\infty}}{(t / d e, t / d f, t / e f, a t q / b c d e f; q)_{\infty}} d_{q} t \\ = \frac{1}{d e f - a q} \frac{(a q, a q / b c, d, e, f, a q / d e, a q / d f, a q / e f; q)_{\infty}}{(a q / b, a q / c, a q / d, a q / e, a q / f, a q^{2} / d e f, d e f / a, q; q)_{\infty}} \\ \cdot (\frac{b d e f}{a} - a q) \frac{(b q / a, b q / c, b q / d, b q / e, b q / f, a^{2} q^{2} / b d e f, b d e f / a^{2}, q; q)_{\infty}}{(b^{2} q / a, b d / a, b e / a, b f / a, q / c, a q / d e, a q / d f, a q / e f; q)_{\infty}} W (b^{2} / a; b, b c / a, b d / a, b e / a, b f / a; \frac{a^{2} q^{2}}{b c d e f}) \\ + \frac{1}{d e f - a q} \frac{(a q, a q / b c, d, e, f, a q / d e, a q / d f, a q / e f; q)_{\infty}}{(a q / b, a q / c, a q / d, a q / e, a q / f, a q^{2} / d e f, d e f / a, q; q)_{\infty}} \\ \cdot (d e f - \frac{b d e f}{a}) \frac{(a q / c, d e f / a, e q / c, f q / c, b e f / a, b q / a, a q / b, q; q)_{\infty}}{(e f q / c, d, e, f, a q / b c, b e / a, b f / a, q / c; q)_{\infty}} W (e f / c; e f / a, a q / c d, a q / b c, e, f; \frac{b d}{a}) \\ = \frac{(a q, a q / d e, a q / d f, a q / e f, e q / c, f q / c, b e f / a, b / a; q)_{\infty}}{(a q / d, a q / e, a q / f, a q / d e f, e f q / c, b e / a, b f / a, q / c; q)_{\infty}} W (e f / c; e f / a, a q / c d, a q / b c, e, f; \frac{b d}{a}) \\ + \frac{(a q, a q / b c, d, e, f, b q / a, b q / c, b q / d, b q / e, b q / f, a^{2} q^{2} / b d e f, b d e f / a^{2} q; q)_{\infty}}{(a q / b, a q / c, a q / d, a q / e, a q / f, a q^{2} / d e f, d e f / a q, b^{2} q / a, b d / a, b e / a, b f / a, q / c; q)_{\infty}} \\ \cdot W (b^{2} / a; b, b c / a, b d / a, b e / a, b f / a; \frac{a^{2} q^{2}}{b c d e f}) \end{aligned}$
と示される.