Baileyのterminating 10φ9変換公式

$w = a^{2} q / b c d, a^{3} q^{n + 2} = b c d e f g$ のときに成り立つ, Baileyによるterminating $_{10} ϕ_{9}$ の変換公式
$\begin{aligned} _{10} ϕ_{9} [\begin{array}{c} a, \sqrt{a} q, - \sqrt{a} q, b, c, d, e, f, g, q^{- n} \\ \sqrt{a}, - \sqrt{a}, a q / b, a q / c, a q / d, a q / e, a q / f, a q / g, a q^{n + 1} \end{array}; q] \\ = \frac{(a q, a q / e f, w q / e, w q / f; q)_{n}}{(a q / e, a q / f, w q, w q / e f; q)_{n}}_{10} ϕ_{9} [\begin{array}{c} w, \sqrt{w} q, - \sqrt{w} q, b, w b / a, w c / a, w d / a, e, f, g, q^{- n} \\ \sqrt{w}, - \sqrt{w}, a q / b, a q / c, a q / d, w q / e, w q / f, w q / g, w q^{n + 1} \end{array}; q] \end{aligned}$
を示す.

Baileyのterminating

_{10} ϕ_{9}

変換公式

$n$ を非負整数, $w = a^{2} q / b c d, a^{3} q^{n + 2} = b c d e f g$ とするとき,
$\begin{aligned} _{10} ϕ_{9} [\begin{array}{c} a, \sqrt{a} q, - \sqrt{a} q, b, c, d, e, f, g, q^{- n} \\ \sqrt{a}, - \sqrt{a}, a q / b, a q / c, a q / d, a q / e, a q / f, a q / g, a q^{n + 1} \end{array}; q] \\ = \frac{(a q, a q / e f, w q / e, w q / f; q)_{n}}{(a q / e, a q / f, w q, w q / e f; q)_{n}}_{10} ϕ_{9} [\begin{array}{c} w, \sqrt{w} q, - \sqrt{w} q, w b / a, w c / a, w d / a, e, f, g, q^{- n} \\ \sqrt{w}, - \sqrt{w}, a q / b, a q / c, a q / d, w q / e, w q / f, w q / g, w q^{n + 1} \end{array}; q] \end{aligned}$
が成り立つ.

まず, Jacksonの $_{8} ϕ_{7}$ 和公式より, $w = a^{2} q / b c d$ としたとき,
$\begin{aligned} _{8} ϕ_{7} [\begin{array}{c} w, \sqrt{w} q, - \sqrt{w} q, w b / a, w c / a, w d / a, a q^{n}, q^{- n} \\ \sqrt{w}, - \sqrt{w}, a q / b, a q / c, a q / d, w q^{1 - n} / a, w q^{n + 1} \end{array}; q] & = \frac{(b, c, d, w q; q)_{n}}{(a q / b, a q / c, a q / d, a / w; q)_{n}} \end{aligned}$
より,
$\begin{aligned} \frac{(a, b, c, d; q)_{n}}{(a q / b, a q / c, a q / d, q; q)_{n}} & = \frac{(a, a / w; q)_{n}}{(w q, q; q)_{n}} \sum_{k = 0}^{n} \frac{(w, \sqrt{w} q, - \sqrt{w} q, w b / a, w c / a, w d / a, a q^{n}, q^{- n})_{k}}{(\sqrt{w}, - \sqrt{w}, a q / b, a q / c, a q / d, w q^{1 - n} / a, w q^{n + 1}, q; q)_{k}} q^{k} \\ = \sum_{k = 0}^{n} \frac{(w, \sqrt{w} q, - \sqrt{w} q, w b / a, w c / a, w d / a)_{k} (a; q)_{n + k} (a / w; q)_{n - k}}{(\sqrt{w}, - \sqrt{w}, a q / b, a q / c, a q / d, q; q)_{k} (w q; q)_{n + k} (q; q)_{n - k}} {(\frac{a}{w})}^{k} \end{aligned}$
であるから,
$\begin{aligned} _{10} ϕ_{9} [\begin{array}{c} a, \sqrt{a} q, - \sqrt{a} q, b, c, d, e, f, g, q^{- n} \\ \sqrt{a}, - \sqrt{a}, a q / b, a q / c, a q / d, a q / e, a q / f, a q / g, a q^{n + 1} \end{array}; q] \\ = \sum_{0 \leq k} \frac{(\sqrt{a} q, - \sqrt{a} q, e, f, g, q^{- n})_{k}}{(\sqrt{a}, - \sqrt{a}, a q / e, a q / f, a q / g, a q^{n + 1}; q)_{k}} q^{k} \sum_{j = 0}^{k} \frac{(w, \sqrt{w} q, - \sqrt{w} q, w b / a, w c / a, w d / a; q)_{j} (a; q)_{k + j} (a / w; q)_{k - j}}{(\sqrt{w}, - \sqrt{w}, a q / b, a q / c, a q / d, q; q)_{j} (w q; q)_{k + j} (q; q)_{k - j}} {(\frac{a}{w})}^{j} \\ = \sum_{j = 0}^{n} \frac{(w, \sqrt{w} q, - \sqrt{w} q, w b / a, w c / a, w d / a; q)_{j}}{(\sqrt{w}, - \sqrt{w}, a q / b, a q / c, a q / d, q; q)_{j}} {(\frac{a}{w})}^{j} \sum_{k = j}^{n} \frac{(\sqrt{a} q, - \sqrt{a} q, e, f, g, q^{- n})_{k} (a; q)_{k + j} (a / w; q)_{k - j}}{(\sqrt{a}, - \sqrt{a}, a q / e, a q / f, a q / g, a q^{n + 1}; q)_{k} (w q; q)_{k + j} (q; q)_{k - j}} q^{k} \\ = \sum_{j = 0}^{n} \frac{(w, \sqrt{w} q, - \sqrt{w} q, w b / a, w c / a, w d / a, \sqrt{a} q, - \sqrt{a} q, e, f, g, q^{- n}; q)_{j} (a; q)_{2 j}}{(\sqrt{w}, - \sqrt{w}, a q / b, a q / c, a q / d, \sqrt{a}, - \sqrt{a}, a q / e, a q / f, a q / g, a q^{n + 1}, q; q)_{j} (w q; q)_{2 j}} {(\frac{a q}{w})}^{j} \\ \cdot \sum_{k = 0}^{n - j} \frac{(a q^{2 j}, \sqrt{a} q^{j + 1}, - \sqrt{a} q^{j + 1}, e q^{j}, f q^{j}, g q^{j}, a / w, q^{j - n})_{k}}{(\sqrt{a} q^{j}, - \sqrt{a} q^{j}, a q^{j + 1} / e, a q^{j + 1} / f, a q^{j + 1} / g, a q^{n + j + 1}, w q^{2 j + 1}, q; q)_{k}} q^{k} \end{aligned}$
ここで, Jacksonの $_{8} ϕ_{7}$ 和公式より, $a^{3} q^{n + 2} = b c d e f g$ のとき,
$\begin{aligned} \sum_{k = 0}^{n - j} \frac{(a q^{2 j}, \sqrt{a} q^{j + 1}, - \sqrt{a} q^{j + 1}, e q^{j}, f q^{j}, g q^{j}, a / w, q^{j - n})_{k}}{(\sqrt{a} q^{j}, - \sqrt{a} q^{j}, a q^{j + 1} / e, a q^{j + 1} / f, a q^{j + 1} / g, a q^{n + j + 1}, w q^{2 j + 1}, q; q)_{k}} q^{k} \\ = \frac{(a q^{2 j + 1}, a q / e f, a q / e g, a q / f g; q)_{n - j}}{(a q^{j + 1} / e, a q^{j + 1} / f, a q^{j + 1} / g, a q^{1 - j} / e f g; q)_{n - j}} \\ = \frac{(a q; q)_{n + j}}{(a q; q)_{2 j}} \frac{(a q / e, a q / f, a q / g; q)_{j}}{(a q / e, a q / f, a q / g; q)_{n}} \frac{(e q^{- n} / w, f q^{- n} / w, g q^{- n} / w; q)_{n - j}}{(q^{- n - j} / w; q)_{n - j}} \\ = \frac{(a q; q)_{n + j}}{(a q; q)_{2 j}} \frac{(a q / e, a q / f, a q / g; q)_{j}}{(a q / e, a q / f, a q / g; q)_{n}} \frac{(w q^{j + 1} / e, w q^{j + 1} / f, w q^{j + 1} / g; q)_{n - j}}{(w q^{2 j + 1}; q)_{n - j}} {(\frac{e f g q^{j - 2 n}}{w^{2}})}^{n - j} q^{2 (\binom{n - j}{2})} \\ = \frac{(a q; q)_{n + j}}{(a q; q)_{2 j}} \frac{(a q / e, a q / f, a q / g; q)_{j}}{(a q / e, a q / f, a q / g; q)_{n}} \frac{(w q^{j + 1} / e, w q^{j + 1} / f, w q^{j + 1} / g; q)_{n - j}}{(w q^{2 j + 1}; q)_{n - j}} {(\frac{a q^{1 + j - n}}{w})}^{n - j} q^{2 (\binom{n - j}{2})} \\ = \frac{(a q; q)_{n + j}}{(a q; q)_{2 j}} \frac{(a q / e, a q / f, a q / g; q)_{j}}{(a q / e, a q / f, a q / g; q)_{n}} \frac{(w q / e, w q / f, w q / g; q)_{n}}{(w q / e, w q / f, w q / g; q)_{j}} \frac{(w q; q)_{2 j}}{(w q; q)_{n + j}} {(\frac{a}{w})}^{n - j} \end{aligned}$
より,
$\begin{aligned} _{10} ϕ_{9} [\begin{array}{c} a, \sqrt{a} q, - \sqrt{a} q, b, c, d, e, f, g, q^{- n} \\ \sqrt{a}, - \sqrt{a}, a q / b, a q / c, a q / d, a q / e, a q / f, a q / g, a q^{n + 1} \end{array}; q] \\ = \frac{(a q, w q / e, w q / f, w q / g; q)_{n}}{(w q, a q / e, a q / f, a q / g; q)_{n}} \sum_{j = 0}^{n} \frac{(w, \sqrt{w} q, - \sqrt{w} q, w b / a, w c / a, w d / a, \sqrt{a} q, - \sqrt{a} q, e, f, g, q^{- n}; q)_{j} (a; q)_{2 j}}{(\sqrt{w}, - \sqrt{w}, a q / b, a q / c, a q / d, \sqrt{a}, - \sqrt{a}, a q / e, a q / f, a q / g, a q^{n + 1}, q; q)_{j} (w q; q)_{2 j}} \\ \cdot \frac{(a q / e, a q / f, a q / g, a q^{n + 1}; q)_{j}}{(w q / e, w q / f, w q / g, w q^{n + 1}; q)_{j}} \frac{(w q; q)_{2 j}}{(a q; q)_{2 j}} {(\frac{a}{w})}^{n - j} \\ = \frac{(a q, w q / e, w q / f, w q / g; q)_{n}}{(w q, a q / e, a q / f, a q / g; q)_{n}} {(\frac{a}{w})}^{n} \sum_{j = 0}^{n} \frac{(w, \sqrt{w} q, - \sqrt{w} q, w b / a, w c / a, w d / a, e, f, g, q^{- n}; q)_{j}}{(\sqrt{w}, - \sqrt{w}, a q / b, a q / c, a q / d, w q / e, w q / f, w q / g, w q^{n + 1}, q; q)_{j}} q^{j} \end{aligned}$
最後に, $g = a w q^{n + 1} / e f$ であることから,
$\begin{aligned} \frac{(w q / g; q)_{n}}{(a q / g; q)_{n}} & = \frac{(e f q^{- n} / a; q)_{n}}{(e f q^{- n} / w; q)_{n}} = \frac{(a q / e f; q)_{n}}{(w q / e f; q)_{n}} {(\frac{w}{a})}^{n} \end{aligned}$
であることから定理を得る.

$a, b, c, d, e, f$ を固定して, $n \to \infty$ とすると, 以下のnon-terminatingな $_{8} ϕ_{7}$ の変換公式を得る.

$w = a^{2} q / b c d$ としたとき,
$\begin{aligned} _{8} ϕ_{7} [\begin{array}{c} a, \sqrt{a} q, - \sqrt{a} q, b, c, d, e, f \\ \sqrt{a}, - \sqrt{a}, a q / b, a q / c, a q / d, a q / e, a q / f \end{array}; \frac{a^{2} q^{2}}{b c d e f}] & = \frac{(a q, a q / e f, w q / e, w q / f; q)_{\infty}}{(a q / e, a q / f, w q, w q / e f; q)_{\infty}}_{8} ϕ_{7} [\begin{array}{c} w, \sqrt{w} q, - \sqrt{w} q, w b / a, w c / a, w d / a, e, f \\ \sqrt{w}, - \sqrt{w}, a q / b, a q / c, a q / d, w q / e, w q / f \end{array}; \frac{a q}{e f}] \end{aligned}$
が成り立つ.