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Rahmanの超幾何積分

前回の記事で, Rahmanによる積分
$\begin{array}{r} \frac{1}{2 π} \int_{- 1}^{1} {| \frac{(e^{2 i θ}, a b c s t e^{i θ}; q)_{\infty}}{(a e^{i θ}, b e^{i θ}, c e^{i θ}, s e^{i θ}, t e^{i θ}; q)_{\infty}} |}^{2} \frac{d x}{\sqrt{1 - x^{2}}} = \frac{(a b c s, a b c t, a b s t, a c s t, b c s t; q)_{\infty}}{(q, a b, a c, b c, a s, b s, c s, a t, b t, c t, s t; q)_{\infty}}, x = \cos θ \end{array}$
を示した. それを用いて, 以下の一般化を示す.

Rahman(1986)

$A B C q = (a b c s t)^{2}$ のとき,
$\begin{aligned} \frac{1}{2 π} \int_{- 1}^{1} {| \frac{(e^{2 i θ}, a b c s t e^{i θ} / B, a b c s t e^{i θ} / C; q)_{\infty}}{(a e^{i θ}, b e^{i θ}, c e^{i θ}, s e^{i θ}, t e^{i θ}, A q e^{i θ} / a b c s t; q)_{\infty}} |}^{2} \frac{d x}{\sqrt{1 - x^{2}}} \\ = \frac{(a b c s t^{2} / B, a b c s t^{2} / C, A B q / a b c s t^{2}, A C q / a b c s t^{2}, a b c t, a b s t, a c s t, b c s t; q)_{\infty}}{(a b c s t^{2}, A q / a b c s t^{2}, a b c s t^{2} / B C, q, a b, a c, b c, a s, b s, c s, a t, b t, c t, s t; q)_{\infty}} \\ \cdot_{10} ϕ_{9} [\begin{array}{c} a b c s t^{2} / q, q \sqrt{a b c s t^{2} / q}, - q \sqrt{a b c s t^{2} / q}, A, B, C, a t, b t, c t, s t \\ \sqrt{a b c s t^{2} / q}, - \sqrt{a b c s t^{2} / q}, a b c s t^{2} / A, a b c s t^{2} / B, a b c s t^{2} / C, a b c t, a b s t, a c s t, b c s t \end{array}; q] \\ + \frac{(A q / B, A q / C, B, C, A q / a t, A q / b t, A q / c t, A q / s t; q)_{\infty}}{(a b c s t^{2} / B C, A^{2} q^{2} / a b c s t^{2}, a b c s t^{2} / A q, q, a b, a c, b c, a s, b s, c s, A q / a b s t, A q / a c s t, A q / b c s t, A q / a b c t; q)_{\infty}} \\ \cdot_{10} ϕ_{9} [\begin{array}{c} A^{2} q / a b c s t^{2}, q \sqrt{A^{2} q / a b c s t^{2}}, - q \sqrt{A^{2} q / a b c s t^{2}}, A, A B q / a b c s t^{2}, A C q / a b c s t^{2}, A q / a b s t, A q / a c s t, A q / b c s t, A q / a b c t \\ \sqrt{A^{2} q / a b c s t^{2}}, - \sqrt{A^{2} q / a b c s t^{2}}, A / a b c s t^{2}, A q / B, A q / C, A q / a t, A q / b t, A q / c t, A q / s t \end{array}; q] \end{aligned}$
が成り立つ. ただし, $x = \cos θ$ である.

$A B C q = (a b c s t)^{2}$ として, non-terminating Jacksonの $_{8} ϕ_{7}$ 和公式より,
$\begin{aligned} _{8} ϕ_{7} [\begin{array}{c} a b c s t^{2} / q, q \sqrt{a b c s t^{2} / q}, - q \sqrt{a b c s t^{2} / q}, A, B, C, t e^{i θ}, t e^{- i θ} \\ \sqrt{a b c s t^{2} / q}, - \sqrt{a b c s t^{2} / q}, a b c s t^{2} / A, a b c s t^{2} / B, a b c s t^{2} / C, a b c s t e^{i θ}, a b c s t e^{- i θ} \end{array}; q] \\ + \frac{(A q / B, A q / C, A q e^{i θ} / t, A q e^{- i θ} / t, a b c s t^{2}, B, C, t e^{i θ}, t e^{- i θ}, A q / a b c s t^{2}; q)_{\infty}}{(A^{2} q^{2} / a b c s t^{2}, A B q / a b c s t^{2}, A C q / a b c s t^{2}, A q e^{i θ} / a b c s t, A q e^{- i θ} / a b c s t; q)_{\infty}} \\ \cdot \frac{1}{(a b c s t^{2} / B, a b c s t^{2} / C, a b c s t e^{i θ}, a b c s t e^{- i θ}, a b c s t^{2} / A q; q)_{\infty}} \\ \cdot_{8} ϕ_{7} [\begin{array}{c} A^{2} q / a b c s t^{2}, q \sqrt{A^{2} q / a b c s t^{2}}, - q \sqrt{A^{2} q / a b c s t^{2}}, A, A B q / a b c s t^{2}, A C q / a b c s t^{2}, A q t e^{i θ} / a b c s t, A q e^{- i θ} / a b c s t \\ \sqrt{A^{2} q / a b c s t^{2}}, - \sqrt{A^{2} q / a b c s t^{2}}, A / a b c s t^{2}, A q / B, A q / C, A q e^{i θ} / t, A q e^{- i θ} / t \end{array}; q] \\ = \frac{(a b c s t^{2}, A q / a b c s t^{2}, a b c s t^{2} / B C, a b c s t e^{i θ} / B, a b c s t e^{- i θ} / B, a b c s t e^{i θ} / C, a b c s t e^{- i θ} / C, a b c s; q)_{\infty}}{(a b c s t^{2} / B, a b c s t^{2} / C, a b c s t e^{i θ}, a b c s t e^{- i θ}, A B q / a b c s t^{2}, A C q / a b c s t^{2}, A q e^{i θ} / a b c s t, A q e^{- i θ} / a b c s t; q)_{\infty}} \end{aligned}$
ここで, これに
$\begin{array}{r} \frac{1}{2 π} {| \frac{(e^{2 i θ}, a b c s t e^{i θ}; q)_{\infty}}{(a e^{i θ}, b e^{i θ}, c e^{i θ}, s e^{i θ}, t e^{i θ}; q)_{\infty}} |}^{2} \frac{1}{\sqrt{1 - x^{2}}} \end{array}$
を掛けて積分すると,
$\begin{aligned} \frac{1}{2 π} \int_{- 1}^{1} {| \frac{(e^{2 i θ}, a b c s t e^{i θ}; q)_{\infty}}{(a e^{i θ}, b e^{i θ}, c e^{i θ}, s e^{i θ}, t e^{i θ}; q)_{\infty}} |}^{2} \frac{(t e^{i θ}, t e^{- i θ}; q)_{r}}{(a b c s t e^{i θ}, a b c s t e^{- i θ}; q)_{r}} \frac{d x}{\sqrt{1 - x^{2}}} \\ = \frac{(a b c s, a b c t, a b s t, a c s t, b c s t; q)_{\infty}}{(q, a b, a c, b c, a s, b s, c s, a t, b t, c t, s t; q)_{\infty}} \frac{(a t, b t, c t, s t; q)_{r}}{(a b c t, a b s t, a c s t, b c s t; q)_{r}} \\ \frac{1}{2 π} \int_{- 1}^{1} {| \frac{(e^{2 i θ}, a b c s t e^{i θ}; q)_{\infty}}{(a e^{i θ}, b e^{i θ}, c e^{i θ}, s e^{i θ}, t e^{i θ}; q)_{\infty}} |}^{2} \frac{(A q e^{i θ} / t, A q e^{- i θ} / t, t e^{i θ}, t e^{- i θ}; q)_{\infty}}{(A q e^{i θ} / a b c s t, A q e^{- i θ} / a b c s t, a b c s t e^{i θ}, a b c s t e^{- i θ}; q)_{\infty}} \\ \cdot \frac{(A q e^{i θ} / a b c s t^{2}, A q e^{- i θ} / a b c s t^{2}; q)_{r}}{(A q e^{i θ} / t, A q e^{- i θ} / t; q)_{r}} \frac{d x}{\sqrt{1 - x^{2}}} \\ = \frac{1}{2 π} \int_{- 1}^{1} {| \frac{(e^{2 i θ}, A q^{r + 1} e^{i θ} / t; q)_{\infty}}{(a e^{i θ}, b e^{i θ}, c e^{i θ}, s e^{i θ}, A q^{r + 1} e^{i θ} / a b c s t; q)_{\infty}} |}^{2} \frac{d x}{\sqrt{1 - x^{2}}} \\ = \frac{(a b c s, A q / a t, A q / b t, A q / c t, A q / s t; q)_{\infty}}{(q, a b, a c, b c, a s, b s, c s, A q / a b s t, A q / a c s t, A q / b c s t, A q / a b c t; q)_{\infty}} \\ \cdot \frac{(A q / a b s t, A q / a c s t, A q / b c s t, A q / a b c t; q)_{r}}{(A q / a t, A q / b t, A q / c t, A q / s t; q)_{r}} \end{aligned}$
が成り立つから,

$\begin{aligned} \frac{1}{2 π} \frac{(a b c s t^{2}, A q / a b c s t^{2}, a b c s t^{2} / B C, a b c s; q)_{\infty}}{(a b c s t^{2} / B, a b c s t^{2} / C, A B q / a b c s t^{2}, A C q / a b c s t^{2}; q)_{\infty}} \int_{- 1}^{1} {| \frac{(e^{2 i θ}, a b c s t e^{i θ} / B, a b c s t e^{i θ} / C; q)_{\infty}}{(a e^{i θ}, b e^{i θ}, c e^{i θ}, s e^{i θ}, t e^{i θ}, A q e^{i θ} / a b c s t; q)_{\infty}} |}^{2} \frac{d x}{\sqrt{1 - x^{2}}} \\ = \frac{(a b c s, a b c t, a b s t, a c s t, b c s t; q)_{\infty}}{(q, a b, a c, b c, a s, b s, c s, a t, b t, c t, s t; q)_{\infty}} \\ \cdot_{10} ϕ_{9} [\begin{array}{c} a b c s t^{2} / q, q \sqrt{a b c s t^{2} / q}, - q \sqrt{a b c s t^{2} / q}, A, B, C, a t, b t, c t, s t \\ \sqrt{a b c s t^{2} / q}, - \sqrt{a b c s t^{2} / q}, a b c s t^{2} / A, a b c s t^{2} / B, a b c s t^{2} / C, a b c t, a b s t, a c s t, b c s t \end{array}; q] \\ + \frac{(A q / B, A q / C, a b c s t^{2}, B, C, A q / a b c s t^{2}; q)_{\infty}}{(A^{2} q^{2} / a b c s t^{2}, A B q / a b c s t^{2}, A C q / a b c s t^{2}, a b c s t^{2} / B, a b c s t^{2} / C, a b c s t^{2} / A q; q)_{\infty}} \\ \cdot \frac{(a b c s, A q / a t, A q / b t, A q / c t, A q / s t; q)_{\infty}}{(q, a b, a c, b c, a s, b s, c s, A q / a b s t, A q / a c s t, A q / b c s t,, A q / a b c t; q)_{\infty}} \\ \cdot_{10} ϕ_{9} [\begin{array}{c} A^{2} q / a b c s t^{2}, q \sqrt{A^{2} q / a b c s t^{2}}, - q \sqrt{A^{2} q / a b c s t^{2}}, A, A B q / a b c s t^{2}, A C q / a b c s t^{2}, A q / a b s t, A q / a c s t, A q / b c s t, A q / a b c t \\ \sqrt{A^{2} q / a b c s t^{2}}, - \sqrt{A^{2} q / a b c s t^{2}}, A / a b c s t^{2}, A q / B, A q / C, A q / a t, A q / b t, A q / c t, A q / s t \end{array}; q] \end{aligned}$
より,
$\begin{aligned} \frac{1}{2 π} \int_{- 1}^{1} {| \frac{(e^{2 i θ}, a b c s t e^{i θ} / B, a b c s t e^{i θ} / C; q)_{\infty}}{(a e^{i θ}, b e^{i θ}, c e^{i θ}, s e^{i θ}, t e^{i θ}, A q e^{i θ} / a b c s t; q)_{\infty}} |}^{2} \frac{d x}{\sqrt{1 - x^{2}}} \\ = \frac{(a b c s t^{2} / B, a b c s t^{2} / C, A B q / a b c s t^{2}, A C q / a b c s t^{2}; q)_{\infty}}{(a b c s t^{2}, A q / a b c s t^{2}, a b c s t^{2} / B C, a b c s; q)_{\infty}} \frac{(a b c s, a b c t, a b s t, a c s t, b c s t; q)_{\infty}}{(q, a b, a c, b c, a s, b s, c s, a t, b t, c t, s t; q)_{\infty}} \\ \cdot_{10} ϕ_{9} [\begin{array}{c} a b c s t^{2} / q, q \sqrt{a b c s t^{2} / q}, - q \sqrt{a b c s t^{2} / q}, A, B, C, a t, b t, c t, s t \\ \sqrt{a b c s t^{2} / q}, - \sqrt{a b c s t^{2} / q}, a b c s t^{2} / A, a b c s t^{2} / B, a b c s t^{2} / C, a b c t, a b s t, a c s t, b c s t \end{array}; q] \\ + \frac{(a b c s t^{2} / B, a b c s t^{2} / C, A B q / a b c s t^{2}, A C q / a b c s t^{2}; q)_{\infty}}{(a b c s t^{2}, A q / a b c s t^{2}, a b c s t^{2} / B C, a b c s; q)_{\infty}} \\ \cdot \frac{(A q / B, A q / C, a b c s t^{2}, B, C, A q / a b c s t^{2}; q)_{\infty}}{(A^{2} q^{2} / a b c s t^{2}, A B q / a b c s t^{2}, A C q / a b c s t^{2}, a b c s t^{2} / B, a b c s t^{2} / C, a b c s t^{2} / A q; q)_{\infty}} \\ \cdot \frac{(a b c s, A q / a t, A q / b t, A q / c t, A q / s t; q)_{\infty}}{(q, a b, a c, b c, a s, b s, c s, A q / a b s t, A q / a c s t, A q / b c s t,, A q / a b c t; q)_{\infty}} \\ \cdot_{10} ϕ_{9} [\begin{array}{c} A^{2} q / a b c s t^{2}, q \sqrt{A^{2} q / a b c s t^{2}}, - q \sqrt{A^{2} q / a b c s t^{2}}, A, A B q / a b c s t^{2}, A C q / a b c s t^{2}, A q / a b s t, A q / a c s t, A q / b c s t, A q / a b c t \\ \sqrt{A^{2} q / a b c s t^{2}}, - \sqrt{A^{2} q / a b c s t^{2}}, A / a b c s t^{2}, A q / B, A q / C, A q / a t, A q / b t, A q / c t, A q / s t \end{array}; q] \\ = \frac{(a b c s t^{2} / B, a b c s t^{2} / C, A B q / a b c s t^{2}, A C q / a b c s t^{2}, a b c t, a b s t, a c s t, b c s t; q)_{\infty}}{(a b c s t^{2}, A q / a b c s t^{2}, a b c s t^{2} / B C, q, a b, a c, b c, a s, b s, c s, a t, b t, c t, s t; q)_{\infty}} \\ \cdot_{10} ϕ_{9} [\begin{array}{c} a b c s t^{2} / q, q \sqrt{a b c s t^{2} / q}, - q \sqrt{a b c s t^{2} / q}, A, B, C, a t, b t, c t, s t \\ \sqrt{a b c s t^{2} / q}, - \sqrt{a b c s t^{2} / q}, a b c s t^{2} / A, a b c s t^{2} / B, a b c s t^{2} / C, a b c t, a b s t, a c s t, b c s t \end{array}; q] \\ \cdot \frac{(A q / B, A q / C, B, C, A q / a t, A q / b t, A q / c t, A q / s t; q)_{\infty}}{(a b c s t^{2} / B C, A^{2} q^{2} / a b c s t^{2}, a b c s t^{2} / A q, q, a b, a c, b c, a s, b s, c s, A q / a b s t, A q / a c s t, A q / b c s t, A q / a b c t; q)_{\infty}} \\ \cdot_{10} ϕ_{9} [\begin{array}{c} A^{2} q / a b c s t^{2}, q \sqrt{A^{2} q / a b c s t^{2}}, - q \sqrt{A^{2} q / a b c s t^{2}}, A, A B q / a b c s t^{2}, A C q / a b c s t^{2}, A q / a b s t, A q / a c s t, A q / b c s t, A q / a b c t \\ \sqrt{A^{2} q / a b c s t^{2}}, - \sqrt{A^{2} q / a b c s t^{2}}, A / a b c s t^{2}, A q / B, A q / C, A q / a t, A q / b t, A q / c t, A q / s t \end{array}; q] \end{aligned}$
となって定理が示された.

$u = A q / a b c s t, v = a b c s t / B, w = a b c s t / C$ として, $A, B, C$ を消去すると
$\begin{aligned} \frac{1}{2 π} \int_{- 1}^{1} {| \frac{(e^{2 i θ}, v e^{i θ}, w e^{i θ}; q)_{\infty}}{(a e^{i θ}, b e^{i θ}, c e^{i θ}, s e^{i θ}, t e^{i θ}, u e^{i θ}; q)_{\infty}} |}^{2} \frac{d x}{\sqrt{1 - x^{2}}} \\ = \frac{(v t, w t, w / t, v / t, a b c t, a b s t, a c s t, b c s t; q)_{\infty}}{(t u, v w t / u, u / t, q, a b, a c, b c, a s, b s, c s, a t, b t, c t, s t; q)_{\infty}} \\ \cdot_{10} ϕ_{9} [\begin{array}{c} v w t / u q, q \sqrt{v w t / u q}, - q \sqrt{v w t / u q}, v w / q, w / u, v / u, a t, b t, c t, s t \\ \sqrt{v w t / u q}, - \sqrt{v w t / u q}, t q / u, v t, w t, a b c t, a b s t, a c s t, b c s t \end{array}; q] \\ + \frac{(u v, u w, w / u, v / u, a b c u, a b s u, a c s u, b c s u; q)_{\infty}}{(u t, u v w / t, t / u, q, a b, a c, b c, a s, b s, c s, a u, b u, c u, s u; q)_{\infty}} \\ \cdot_{10} ϕ_{9} [\begin{array}{c} u v w / t q, q \sqrt{u v w / t q}, - q \sqrt{u v w / t q}, v w / q, w / t, v / t, a u, b u, c u, s u \\ \sqrt{u v w / t q}, - \sqrt{u v w / t q}, u q / t, v u, w u, a b c u, a b s u, a c s u, b c s u \end{array}; q] \end{aligned}$
となって右辺が $t, u$ に関して対称的な形になり, 条件は $a b c s t u = v w$ となる. 以下に定理として書いておく.

$a b c s t u = v w$ のとき,
$\begin{aligned} \frac{1}{2 π} \int_{- 1}^{1} {| \frac{(e^{2 i θ}, v e^{i θ}, w e^{i θ}; q)_{\infty}}{(a e^{i θ}, b e^{i θ}, c e^{i θ}, s e^{i θ}, t e^{i θ}, u e^{i θ}; q)_{\infty}} |}^{2} \frac{d x}{\sqrt{1 - x^{2}}} \\ = \frac{(v t, w t, w / t, v / t, a b c t, a b s t, a c s t, b c s t; q)_{\infty}}{(t u, v w t / u, u / t, q, a b, a c, b c, a s, b s, c s, a t, b t, c t, s t; q)_{\infty}} \\ \cdot_{10} ϕ_{9} [\begin{array}{c} v w t / u q, q \sqrt{v w t / u q}, - q \sqrt{v w t / u q}, v w / q, w / u, v / u, a t, b t, c t, s t \\ \sqrt{v w t / u q}, - \sqrt{v w t / u q}, t q / u, v t, w t, a b c t, a b s t, a c s t, b c s t \end{array}; q] \\ + \frac{(u v, u w, w / u, v / u, a b c u, a b s u, a c s u, b c s u; q)_{\infty}}{(u t, u v w / t, t / u, q, a b, a c, b c, a s, b s, c s, a u, b u, c u, s u; q)_{\infty}} \\ \cdot_{10} ϕ_{9} [\begin{array}{c} u v w / t q, q \sqrt{u v w / t q}, - q \sqrt{u v w / t q}, v w / q, w / t, v / t, a u, b u, c u, s u \\ \sqrt{u v w / t q}, - \sqrt{u v w / t q}, u q / t, v u, w u, a b c u, a b s u, a c s u, b c s u \end{array}; q] \end{aligned}$
が成り立つ.