文献あり

Nassrallah-Rahman積分

前回の記事と同様に,
$\begin{aligned} w (x; a, b, c, d) & := \frac{h (x, 1) h (x, - 1) h (x, \sqrt{q}) h (x, - \sqrt{q})}{\sqrt{1 - x^{2}} h (x, a) h (x, b) h (x, c) h (x, d)} \\ h (x, a) & := \prod_{k = 0}^{\infty} (1 - 2 a x q^{k} + a^{2} q^{2 k}) \end{aligned}$
とする. Askey-Wilson積分は
$\begin{aligned} \int_{- 1}^{1} w (x; a, b, c, d) d x & = \frac{2 π (a b c d; q)_{\infty}}{(a b, a c, a d, b c, b d, c d, q; q)_{\infty}} \end{aligned}$
で与えられる. その一般化として, 以下はNassrallah-Rahman積分として知られているものである.

Nassrallah-Rahman(1985)

$\begin{aligned} \frac{(q, a b, a c, a d, b c, b d, c d; q)_{\infty}}{2 π (a b c d; q)_{\infty}} \int_{- 1}^{1} {| \frac{(A t e^{i θ}; q)_{\infty}}{(t e^{i θ}; q)_{\infty}} |}^{2} w (x; a, b, c, d) d x \\ = \frac{(a c d t, A a t, A c t, A d t; q)_{\infty}}{(A a c d t, a t, c t, d t; q)_{\infty}}_{8} ϕ_{7} [\begin{array}{c} A a c d t / q, q \sqrt{A a c d t / q}, - q \sqrt{A a c d t / q}, A t / b, A, a c, a d, c d \\ \sqrt{A a c d t / q}, - \sqrt{A a c d t / q}, a b c d, a c d t, A a t, A d t, A c t \end{array}; b t] \end{aligned}$

non-terminating $q$ -Saalschützの和公式
$\begin{aligned} _{3} ϕ_{2} [\begin{array}{c} a, b, c \\ e, f \end{array}; q] + \frac{(q / e, a, b, c, f q / e; q)_{\infty}}{(e / q, a q / e, b q / e, c q / e, f; q)_{\infty}}_{3} ϕ_{2} [\begin{array}{c} a q / e, b q / e, c q / e \\ q^{2} / e, f q / e \end{array}; q] \\ = \frac{(q / e, f / a, f / b, f / c; q)_{\infty}}{(a q / e, b q / e, c q / e, f; q)_{\infty}}, a b c q = e f \end{aligned}$
から始める. $a \mapsto A, b \mapsto a e^{i θ}, c \mapsto a e^{- i θ}, e \mapsto E$ とすると,
$\begin{aligned} _{3} ϕ_{2} [\begin{array}{c} A, a e^{i θ}, a e^{- i θ} \\ E, A a^{2} q / E \end{array}; q] + \frac{(q / E, A, a e^{i θ}, a e^{- i θ}, A a^{2} q^{2} / E^{2}; q)_{\infty}}{(E / q, A q / E, a q e^{i θ} / E, a q e^{- i θ} / E, A a^{2} q / E; q)_{\infty}}_{3} ϕ_{2} [\begin{array}{c} A q / E, a q e^{i θ} / E, a q e^{- i θ} / E \\ q^{2} / E, A a^{2} q^{2} / E^{2} \end{array}; q] \\ = \frac{(q / E, a^{2} q / E, A a q e^{i θ} / E, A a q e^{- i θ} / E; q)_{\infty}}{(A q / E, a q e^{i θ} / E, a q e^{- i θ} / E, A a^{2} q / E; q)_{\infty}} \end{aligned}$
両辺に $w (x; a, b, c, d)$ を掛けて $[- 1, 1]$ で積分すると, Askey-Wilson積分より,
$\begin{aligned} \int_{- 1}^{1}_{3} ϕ_{2} [\begin{array}{c} A, a e^{i θ}, a e^{- i θ} \\ E, A a^{2} q / E \end{array}; q] w (x; a, b, c, d) d x \\ = \frac{2 π (a b c d; q)_{\infty}}{(a b, a c, a d, b c, b d, c d, q; q)_{\infty}}_{4} ϕ_{3} [\begin{array}{c} A, a b, a c, a d \\ E, A a^{2} q / E, a b c d \end{array}; q] \\ \int_{- 1}^{1} \frac{(a e^{i θ}, a e^{- i θ}; q)_{\infty}}{(a q e^{i θ} / E, a q e^{- i θ} / E; q)_{\infty}}_{3} ϕ_{2} [\begin{array}{c} A q / E, a q e^{i θ} / E, a q e^{- i θ} / E \\ q^{2} / E, A a^{2} q^{2} / E^{2} \end{array}; q] w (x; a, b, c, d) d x \\ = \int_{- 1}^{1}_{3} ϕ_{2} [\begin{array}{c} A q / E, a q e^{i θ} / E, a q e^{- i θ} / E \\ q^{2} / E, A a^{2} q^{2} / E^{2} \end{array}; q] w (x; a q / E, b, c, d) d x \\ = \frac{2 π (a b c d q / E; q)_{\infty}}{(a b q / E, a c q / E, a d q / E, b c, b d, c d, q; q)_{\infty}}_{4} ϕ_{3} [\begin{array}{c} A q / E, a b q / E, a c q / E, a d q / E \\ q^{2} / E, A a^{2} q^{2} / E^{2}, a b c d q / E \end{array}; q] \end{aligned}$
であるから,
$\begin{aligned} \frac{(q / E, a^{2} q / E; q)_{\infty}}{(A q / E, A a^{2} q / E; q)_{\infty}} \int_{- 1}^{1} {| \frac{(A a q e^{i θ} / E; q)_{\infty}}{(a q e^{i θ} / E; q)_{\infty}} |}^{2} w (x; a, b, c, d) d x \\ = \frac{2 π (a b c d; q)_{\infty}}{(a b, a c, a d, b c, b d, c d, q; q)_{\infty}}_{4} ϕ_{3} [\begin{array}{c} A, a b, a c, a d \\ E, A a^{2} q / E, a b c d \end{array}; q] \\ + \frac{(q / E, A, A a^{2} q^{2} / E^{2}; q)_{\infty}}{(E / q, A q / E, A a^{2} q / E; q)_{\infty}} \frac{2 π (a b c d q / E; q)_{\infty}}{(a b q / E, a c q / E, a d q / E, b c, b d, c d, q; q)_{\infty}}_{4} ϕ_{3} [\begin{array}{c} A q / E, a b q / E, a c q / E, a d q / E \\ q^{2} / E, A a^{2} q^{2} / E^{2}, a b c d q / E \end{array}; q] \\ = \frac{2 π (a b c d; q)_{\infty}}{(a b, a c, a d, b c, b d, c d, q; q)_{\infty}}_{4} ϕ_{3} [\begin{array}{c} A, a b, a c, a d \\ E, A a^{2} q / E, a b c d \end{array}; q] \\ + \frac{2 π (a b c d; q)_{\infty}}{(a b, a c, a d, b c, b d, c d, q; q)_{\infty}} \frac{(q / E, A, A a^{2} q^{2} / E^{2}; q)_{\infty}}{(E / q, A q / E, A a^{2} q / E; q)_{\infty}} \frac{(a b, a c, a d, a b c d q / E; q)_{\infty}}{(a b c d, a b q / E, a c q / E, a d q / E; q)_{\infty}}_{4} ϕ_{3} [\begin{array}{c} A q / E, a b q / E, a c q / E, a d q / E \\ q^{2} / E, A a^{2} q^{2} / E^{2}, a b c d q / E \end{array}; q] \end{aligned}$
ここで, non-terminating $q$ -Whippleの変換公式より,
$\begin{aligned} _{4} ϕ_{3} [\begin{array}{c} A, a b, a c, a d \\ E, A a^{2} q / E, a b c d \end{array}; q] \\ + \frac{(q / E, A, A a^{2} q^{2} / E^{2}; q)_{\infty}}{(E / q, A q / E, A a^{2} q / E; q)_{\infty}} \frac{(a b, a c, a d, a b c d q / E; q)_{\infty}}{(a b c d, a b q / E, a c q / E, a d q / E; q)_{\infty}}_{4} ϕ_{3} [\begin{array}{c} A q / E, a b q / E, a c q / E, a d q / E \\ q^{2} / E, A a^{2} q^{2} / E^{2}, a b c d q / E \end{array}; q] \\ = \frac{(a^{2} c d q / E, A a c q / E, A a d q / E, q / E; q)_{\infty}}{(A a^{2} c d q / E, a c q / E, a d q / E, A q / E; q)_{\infty}} \\ \cdot_{8} ϕ_{7} [\begin{array}{c} A a^{2} c d / E, q \sqrt{A a^{2} c d / E}, - q \sqrt{A a^{2} c d / E}, c d, A a q / b E, A, a c, a d \\ \sqrt{A a^{2} c d / E}, - \sqrt{A a^{2} c d / E}, A a^{2} q / E, a b c d q, a^{2} c d q / E, A a d q / E, A a c q / E \end{array}; \frac{a b q}{E}] \end{aligned}$
だから, これを代入して,
$\begin{aligned} \frac{(a b, a c, a d, b c, b d, c d, q; q)_{\infty}}{2 π (a b c d; q)_{\infty}} \int_{- 1}^{1} {| \frac{(A a q e^{i θ} / E; q)_{\infty}}{(a q e^{i θ} / E; q)_{\infty}} |}^{2} w (x; a, b, c, d) d x \\ = \frac{(A q / E, A a^{2} q / E; q)_{\infty}}{(q / E, a^{2} q / E; q)_{\infty}} \frac{(a^{2} c d q / E, A a c q / E, A a d q / E, q / E; q)_{\infty}}{(A a^{2} c d q / E, a c q / E, a d q / E, A q / E; q)_{\infty}} \\ \cdot_{8} ϕ_{7} [\begin{array}{c} A a^{2} c d / E, q \sqrt{A a^{2} c d / E}, - q \sqrt{A a^{2} c d / E}, c d, A a q / b E, A, a c, a d \\ \sqrt{A a^{2} c d / E}, - \sqrt{A a^{2} c d / E}, A a^{2} q / E, a b c d q, a^{2} c d q / E, A a d q / E, A a c q / E \end{array}; \frac{a b q}{E}] \\ = \frac{(A a^{2} q / E, a^{2} c d q / E, A a c q / E, A a d q / E; q)_{\infty}}{(a^{2} q / E, A a^{2} c d q / E, a c q / E, a d q / E; q)_{\infty}} \\ \cdot_{8} ϕ_{7} [\begin{array}{c} A a^{2} c d / E, q \sqrt{A a^{2} c d / E}, - q \sqrt{A a^{2} c d / E}, c d, A a q / b E, A, a c, a d \\ \sqrt{A a^{2} c d / E}, - \sqrt{A a^{2} c d / E}, A a^{2} q / E, a b c d q, a^{2} c d q / E, A a d q / E, A a c q / E \end{array}; \frac{a b q}{E}] \end{aligned}$
ここで, $a q / E \mapsto t$ と置き換えて定理を得る.

$w$ を明示的に書くと,
$\begin{aligned} \frac{(q, a b, a c, a d, b c, b d, c d; q)_{\infty}}{2 π (a b c d; q)_{\infty}} \int_{- 1}^{1} {| \frac{(e^{2 i θ}, A t e^{i θ}; q)_{\infty}}{(a e^{i θ}, b e^{i θ}, c e^{i θ}, d e^{i θ}, t e^{i θ}; q)_{\infty}} |}^{2} \frac{d x}{\sqrt{1 - x^{2}}} \\ = \frac{(a c d t, A a t, A c t, A d t; q)_{\infty}}{(A a c d t, a t, c t, d t; q)_{\infty}}_{8} ϕ_{7} [\begin{array}{c} A a c d t / q, q \sqrt{A a c d t / q}, - q \sqrt{A a c d t / q}, A t / b, A, a c, a d, c d \\ \sqrt{A a c d t / q}, - \sqrt{A a c d t / q}, a b c d, a c d t, A a t, A d t, A c t \end{array}; b t] \end{aligned}$
となる. さらに, $A t \mapsto f$ とすると,
$\begin{aligned} \frac{(q, a b, a c, a d, b c, b d, c d; q)_{\infty}}{2 π (a b c d; q)_{\infty}} \int_{- 1}^{1} {| \frac{(e^{2 i θ}, f e^{i θ}; q)_{\infty}}{(a e^{i θ}, b e^{i θ}, c e^{i θ}, d e^{i θ}, t e^{i θ}; q)_{\infty}} |}^{2} \frac{d x}{\sqrt{1 - x^{2}}} \\ = \frac{(a c d t, a f, c f, d f; q)_{\infty}}{(a c d f, a t, c t, d t; q)_{\infty}}_{8} ϕ_{7} [\begin{array}{c} a c d f / q, q \sqrt{a c d f / q}, - q \sqrt{a c d f / q}, f / b, f / t, a c, a d, c d \\ \sqrt{a c d f / q}, - \sqrt{a c d f / q}, a b c d, a c d t, a f, d f, c f \end{array}; b t] \end{aligned}$
となる. 右辺の $q$ 超幾何級数が $a, c, d$ に関して対称であることと, $b, t$ に関して対称であることに着目し, $d \mapsto b, b \mapsto s$ と文字を置き換えると
$\begin{aligned} \frac{(q, a b, a c, b c; q)_{\infty}}{2 π} \int_{- 1}^{1} {| \frac{(e^{2 i θ}, f e^{i θ}; q)_{\infty}}{(a e^{i θ}, b e^{i θ}, c e^{i θ}, s e^{i θ}, t e^{i θ}; q)_{\infty}} |}^{2} \frac{d x}{\sqrt{1 - x^{2}}} \\ = \frac{(a b c s, a b c t, a f, b f, c f; q)_{\infty}}{(a b c f, a s, b s, c s, a t, b t, c t; q)_{\infty}}_{8} ϕ_{7} [\begin{array}{c} a b c f / q, q \sqrt{a b c f / q}, - q \sqrt{a b c f / q}, f / s, f / t, a b, a c, b c \\ \sqrt{a b c f / q}, - \sqrt{a b c f / q}, a b c s, a b c t, a f, b f, c f \end{array}; s t] \end{aligned}$
と書くことができ, より覚えやすくなると思う.

Rahman(1986)

$\begin{array}{r} \frac{1}{2 π} \int_{- 1}^{1} {| \frac{(e^{2 i θ}, a b c s t e^{i θ}; q)_{\infty}}{(a e^{i θ}, b e^{i θ}, c e^{i θ}, s e^{i θ}, t e^{i θ}; q)_{\infty}} |}^{2} \frac{d x}{\sqrt{1 - x^{2}}} = \frac{(a b c s, a b c t, a b s t, a c s t, b c s t; q)_{\infty}}{(q, a b, a c, b c, a s, b s, c s, a t, b t, c t, s t; q)_{\infty}}, x = \cos θ \end{array}$

$f = a b c s t$ として, Nassrallah-Rahman積分より,
$\begin{aligned} \frac{1}{2 π} \int_{- 1}^{1} {| \frac{(e^{2 i θ}, f e^{i θ}; q)_{\infty}}{(a e^{i θ}, b e^{i θ}, c e^{i θ}, s e^{i θ}, t e^{i θ}; q)_{\infty}} |}^{2} \frac{d x}{\sqrt{1 - x^{2}}} \\ = \frac{1}{(q, a b, a c, b c; q)_{\infty}} \frac{(a b c s, a b c t, a f, b f, c f; q)_{\infty}}{(a b c f, a s, b s, c s, a t, b t, c t; q)_{\infty}}_{6} ϕ_{5} [\begin{array}{c} a b c f / q, q \sqrt{a b c f / q}, - q \sqrt{a b c f / q}, a b, a c, b c \\ \sqrt{a b c f / q}, - \sqrt{a b c f / q}, a f, b f, c f \end{array}; s t] \end{aligned}$
ここで, Rogersの $_{6} ϕ_{5}$ 和公式より,
$\begin{aligned} _{6} ϕ_{5} [\begin{array}{c} a b c f / q, q \sqrt{a b c f / q}, - q \sqrt{a b c f / q}, a b, a c, b c \\ \sqrt{a b c f / q}, - \sqrt{a b c f / q}, a f, b f, c f \end{array}; s t] & = \frac{(a b c f, f / a, f / b, f / c; q)_{\infty}}{(a f, b f, c f, f / a b c; q)_{\infty}} \end{aligned}$
であるから, これを代入して,
$\begin{aligned} \frac{1}{2 π} \int_{- 1}^{1} {| \frac{(e^{2 i θ}, f e^{i θ}; q)_{\infty}}{(a e^{i θ}, b e^{i θ}, c e^{i θ}, s e^{i θ}, t e^{i θ}; q)_{\infty}} |}^{2} \frac{d x}{\sqrt{1 - x^{2}}} \\ = \frac{1}{(q, a b, a c, b c; q)_{\infty}} \frac{(a b c s, a b c t, a f, b f, c f; q)_{\infty}}{(a b c f, a s, b s, c s, a t, b t, c t; q)_{\infty}} \frac{(a b c f, f / a, f / b, f / c; q)_{\infty}}{(a f, b f, c f, f / a b c; q)_{\infty}} \\ = \frac{(a b c s, a b c t, a b s t, a c s t, b c s t; q)_{\infty}}{(q, a b, a c, b c, a s, b s, c s, a t, b t, c t, s t; q)_{\infty}} \end{aligned}$
となって定理が示される.

これは, 総乗の記法を用いると,
$\begin{array}{r} \frac{1}{2 π} \int_{- 1}^{1} {| \frac{(e^{2 i θ}, t_{1} t_{2} t_{3} t_{4} t_{5} e^{i θ}; q)_{\infty}}{\prod_{i = 1}^{5} (t_{i} e^{i θ}; q)_{\infty}} |}^{2} \frac{d x}{\sqrt{1 - x^{2}}} = \frac{\prod_{i = 1}^{5} (t_{1} t_{2} t_{3} t_{4} t_{5} / t_{i}; q)_{\infty}}{(q; q)_{\infty} \prod_{1 \leq i \leq j \leq 5} (t_{i} t_{j}; q)_{\infty}} \end{array}$
と美しく書くことができる. これはAskey-Wilson積分の一般化であり, 実際に $t_{5} = 0$ とするとAskey-Wilson積分に一致する.

古典極限

定理1の古典極限は以下のようになる.

$\begin{aligned} \frac{Γ (a + b + c + d)}{2 π Γ (a + b) Γ (a + c) Γ (a + d) Γ (b + c) Γ (b + d) Γ (c + d)} \\ \cdot \int_{0}^{\infty} {| \frac{Γ (a + i x) Γ (b + i x) Γ (c + i x) Γ (d + i x) Γ (t + i x)}{Γ (2 i x) Γ (A + t + i x)} |}^{2} d x \\ = \frac{Γ (A + a + c + d + t) Γ (a + t) Γ (c + t) Γ (d + t)}{Γ (a + c + d + t) Γ (A + a + t) Γ (A + c + t) Γ (A + d + t)} \\ \cdot_{7} F_{6} [\begin{array}{c} A + a + c + d + t - 1, 1 + \frac{A + a + c + d + t - 1}{2}, A + t - b, A, a + c, a + d, c + d \\ \frac{A + a + c + d + t - 1}{2}, a + b + c + d, a + c + d + t, A + a + t, A + d + t, A + c + t \end{array}; 1] \end{aligned}$

先ほどのように整理すると,
$\begin{aligned} \frac{1}{2 π Γ (a + b) Γ (a + c) Γ (b + c)} \int_{0}^{\infty} {| \frac{Γ (a + i x) Γ (b + i x) Γ (c + i x) Γ (d + i x) Γ (t + i x)}{Γ (2 i x) Γ (f + i x)} |}^{2} d x \\ = \frac{Γ (a + b + c + f) Γ (a + s) Γ (b + s) Γ (c + s) Γ (a + t) Γ (b + t) Γ (c + t)}{Γ (a + b + c + s) Γ (a + b + c + t) Γ (a + f) Γ (b + f) Γ (c + f)} \\ \cdot_{7} F_{6} [\begin{array}{c} a + b + c + f - 1, 1 + \frac{a + b + c + f - 1}{2}, f - s, f - t, a + b, a + c, b + c \\ \frac{a + b + c + f - 1}{2}, a + b + c + s, a + b + c + t, a + f, b + f, c + f \end{array}; 1] \end{aligned}$
と書くことができる.

定理2の古典極限は以下のようになる.

$\begin{aligned} \frac{1}{2 π} \int_{0}^{\infty} {| \frac{Γ (a + i x) Γ (b + i x) Γ (c + i x) Γ (s + i x) Γ (t + i x)}{Γ (2 i x) Γ (a + b + c + s + t + i x)} |}^{2} d x \\ = \frac{Γ (a + b) Γ (a + c) Γ (b + c) Γ (a + s) Γ (b + s) Γ (c + s) Γ (a + t) Γ (b + t) Γ (c + t) Γ (s + t)}{Γ (a + b + c + s) Γ (a + b + c + t) Γ (a + b + s + t) Γ (a + c + s + t) Γ (b + c + s + t)} \end{aligned}$

参考文献

[1]

B. Nassrallah, M. Rahman, Projection formulas, a reproducing kernel and a generating function for $q$-Wilson polynomials, SIAM J. Math. Anal., 1985, 186–197.

[2]

M. Rahman, An integral representation of a 10φ9 and continuous bi-orthogonal 10φ9 rational functions., Canad. J. Math., 1986, 605-618

投稿日：4月22日

更新日：4月22日