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Burchnall-Chaundyの作用素とAppellの超幾何級数

Burchnall-Chaundyの作用素

$x, y$ に関するべき級数に対して, Burchnall-Chaundyの作用素 $Δ (h), \nabla (h)$ を
$\begin{aligned} Δ (h) x^{n} y^{m} & := \frac{(h)_{n + m}}{(h)_{n} (h)_{m}} x^{n} y^{m} \\ \nabla (h) x^{n} y^{m} & := \frac{(h)_{n} (h)_{m}}{(h)_{n + m}} x^{n} y^{m} \end{aligned}$
によって導入する. これは, $θ_{x}, θ_{y}$ をそれぞれ $x, y$ に関するEuler作用素
$\begin{aligned} θ_{x} x^{n} & := n x^{n} \\ θ_{y} y^{m} & := m y^{m} \end{aligned}$
として,
$\begin{aligned} Δ (h) & = \frac{Γ (h) Γ (θ_{x} + θ_{y} + h)}{Γ (θ_{x} + h) Γ (θ_{y} + h)} \\ \nabla (h) & = \frac{Γ (h) Γ (θ_{x} + θ_{y} + h)}{Γ (θ_{x} + h) Γ (θ_{y} + h)} \end{aligned}$
と表すことができる. 定義から, $Δ (h), \nabla (h)$ は互いに逆変換になっていることが分かる.

Appellの超幾何級数

Appellの超幾何級数は
$\begin{aligned} F_{1} (\begin{array}{c} a; b_{1}, b_{2} \\ c \end{array}; x, y) & := \sum_{0 \leq n, m} \frac{(a)_{n + m} (b_{1})_{n} (b_{2})_{m}}{(c)_{n + m} n! m!} x^{n} y^{m} \\ F_{2} (\begin{array}{c} a; b_{1}, b_{2} \\ c_{1}, c_{2} \end{array}; x, y) & := \sum_{0 \leq n, m} \frac{(a)_{n + m} (b_{1})_{n} (b_{2})_{m}}{(c_{1})_{n} (c_{2})_{m} n! m!} x^{n} y^{m} \\ F_{3} (\begin{array}{c} a_{1}, a_{2}; b_{1}, b_{2} \\ c \end{array}; x, y) & := \sum_{0 \leq n, m} \frac{(a_{1}, b_{1})_{n} (a_{2}, b_{2})_{m}}{(c)_{n + m} n! m!} x^{n} y^{m} \\ F_{4} (\begin{array}{c} a; b \\ c_{1}, c_{2} \end{array}; x, y) & := \sum_{0 \leq n, m} \frac{(a, b)_{n + m}}{(c_{1})_{n} (c_{2})_{m} n! m!} x^{n} y^{m} \end{aligned}$
によって定義される. 定義通りに計算することによって, 以下の等式が得られる.

$\begin{aligned} F_{1} (\begin{array}{c} a; b_{1}, b_{2} \\ c \end{array}; x, y) & = \nabla (a) Δ (c) F (\begin{array}{c} a, b_{1} \\ c \end{array}; x) F (\begin{array}{c} a, b_{2} \\ c \end{array}; y) \\ = \nabla (a) F_{3} (\begin{array}{c} a, a; b_{1}, b_{2} \\ c \end{array}; x, y) \\ = Δ (c) F_{2} (\begin{array}{c} a; b_{1}, b_{2} \\ c, c \end{array}; x, y) \\ F_{2} (\begin{array}{c} a; b_{1}, b_{2} \\ c_{1}, c_{2} \end{array}; x, y) & = \nabla (a) F (\begin{array}{c} a, b_{1} \\ c_{1} \end{array}; x) F (\begin{array}{c} a, b_{2} \\ c_{2} \end{array}; y) \\ F_{3} (\begin{array}{c} a_{1}, a_{2}; b_{1}, b_{2} \\ c \end{array}; x, y) & = Δ (c) F (\begin{array}{c} a_{1}, b_{1} \\ c \end{array}; x) F (\begin{array}{c} a_{2}, b_{2} \\ c \end{array}; y) \\ F_{4} (\begin{array}{c} a; b \\ c_{1}, c_{2} \end{array}; x, y) & = \nabla (b) F_{2} (\begin{array}{c} a; b, b \\ c_{1}, c_{2} \end{array}; x, y) \\ F (\begin{array}{c} a, b \\ c \end{array}; x + y) & = \nabla (b) F_{1} (\begin{array}{c} a; b, b \\ c \end{array}; x, y) \\ = Δ (c) F_{4} (\begin{array}{c} a; b \\ c_{1}, c_{2} \end{array}; x, y) \\ = \nabla (b) F_{2} (\begin{array}{c} a; b, b \\ c, c \end{array}; x, y) \\ = \nabla (b) Δ (c) F_{2} (\begin{array}{c} a; b, b \\ c, c \end{array}; x, y) \end{aligned}$

Gaussの超幾何定理より,

$\begin{aligned} \frac{Γ (h) Γ (n + m + h)}{Γ (n + h) Γ (m + h)} & = \sum_{0 \leq r} \frac{(- n, - m)_{r}}{r! (h)_{r}} \\ \frac{Γ (n + h) Γ (m + h)}{Γ (h) Γ (n + m + h)} & = \sum_{0 \leq r} \frac{(- n, - m)_{r}}{r! (- h - n - m + 1)_{r}} \end{aligned}$
が得られる. また, Dougallの $_{5} F_{4}$ の和公式から
$\begin{aligned} \frac{Γ (n + h) Γ (m + h)}{Γ (h) Γ (n + m + h)} & = \sum_{0 \leq r} \frac{(- 1)^{r} (h)_{2 r} (- n, - m)_{r}}{r! (h + r - 1)_{r} (m + h, n + h)_{r}} \\ \frac{Γ (h) Γ (n + m + h) Γ (m + k) Γ (n + k)}{Γ (n + h) Γ (m + h) Γ (k) Γ (m + n + k)} & = \sum_{0 \leq r} \frac{(k)_{2 r} (k - h, - n, - m)_{r}}{r! (k + r - 1)_{r} (m + k, n + k, h)_{r}} \end{aligned}$
が得られる. また, Saalschutzの和公式により,
$\begin{aligned} \frac{Γ (h) Γ (n + m + h) Γ (m + k) Γ (n + k)}{Γ (n + h) Γ (m + h) Γ (k) Γ (m + n + k)} & = \sum_{0 \leq r} \frac{(h - k, - n, - m)_{r}}{r! (h, - k - n - m + 1)_{r}} \end{aligned}$
という表示も得られる. これらの公式において, $n, m$ を $θ_{x}, θ_{y}$ に置き換えることによって作用素 $\nabla (h), Δ (h)$ の表示を得ることができる. 具体例を挙げると,
$\begin{aligned} Δ (h) & = \sum_{0 \leq r} \frac{(- θ_{x}, - θ_{y})_{r}}{r! (- h - θ_{x} - θ_{y} + 1)_{r}} \\ \nabla (h) & = \sum_{0 \leq r} \frac{(- θ_{x}, - θ_{y})_{r}}{r! (h)_{r}} \end{aligned}$
のようになる.

18個の展開公式

$\begin{aligned} F_{2} (\begin{array}{c} a; b_{1}, b_{2} \\ c_{1}, c_{2} \end{array}; x, y) & = \sum_{0 \leq r} \frac{(a, b_{1}, b_{2})_{r}}{r! (c_{1}, c_{2})_{r}} (x y)^{r} F (\begin{array}{c} a + r, b_{1} + r \\ c_{1} + r \end{array}; x) F (\begin{array}{c} a + r, b_{2} + r \\ c_{2} + r \end{array}; y) \\ F (\begin{array}{c} a, b_{1} \\ c_{1} \end{array}; x) F (\begin{array}{c} a, b_{2} \\ c_{2} \end{array}; y) & = \sum_{0 \leq r} (- 1)^{r} \frac{(a, b_{1}, b_{2})_{r}}{r! (c_{1}, c_{2})_{r}} (x y)^{r} F_{2} (\begin{array}{c} a + r; b_{1} + r, b_{2} + r \\ c_{1} + r, c_{2} + r \end{array}; x, y) \\ F_{3} (\begin{array}{c} a_{1}, a_{2}; b_{1}, b_{2} \\ c \end{array}; x, y) & = \sum_{0 \leq r} (- 1)^{r} \frac{(a_{1}, a_{2}, b_{1}, b_{2})_{r}}{r! (c + r - 1)_{r} (c)_{2 r}} (x y)^{r} F (\begin{array}{c} a_{1} + r, b_{1} + r \\ c \end{array}; x) F (\begin{array}{c} a_{2} + r, b_{2} + r \\ c \end{array}; y) \\ F (\begin{array}{c} a_{1}, b_{1} \\ c \end{array}; x) F (\begin{array}{c} a_{2}, b_{2} \\ c \end{array}; y) & = \sum_{0 \leq r} \frac{(a_{1}, a_{2}, b_{1}, b_{2})_{r}}{r! (c)_{r} (c)_{2 r}} (x y)^{r} F_{3} (\begin{array}{c} a_{1} + r, a_{2} + r; b_{1} + r, b_{2} + r \\ c + 2 r \end{array}; x, y) \\ F_{1} (\begin{array}{c} a; b_{1}, b_{2} \\ c \end{array}; x, y) & = \sum_{0 \leq r} \frac{(a, b_{1}, b_{2}, c - a)_{r}}{r! (c + r - 1)_{r} (c)_{2 r}} (x y)^{r} F (\begin{array}{c} a + r, b_{1} + r \\ c + 2 r \end{array}; x) F (\begin{array}{c} a + r, b_{2} + r \\ c + 2 r \end{array}; y) \\ F (\begin{array}{c} a, b_{1} \\ c \end{array}; x) F (\begin{array}{c} a, b_{2} \\ c \end{array}; y) & = \sum_{0 \leq r} (- 1)^{r} \frac{(a, b_{1}, b_{2}, c - a)_{r}}{r! (c)_{r} (c)_{2 r}} (x y)^{r} F_{1} (\begin{array}{c} a + r; b_{1} + r, b_{2} + r \\ c + 2 r \end{array}; x, y) \\ F_{1} (\begin{array}{c} a; b_{1}, b_{2} \\ c \end{array}; x, y) & = \sum_{0 \leq r} \frac{(a, b_{1}, b_{2})_{r}}{r! (c)_{2 r}} (x y)^{r} F_{3} (\begin{array}{c} a + r, a + r; b_{1} + r, b_{2} + r \\ c + 2 r \end{array}; x, y) \\ F_{3} (\begin{array}{c} a, a; b_{1}, b_{2} \\ c \end{array}; x, y) & = \sum_{0 \leq r} (- 1)^{r} \frac{(a, b_{1}, b_{2})_{r}}{r! (c)_{2 r}} (x y)^{r} F_{1} (\begin{array}{c} a + r; b_{1} + r, b_{2} + r \\ c + 2 r \end{array}; x, y) \\ F_{1} (\begin{array}{c} a; b_{1}, b_{2} \\ c \end{array}; x, y) & = \sum_{0 \leq r} (- 1)^{r} \frac{(a)_{2 r} (b_{1}, b_{2})_{r}}{r! (c + r - 1)_{r} (c)_{2 r}} (x y)^{r} F_{2} (\begin{array}{c} a + 2 r; b_{1} + r, b_{2} + r \\ c + 2 r, c + 2 r \end{array}; x, y) \\ F_{2} (\begin{array}{c} a; b_{1}, b_{2} \\ c, c \end{array}; x, y) & = \sum_{0 \leq r} \frac{(a)_{2 r} (b_{1}, b_{2})_{r}}{r! (c)_{r} (c)_{2 r}} (x y)^{r} F_{1} (\begin{array}{c} a + 2 r; b_{1} + r, b_{2} + r \\ c + 2 r \end{array}; x, y) \\ F_{4} (\begin{array}{c} a; b \\ c_{1}, c_{2} \end{array}; x, y) & = \sum_{0 \leq r} \frac{(a)_{2 r} (b)_{r}}{r! (c_{1}, c_{2})_{r}} (x y)^{r} F_{2} (\begin{array}{c} a + 2 r; b + r, b + r \\ c_{1} + r, c_{2} + r \end{array}; x, y) \\ F_{2} (\begin{array}{c} a; b, b \\ c_{1}, c_{2} \end{array}; x, y) & = \sum_{0 \leq r} (- 1)^{r} \frac{(a)_{2 r} (b)_{r}}{r! (c_{1}, c_{2})_{r}} (x y)^{r} F_{4} (\begin{array}{c} a + 2 r; b + r \\ c_{1} + r, c_{2} + r \end{array}; x, y) \\ F (\begin{array}{c} a, b \\ c \end{array}; x + y) & = \sum_{0 \leq r} \frac{(a)_{2 r} (b)_{r}}{r! (c)_{2 r}} (x y)^{r} F_{1} (\begin{array}{c} a + 2 r; b + r, b + r \\ c + 2 r \end{array}; x, y) \\ F_{1} (\begin{array}{c} a; b_{1}, b_{2} \\ c \end{array}; x, y) & = \sum_{0 \leq r} (- 1)^{r} \frac{(a)_{2 r} (b)_{r}}{r! (c)_{2 r}} (x y)^{r} F (\begin{array}{c} a + 2 r; b + r \\ c + 2 r \end{array}; x + y) \\ F (\begin{array}{c} a, b \\ c \end{array}; x + y) & = \sum_{0 \leq r} (- 1)^{r} \frac{(a, b)_{2 r}}{r! (c + r - 1)_{r} (c)_{2 r}} (x y)^{r} F_{4} (\begin{array}{c} a + 2 r; b + 2 r \\ c + 2 r, c + 2 r \end{array}; x, y) \\ F_{4} (\begin{array}{c} a; b \\ c, c \end{array}; x, y) & = \sum_{0 \leq r} (- 1)^{r} \frac{(a, b)_{2 r}}{r! (c)_{r} (c)_{2 r}} (x y)^{r} F (\begin{array}{c} a + 2 r; b + 2 r \\ c + 2 r \end{array}; x + y) \\ F (\begin{array}{c} a, b \\ c \end{array}; x + y) & = \sum_{0 \leq r} \frac{(a)_{2 r} (b, c - b)_{r}}{r! (c + r - 1)_{r} (c)_{2 r}} (x y)^{r} F_{2} (\begin{array}{c} a + 2 r; b + r, b + r \\ c + 2 r, c + 2 r \end{array}; x, y) \\ F_{2} (\begin{array}{c} a; b, b \\ c, c \end{array}; x, y) & = \sum_{0 \leq r} (- 1)^{r} \frac{(a)_{2 r} (b, c - b)_{r}}{r! (c)_{r} (c)_{2 r}} (x y)^{r} F (\begin{array}{c} a + 2 r; b + r \\ c + 2 r \end{array}; x + y) \end{aligned}$

数が多いので, 例として1つ目の公式を考えると,
$\begin{aligned} F_{2} (\begin{array}{c} a; b_{1}, b_{2} \\ c_{1}, c_{2} \end{array}; x, y) & = \nabla (a) F (\begin{array}{c} a, b_{1} \\ c_{1} \end{array}; x) F (\begin{array}{c} a, b_{2} \\ c_{2} \end{array}; y) \\ = \sum_{0 \leq r} \frac{(- θ_{x}, - θ_{y})_{r}}{r! (a)_{r}} F (\begin{array}{c} a, b_{1} \\ c_{1} \end{array}; x) F (\begin{array}{c} a, b_{2} \\ c_{2} \end{array}; y) \end{aligned}$
であり,
$\begin{array}{r} (- θ_{x})_{r} F (\begin{array}{c} a, b \\ c \end{array}; x) = (- 1)^{r} \frac{(a, b)_{r}}{(c)_{r}} x^{r} F (\begin{array}{c} a + r, b + r \\ c + r \end{array}; x) \end{array}$
と計算できることから示すことができる. 同様に他の公式も, Gaussの超幾何定理, Dougallの和公式, Saalschutzの和公式から従う作用素の展開を用いることによって示すことができる.