３変数付き多重級数の巡回和公式の発見について

多重ゼータ値,パラメータ付き多重級数,論文解説

1538

はじめに

どうも, 色数です.
12/5 に出した論文　 https://arxiv.org/pdf/2412.04089 　の僕が先生に送った下書きのようなものとちょっとした解説を公開いたします.

多重ゼータ値とは

簡単に多重ゼータ値について紹介いたします.
多重ゼータ値は Riemann のゼータ関数の一つの一般化として知られています.
まず, Riemann のゼータ関数がどのようなものか？

Riemann のゼータ関数は 1644年に提起された「平方数の逆数全ての和は収束するか？仮に収束するとしてそれは幾らの数値に収束するか？」という問い (1735年に Euler により解決) を一般化したものです.

Riemann のゼータ関数

Riemann のゼータ関数は, 実部が $1$ より大きい複素数 $s$ に対し,
次のように定義される.
$ζ (s) : = \sum_{n = 1}^{\infty} \frac{1}{n^{s}} .$

その後, Riemann のゼータ関数は様々な方法で拡張されてきました.
そのうちの一つの方法として次に述べる「多重ゼータ値」が導入され, 近年急速に発展していっています.

多重ゼータ値は次のように定義されます.

多重ゼータ値　(Multiple zeta value)

$r$ を正整数とする.
$r$ 個の正整数の組 $(k_{1}, \dots, k_{r})$ をインデックスと呼び, 特に $k_{r} > 1$ であるとき, 多重ゼータ値は次のように定義される.
$ζ (k_{1}, \dots, k_{r}) : = \sum_{0 < n_{1} < \dots < n_{r}} \frac{1}{n_{1}^{k_{1}} \dots n_{r}^{k_{r}}} .$

このような多重化をすることにより多重ゼータ値には関係式族と呼ばれる多種多様な関係式が発見されていきました.
最も有名で, 魅力的な関係式の一つとして次の「双対性」を紹介しておきます.

双対性 (Duality)

$k = ({1}^{a_{1} - 1}, b_{1} + 1, \dots, {1}^{a_{r} - 1}, b_{r} + 1)$
$k^{†} = ({1}^{b_{r} - 1}, a_{r} + 1, \dots, {1}^{b_{1} - 1}, a_{1} + 1)$ としたとき,
$ζ (k) = ζ (k^{†})$
が成り立つ.

具体例として $k = (3)$ とすると, $k^{†} = (1, 2)$ となるため $ζ (3) = ζ (1, 2)$ が成り立つことがわかります.
証明は余余余さんの記事か NKSさんの記事を参照してください.

巡回和公式

多重ゼータ値の関係式には上の双対性以外にも次の巡回和公式が知られています.

巡回和公式 (Cyclic sum formula)

$\begin{aligned} \sum_{\begin{array}{c} k \in α \\ 0 \leq i \leq k_{a} - 2 \end{array}} ζ (i + 1, k_{1}, \dots, k_{a - 1}, k_{a} - i) & = \sum_{k \in α} ζ (k_{1}, \dots, k_{a - 1}, k_{a} + 1) . \end{aligned}$

長くなってしまうので詳しくは NKSさんの記事を参照してください.
この公式にパラメータを付けるというのが本研究の大まかな内容です.
具体的には Igarashi.M によって
$Z (k; α) : = \sum_{0 \leq m_{1} < \dots < m_{n}} \frac{(α)_{m_{1}}}{m_{1}!} \frac{m_{n}!}{(α)_{m_{n}}} \prod_{i = 1}^{n} \frac{1}{(m_{i} + α)^{k_{i}}}$
という形の多重級数においても巡回和公式が成り立つことが2011年に証明されました.
さらには $α, β$ という２つのパラメータが付いている多重級数においても成り立つことが2020年に証明されました.
その論文において $α, β, γ$ という３つのパラメータがついた場合はどうなるのか？といった問題が提起されました.
今回出した論文はその問題に対する一つのアンサーを与えました.

主結果とその導出

$Z_{I}^{⋆} (k; α, β, γ) : = \sum_{0 \leq m_{1} \leq \dots \leq m_{d}} \frac{(α)_{m_{1}} (β)_{m_{1}}}{(γ)_{m_{1}} m_{1}!} \frac{(γ)_{m_{d}} m_{d}!}{(α)_{m_{d}} (β)_{m_{d}}} \frac{1}{(m_{1} + γ)^{k_{1}}} \prod_{i = 2}^{d} \frac{1}{(m_{i} + α) (m_{i} + β) (m_{i} + γ)^{k_{i} - 2}}$
$Z_{I I}^{⋆} (k; α, β, γ) : = \sum_{0 \leq m_{1} \leq \dots \leq m_{d}} \frac{(α)_{m_{1}} (β)_{m_{1}}}{(γ)_{m_{1}} m_{1}!} \frac{(γ)_{m_{d}} m_{d}!}{(α)_{m_{d}} (β)_{m_{d}}} \prod_{i = 1}^{d} \frac{1}{(m_{i} + α) (m_{i} + β) (m_{i} + γ)^{k_{i} - 2}}$
$Z (a | b | c; α, β, γ) : = \sum_{n = 0}^{\infty} \frac{1}{(n + α)^{a} (n + β)^{b} (n + γ)^{c}}$
とすると
$\begin{aligned} \sum_{i = 1}^{d} \sum_{j = 0}^{k_{i} - 3} Z_{I}^{⋆} (j + 1, k_{i + 1}, \dots, k_{d}, k_{1}, \dots, k_{i - 1}, k_{i} - j; α, β, γ) + (α + β - γ) \sum_{i = 1}^{d} Z_{I I}^{⋆} (k_{i}, \dots, k_{d}, k_{1}, \dots, k_{i}, 2; α, β, γ) & = d Z (d | d + 1 | k - 2 d; α, β, γ) + d Z (d + 1 | d | k - 2 d; α, β, γ) + (k - 2 d) Z (d | d | k - 2 d + 1; α, β, γ) \end{aligned}$
が成り立つ.

証明

$\begin{aligned} \frac{(α)_{m + 1} (β)_{m + 1}}{m! (γ)_{m}} \sum_{n \leq l} \frac{l! (γ)_{l}}{(α)_{l + 1} (β)_{l + 1} (l - m)} & = \begin{matrix} \frac{n! (γ)_{n}}{(α)_{n} (β)_{n}} \sum_{k = 0}^{m} \frac{(α)_{k} (β)_{k}}{k! (γ)_{k} (n - k)} \\ + (γ - α - β) \sum_{k = 0}^{m} \frac{(α)_{k} (β)_{k}}{k! (γ)_{k}} \sum_{n \leq l} \frac{l! (γ)_{l}}{(α)_{l + 1} (β)_{l + 1}} \end{matrix} \end{aligned}$

$S_{m} (α, β, γ) : = \sum_{n < l} \frac{(l - 1)! (γ)_{l - 1}}{(α)_{l} (β)_{l} (l - m)}$ とおくと, 次のように計算できる.
$\begin{aligned} S_{m} (α, β, γ) & = \sum_{n < l} \frac{(l - 1)! (γ)_{l - 1}}{(α)_{l} (β)_{l} (l - m)} \\ = \frac{1}{α + m - 1} \sum_{n < l} \frac{(l - 1)! (γ)_{l - 1}}{(α)_{l - 1} (β)_{l}} (\frac{1}{l - m} - \frac{1}{α + l - 1}) \\ = \begin{matrix} \frac{1}{(α + m - 1) (β + m - 1)} \sum_{n < l} \frac{(l - 1)! (γ)_{l - 1}}{(α)_{l - 1} (β)_{l - 1}} (\frac{1}{l - m} - \frac{1}{β + l - 1}) \\ - \frac{1}{α + m - 1} \sum_{n < l} \frac{(l - 1)! (γ)_{l - 1}}{(α)_{l} (β)_{l}} \end{matrix} \\ = \begin{matrix} \frac{1}{(α + m - 1) (β + m - 1)} \sum_{n < l} \frac{(l - 1)! (γ)_{l - 1}}{(α)_{l - 1} (β)_{l - 1} (l - m)} \\ - \frac{1}{(α + m - 1) (β + m - 1)} \sum_{n < l} \frac{(l - 1)! (γ)_{l - 1}}{(α)_{l - 1} (β)_{l}} \\ - \frac{1}{(α + m - 1)} \sum_{n < l} \frac{(l - 1)! (γ)_{l - 1}}{(α)_{l} (β)_{l}} \end{matrix} \end{aligned}$
(横に長くなってしまったので折り曲げる)
$\begin{aligned} = \begin{matrix} \frac{1}{(α + m - 1) (β + m - 1)} \frac{n! (γ)_{n}}{(α)_{n} (β)_{n} (n + 1 - m)} + \frac{1}{(α + m - 1) (β + m - 1)} \sum_{n < l} \frac{l! (γ)_{l}}{(α)_{l} (β)_{l} (l + 1 - m)} \\ - \frac{1}{(α + m - 1) (β + m - 1)} \sum_{n < l} \frac{(l - 1)! (γ)_{l - 1}}{(α)_{l - 1} (β)_{l}} - \frac{1}{(α + m - 1)} \sum_{n < l} \frac{(l - 1)! (γ)_{l - 1}}{(α)_{l} (β)_{l}} \end{matrix} \\ = \begin{matrix} \frac{1}{(α + m - 1) (β + m - 1)} \frac{n! (γ)_{n}}{(α)_{n} (β)_{n} (n + 1 - m)} \\ + \frac{m - 1}{(α + m - 1) (β + m - 1)} \sum_{n < l} \frac{(l - 1)! (γ)_{l}}{(α)_{l} (β)_{l}} (\frac{1}{l + 1 - m} + \frac{1}{m - 1}) \\ - \frac{1}{(α + m - 1) (β + m - 1)} \sum_{n < l} \frac{(l - 1)! (γ)_{l - 1}}{(α)_{l - 1} (β)_{l}} - \frac{1}{(α + m - 1)} \sum_{n < l} \frac{(l - 1)! (γ)_{l - 1}}{(α)_{l} (β)_{l}} \end{matrix} \end{aligned}$
$\begin{aligned} = \begin{matrix} \frac{1}{(α + m - 1) (β + m - 1)} \frac{n! (γ)_{n}}{(α)_{n} (β)_{n} (n + 1 - m)} + \frac{m - 1}{(α + m - 1) (β + m - 1)} \sum_{n < l} \frac{(l - 1)! (γ)_{l}}{(α)_{l} (β)_{l} (l + 1 - m)} + \\ \frac{1}{(α + m - 1) (β + m - 1)} \sum_{n < l} \frac{(l - 1)! (γ)_{l}}{(α)_{l} (β)_{l}} - \frac{1}{(α + m - 1) (β + m - 1)} \sum_{n < l} \frac{(l - 1)! (γ)_{l - 1}}{(α)_{l - 1} (β)_{l}} \\ - \frac{1}{(α + m - 1)} \sum_{n < l} \frac{(l - 1)! (γ)_{l - 1}}{(α)_{l} (β)_{l}} \end{matrix} \\ = \begin{matrix} \frac{1}{(α + m - 1) (β + m - 1)} \frac{n! (γ)_{n}}{(α)_{n} (β)_{n} (n + 1 - m)} \\ + \frac{(m - 1) (γ + m - 2)}{(α + m - 1) (β + m - 1)} \sum_{n < l} \frac{(l - 1)! (γ)_{l - 1}}{(α)_{l} (β)_{l}} (\frac{1}{l + 1 - m} + \frac{1}{γ + m - 2}) \\ + \frac{1}{(α + m - 1) (β + m - 1)} \sum_{n < l} \frac{(l - 1)! (γ)_{l}}{(α)_{l} (β)_{l}} - \frac{1}{(α + m - 1) (β + m - 1)} \sum_{n < l} \frac{(l - 1)! (γ)_{l - 1}}{(α)_{l - 1} (β)_{l}} \\ - \frac{1}{(α + m - 1)} \sum_{n < l} \frac{(l - 1)! (γ)_{l - 1}}{(α)_{l} (β)_{l}} \end{matrix} \\ = \begin{matrix} \frac{1}{(α + m - 1) (β + m - 1)} \frac{n! (γ)_{n}}{(α)_{n} (β)_{n} (n + 1 - m)} + \frac{(m - 1) (γ + m - 2)}{(α + m - 1) (β + m - 1)} S_{m - 1} (α, β, γ) \\ + \frac{(m - 1)}{(α + m - 1) (β + m - 1)} \sum_{n < l} \frac{(l - 1)! (γ)_{l - 1}}{(α)_{l} (β)_{l}} + \frac{1}{(α + m - 1) (β + m - 1)} \sum_{n < l} \frac{(l - 1)! (γ)_{l}}{(α)_{l} (β)_{l}} \\ - \frac{1}{(α + m - 1) (β + m - 1)} \sum_{n < l} \frac{(l - 1)! (γ)_{l - 1}}{(α)_{l - 1} (β)_{l}} - \frac{1}{(α + m - 1)} \sum_{n < l} \frac{(l - 1)! (γ)_{l - 1}}{(α)_{l} (β)_{l}} \end{matrix} . \end{aligned}$
両辺に $\frac{(α)_{m} (β)_{m}}{(m - 1)! (γ)_{m - 1}}$ を掛け $S_{0} (α, β, γ)$ の項が $0$ になるため補題を得る.

$\begin{aligned} \frac{(α)_{m_{0} + 1} (β)_{m_{0} + 1}}{m_{0}! (γ)_{m_{0}}} \sum_{m_{n - 1} < m_{n}} \frac{m_{n}! (γ)_{m_{n}}}{(α)_{m_{n} + 1} (β)_{m_{n} + 1} (m_{n} - m_{0})} \\ = \begin{matrix} \sum_{k = 0}^{m_{0} - 1} \frac{(α)_{k} (β)_{k}}{k! (γ)_{k}} \frac{m_{n - 1}! (γ)_{m_{n - 1}}}{(α)_{m_{n - 1}} (β)_{m_{n - 1}} (m_{n - 1} - k)} \\ + \frac{m_{n - 1}! (γ)_{m_{n - 1}}}{(α)_{m_{n - 1}} (β)_{m_{n - 1} + 1}} \frac{(α)_{m_{0}} (β)_{m_{0}}}{m_{0}! (γ)_{m_{0}}} \\ + \frac{m_{n - 1}! (γ)_{m_{n - 1}}}{(α)_{m_{n - 1} + 1} (β)_{m_{n - 1} + 1}} \frac{(α)_{m_{0}} (β)_{m_{0} + 1}}{m_{0}! (γ)_{m_{0}}} \\ + (γ - α - β) \sum_{k = 0}^{m_{0}} \frac{(α)_{k} (β)_{k}}{k! (γ)_{k}} \sum_{m_{n - 1} \leq m_{n}} \frac{m_{n}! (γ)_{m_{n}}}{(α)_{m_{n} + 1} (β)_{m_{n} + 1}} \end{matrix} \end{aligned}$

$\begin{aligned} F_{m_{0}} (α, β, γ) & = \frac{(α)_{m_{0} + 1} (β)_{m_{0} + 1}}{m_{0}! (γ)_{m_{0}}} \sum_{m_{n - 1} < m_{n}} \frac{m_{n}! (γ)_{m_{n}}}{(α)_{m_{n} + 1} (β)_{m_{n} + 1} (m_{n} - m_{0})} \\ = \frac{(α)_{m_{0} + 1} (β)_{m_{0} + 1}}{m_{0}! (γ)_{m_{0}}} \sum_{m_{n - 1} \leq m_{n}} \frac{(m_{n} + 1)! (γ)_{m_{n} + 1}}{(α)_{m_{n} + 2} (β)_{m_{n} + 2} (m_{n} + 1 - m_{0})} \\ = \frac{(α)_{m_{0}} (β)_{m_{0} + 1}}{m_{0}! (γ)_{m_{0}}} \sum_{m_{n - 1} \leq m_{n}} \frac{(m_{n} + 1)! (γ)_{m_{n} + 1}}{(α)_{m_{n} + 1} (β)_{m_{n} + 2}} (\frac{1}{m_{n} + 1 - m_{0}} - \frac{1}{α + m_{n} + 1}) \\ = \frac{(α)_{m_{0}} (β)_{m_{0} + 1}}{m_{0}! (γ)_{m_{0}}} \sum_{m_{n - 1} \leq m_{n}} \frac{(m_{n} + 1)! (γ)_{m_{n} + 1}}{(α)_{m_{n} + 1} (β)_{m_{n} + 2} (m_{n} + 1 - m_{0})} - \frac{(α)_{m_{0}} (β)_{m_{0} + 1}}{m_{0}! (γ)_{m_{0}}} \sum_{m_{n - 1} \leq m_{n}} \frac{(m_{n} + 1)! (γ)_{m_{n} + 1}}{(α)_{m_{n} + 2} (β)_{m_{n} + 2}} \\ = \begin{matrix} \frac{(α)_{m_{0}} (β)_{m_{0}}}{m_{0}! (γ)_{m_{0}}} \sum_{m_{n - 1} \leq m_{n}} \frac{(m_{n} + 1)! (γ)_{m_{n} + 1}}{(α)_{m_{n} + 1} (β)_{m_{n} + 1}} (\frac{1}{m_{n} + 1 - m_{0}} - \frac{1}{β + m_{n} + 1}) \\ - \frac{(α)_{m_{0}} (β)_{m_{0} + 1}}{m_{0}! (γ)_{m_{0}}} \sum_{m_{n - 1} \leq m_{n}} \frac{(m_{n} + 1)! (γ)_{m_{n} + 1}}{(α)_{m_{n} + 2} (β)_{m_{n} + 2}} \end{matrix} \\ = \begin{matrix} \frac{(α)_{m_{0}} (β)_{m_{0}}}{m_{0}! (γ)_{m_{0}}} \sum_{m_{n - 1} \leq m_{n}} \frac{(m_{n} + 1)! (γ)_{m_{n} + 1}}{(α)_{m_{n} + 1} (β)_{m_{n} + 1} (m_{n} + 1 - m_{0})} - \frac{(α)_{m_{0}} (β)_{m_{0}}}{m_{0}! (γ)_{m_{0}}} \sum_{m_{n - 1} \leq m_{n}} \frac{(m_{n} + 1)! (γ)_{m_{n} + 1}}{(α)_{m_{n} + 1} (β)_{m_{n} + 2}} \\ - \frac{(α)_{m_{0}} (β)_{m_{0} + 1}}{m_{0}! (γ)_{m_{0}}} \sum_{m_{n - 1} \leq m_{n}} \frac{(m_{n} + 1)! (γ)_{m_{n} + 1}}{(α)_{m_{n} + 2} (β)_{m_{n} + 2}} \end{matrix} \\ = \begin{matrix} \frac{(α)_{m_{0}} (β)_{m_{0}}}{(m_{0} - 1)! (γ)_{m_{0}}} \sum_{m_{n - 1} \leq m_{n}} \frac{m_{n}! (γ)_{m_{n} + 1}}{(α)_{m_{n} + 1} (β)_{m_{n} + 1}} (\frac{1}{m_{n} + 1 - m_{0}} + \frac{1}{m_{0}}) \\ - \frac{(α)_{m_{0}} (β)_{m_{0}}}{m_{0}! (γ)_{m_{0}}} \sum_{m_{n - 1} \leq m_{n}} \frac{(m_{n} + 1)! (γ)_{m_{n} + 1}}{(α)_{m_{n} + 1} (β)_{m_{n} + 2}} - \frac{(α)_{m_{0}} (β)_{m_{0} + 1}}{m_{0}! (γ)_{m_{0}}} \sum_{m_{n - 1} \leq m_{n}} \frac{(m_{n} + 1)! (γ)_{m_{n} + 1}}{(α)_{m_{n} + 2} (β)_{m_{n} + 2}} \end{matrix} \\ = \begin{matrix} \frac{(α)_{m_{0}} (β)_{m_{0}}}{(m_{0} - 1)! (γ)_{m_{0}}} \sum_{m_{n - 1} \leq m_{n}} \frac{m_{n}! (γ)_{m_{n} + 1}}{(α)_{m_{n} + 1} (β)_{m_{n} + 1} (m_{n} + 1 - m_{0})} \\ + \frac{(α)_{m_{0}} (β)_{m_{0}}}{m_{0}! (γ)_{m_{0}}} \sum_{m_{n - 1} \leq m_{n}} \frac{m_{n}! (γ)_{m_{n} + 1}}{(α)_{m_{n} + 1} (β)_{m_{n} + 1}} - \frac{(α)_{m_{0}} (β)_{m_{0}}}{m_{0}! (γ)_{m_{0}}} \sum_{m_{n - 1} \leq m_{n}} \frac{(m_{n} + 1)! (γ)_{m_{n} + 1}}{(α)_{m_{n} + 1} (β)_{m_{n} + 2}} \\ - \frac{(α)_{m_{0}} (β)_{m_{0} + 1}}{m_{0}! (γ)_{m_{0}}} \sum_{m_{n - 1} \leq m_{n}} \frac{(m_{n} + 1)! (γ)_{m_{n} + 1}}{(α)_{m_{n} + 2} (β)_{m_{n} + 2}} \end{matrix} \\ = \begin{matrix} \frac{(α)_{m_{0}} (β)_{m_{0}}}{(m_{0} - 1)! (γ)_{m_{0} - 1}} \sum_{m_{n - 1} \leq m_{n}} \frac{m_{n}! (γ)_{m_{n}}}{(α)_{m_{n} + 1} (β)_{m_{n} + 1}} (\frac{1}{m_{n} + 1 - m_{0}} + \frac{1}{γ + m_{0} - 1}) \\ + \frac{(α)_{m_{0}} (β)_{m_{0}}}{m_{0}! (γ)_{m_{0}}} \sum_{m_{n - 1} \leq m_{n}} \frac{m_{n}! (γ)_{m_{n} + 1}}{(α)_{m_{n} + 1} (β)_{m_{n} + 1}} - \frac{(α)_{m_{0}} (β)_{m_{0}}}{m_{0}! (γ)_{m_{0}}} \sum_{m_{n - 1} \leq m_{n}} \frac{(m_{n} + 1)! (γ)_{m_{n} + 1}}{(α)_{m_{n} + 1} (β)_{m_{n} + 2}} \\ - \frac{(α)_{m_{0}} (β)_{m_{0} + 1}}{m_{0}! (γ)_{m_{0}}} \sum_{m_{n - 1} \leq m_{n}} \frac{(m_{n} + 1)! (γ)_{m_{n} + 1}}{(α)_{m_{n} + 2} (β)_{m_{n} + 2}} \end{matrix} \\ = \begin{matrix} \frac{(α)_{m_{0}} (β)_{m_{0}}}{(m_{0} - 1)! (γ)_{m_{0} - 1}} \sum_{m_{n - 1} \leq m_{n}} \frac{m_{n}! (γ)_{m_{n}}}{(α)_{m_{n} + 1} (β)_{m_{n} + 1} (m_{n} + 1 - m_{0})} \\ + \frac{(α)_{m_{0}} (β)_{m_{0}}}{(m_{0} - 1)! (γ)_{m_{0}}} \sum_{m_{n - 1} \leq m_{n}} \frac{m_{n}! (γ)_{m_{n}}}{(α)_{m_{n} + 1} (β)_{m_{n} + 1}} + \frac{(α)_{m_{0}} (β)_{m_{0}}}{m_{0}! (γ)_{m_{0}}} \sum_{m_{n - 1} \leq m_{n}} \frac{m_{n}! (γ)_{m_{n} + 1}}{(α)_{m_{n} + 1} (β)_{m_{n} + 1}} \\ - \frac{(α)_{m_{0}} (β)_{m_{0}}}{m_{0}! (γ)_{m_{0}}} \sum_{m_{n - 1} \leq m_{n}} \frac{(m_{n} + 1)! (γ)_{m_{n} + 1}}{(α)_{m_{n} + 1} (β)_{m_{n} + 2}} - \frac{(α)_{m_{0}} (β)_{m_{0} + 1}}{m_{0}! (γ)_{m_{0}}} \sum_{m_{n - 1} \leq m_{n}} \frac{(m_{n} + 1)! (γ)_{m_{n} + 1}}{(α)_{m_{n} + 2} (β)_{m_{n} + 2}} \end{matrix} \\ = \begin{matrix} F_{m_{0} - 1} (α, β, γ) + \frac{(α)_{m_{0}} (β)_{m_{0}}}{(m_{0} - 1)! (γ)_{m_{0} - 1}} \frac{m_{n - 1}! (γ)_{m_{n - 1}}}{(α)_{m_{n - 1} + 1} (β)_{m_{n - 1} + 1} (m_{n - 1} + 1 - m_{0})} \\ + \frac{(α)_{m_{0}} (β)_{m_{0}}}{(m_{0} - 1)! (γ)_{m_{0}}} \sum_{m_{n - 1} \leq m_{n}} \frac{m_{n}! (γ)_{m_{n}}}{(α)_{m_{n} + 1} (β)_{m_{n} + 1}} + \frac{(α)_{m_{0}} (β)_{m_{0}}}{m_{0}! (γ)_{m_{0}}} \sum_{m_{n - 1} \leq m_{n}} \frac{m_{n}! (γ)_{m_{n} + 1}}{(α)_{m_{n} + 1} (β)_{m_{n} + 1}} \\ - \frac{(α)_{m_{0}} (β)_{m_{0}}}{m_{0}! (γ)_{m_{0}}} \sum_{m_{n - 1} < m_{n}} \frac{m_{n}! (γ)_{m_{n}}}{(α)_{m_{n}} (β)_{m_{n} + 1}} - \frac{(α)_{m_{0}} (β)_{m_{0} + 1}}{m_{0}! (γ)_{m_{0}}} \sum_{m_{n - 1} < m_{n}} \frac{m_{n}! (γ)_{m_{n}}}{(α)_{m_{n} + 1} (β)_{m_{n} + 1}} \end{matrix} \end{aligned}$
今度は $m_{0} = 0$ から和をとると,
$\begin{aligned} \frac{(α)_{m_{0} + 1} (β)_{m_{0} + 1}}{m_{0}! (γ)_{m_{0}}} \sum_{m_{n - 1} < m_{n}} \frac{m_{n}! (γ)_{m_{n}}}{(α)_{m_{n} + 1} (β)_{m_{n} + 1} (m_{n} - m_{0})} \\ = \begin{matrix} \sum_{k = 0}^{m_{0} - 1} \frac{(α)_{k + 1} (β)_{k + 1}}{k! (γ)_{k}} \frac{m_{n - 1}! (γ)_{m_{n - 1}}}{(α)_{m_{n - 1} + 1} (β)_{m_{n - 1} + 1} (m_{n - 1} - k)} \\ + \sum_{k = 0}^{m_{0} - 1} \frac{(α)_{k + 1} (β)_{k + 1}}{k! (γ)_{k + 1}} \sum_{m_{n - 1} \leq m_{n}} \frac{m_{n}! (γ)_{m_{n}}}{(α)_{m_{n} + 1} (β)_{m_{n} + 1}} + \sum_{k = 0}^{m_{0}} \frac{(α)_{k} (β)_{k}}{k! (γ)_{k}} \sum_{m_{n - 1} \leq m_{n}} \frac{m_{n}! (γ)_{m_{n} + 1}}{(α)_{m_{n} + 1} (β)_{m_{n} + 1}} \\ - \sum_{k = 0}^{m_{0}} \frac{(α)_{k} (β)_{k}}{k! (γ)_{k}} \sum_{m_{n - 1} < m_{n}} \frac{m_{n}! (γ)_{m_{n}}}{(α)_{m_{n}} (β)_{m_{n} + 1}} - \sum_{k = 0}^{m_{0}} \frac{(α)_{k} (β)_{k + 1}}{k! (γ)_{k}} \sum_{m_{n - 1} < m_{n}} \frac{m_{n}! (γ)_{m_{n}}}{(α)_{m_{n} + 1} (β)_{m_{n} + 1}} \end{matrix} \\ = \begin{matrix} \sum_{k = 0}^{m_{0} - 1} \frac{(α)_{k} (β)_{k + 1}}{k! (γ)_{k}} \frac{m_{n - 1}! (γ)_{m_{n - 1}}}{(α)_{m_{n - 1}} (β)_{m_{n - 1} + 1}} (\frac{1}{m_{n - 1} - k} - \frac{1}{α + m_{n - 1}}) \\ + \sum_{k = 0}^{m_{0} - 1} \frac{(α)_{k + 1} (β)_{k + 1}}{k! (γ)_{k + 1}} \sum_{m_{n - 1} \leq m_{n}} \frac{m_{n}! (γ)_{m_{n}}}{(α)_{m_{n} + 1} (β)_{m_{n} + 1}} + \sum_{k = 0}^{m_{0}} \frac{(α)_{k} (β)_{k}}{k! (γ)_{k}} \sum_{m_{n - 1} \leq m_{n}} \frac{m_{n}! (γ)_{m_{n} + 1}}{(α)_{m_{n} + 1} (β)_{m_{n} + 1}} \\ - \sum_{k = 0}^{m_{0}} \frac{(α)_{k} (β)_{k}}{k! (γ)_{k}} \sum_{m_{n - 1} < m_{n}} \frac{m_{n}! (γ)_{m_{n}}}{(α)_{m_{n}} (β)_{m_{n} + 1}} - \sum_{k = 0}^{m_{0}} \frac{(α)_{k} (β)_{k + 1}}{k! (γ)_{k}} \sum_{m_{n - 1} < m_{n}} \frac{m_{n}! (γ)_{m_{n}}}{(α)_{m_{n} + 1} (β)_{m_{n} + 1}} \end{matrix} \\ = \begin{matrix} \sum_{k = 0}^{m_{0} - 1} \frac{(α)_{k} (β)_{k}}{k! (γ)_{k}} \frac{m_{n - 1}! (γ)_{m_{n - 1}}}{(α)_{m_{n - 1}} (β)_{m_{n - 1}}} (\frac{1}{m_{n - 1} - k} - \frac{1}{β + m_{n - 1}}) \\ - \frac{m_{n - 1}! (γ)_{m_{n - 1}}}{(α)_{m_{n - 1} + 1} (β)_{m_{n - 1} + 1}} \sum_{k = 0}^{m_{0} - 1} \frac{(α)_{k} (β)_{k + 1}}{k! (γ)_{k}} \\ + \sum_{k = 0}^{m_{0} - 1} \frac{(α)_{k + 1} (β)_{k + 1}}{k! (γ)_{k + 1}} \sum_{m_{n - 1} \leq m_{n}} \frac{m_{n}! (γ)_{m_{n}}}{(α)_{m_{n} + 1} (β)_{m_{n} + 1}} + \sum_{k = 0}^{m_{0}} \frac{(α)_{k} (β)_{k}}{k! (γ)_{k}} \sum_{m_{n - 1} \leq m_{n}} \frac{m_{n}! (γ)_{m_{n} + 1}}{(α)_{m_{n} + 1} (β)_{m_{n} + 1}} \\ - \sum_{k = 0}^{m_{0}} \frac{(α)_{k} (β)_{k}}{k! (γ)_{k}} \sum_{m_{n - 1} < m_{n}} \frac{m_{n}! (γ)_{m_{n}}}{(α)_{m_{n}} (β)_{m_{n} + 1}} - \sum_{k = 0}^{m_{0}} \frac{(α)_{k} (β)_{k + 1}}{k! (γ)_{k}} \sum_{m_{n - 1} < m_{n}} \frac{m_{n}! (γ)_{m_{n}}}{(α)_{m_{n} + 1} (β)_{m_{n} + 1}} \end{matrix} \end{aligned}$
$\begin{aligned} = \begin{matrix} \sum_{k = 0}^{m_{0} - 1} \frac{(α)_{k} (β)_{k}}{k! (γ)_{k}} \frac{m_{n - 1}! (γ)_{m_{n - 1}}}{(α)_{m_{n - 1}} (β)_{m_{n - 1}} (m_{n - 1} - k)} \\ - \frac{m_{n - 1}! (γ)_{m_{n - 1}}}{(α)_{m_{n - 1}} (β)_{m_{n - 1} + 1}} \sum_{k = 0}^{m_{0} - 1} \frac{(α)_{k} (β)_{k}}{k! (γ)_{k}} \\ - \frac{m_{n - 1}! (γ)_{m_{n - 1}}}{(α)_{m_{n - 1} + 1} (β)_{m_{n - 1} + 1}} \sum_{k = 0}^{m_{0} - 1} \frac{(α)_{k} (β)_{k + 1}}{k! (γ)_{k}} \\ + \sum_{k = 0}^{m_{0} - 1} \frac{(α)_{k + 1} (β)_{k + 1}}{k! (γ)_{k + 1}} \sum_{m_{n - 1} \leq m_{n}} \frac{m_{n}! (γ)_{m_{n}}}{(α)_{m_{n} + 1} (β)_{m_{n} + 1}} + \sum_{k = 0}^{m_{0}} \frac{(α)_{k} (β)_{k}}{k! (γ)_{k}} \sum_{m_{n - 1} \leq m_{n}} \frac{m_{n}! (γ)_{m_{n} + 1}}{(α)_{m_{n} + 1} (β)_{m_{n} + 1}} \\ - \sum_{k = 0}^{m_{0}} \frac{(α)_{k} (β)_{k}}{k! (γ)_{k}} \sum_{m_{n - 1} < m_{n}} \frac{m_{n}! (γ)_{m_{n}}}{(α)_{m_{n}} (β)_{m_{n} + 1}} - \sum_{k = 0}^{m_{0}} \frac{(α)_{k} (β)_{k + 1}}{k! (γ)_{k}} \sum_{m_{n - 1} < m_{n}} \frac{m_{n}! (γ)_{m_{n}}}{(α)_{m_{n} + 1} (β)_{m_{n} + 1}} \end{matrix} \\ = \begin{matrix} \sum_{k = 0}^{m_{0} - 1} \frac{(α)_{k} (β)_{k}}{k! (γ)_{k}} \frac{m_{n - 1}! (γ)_{m_{n - 1}}}{(α)_{m_{n - 1}} (β)_{m_{n - 1}} (m_{n - 1} - k)} \\ - \frac{m_{n - 1}! (γ)_{m_{n - 1}}}{(α)_{m_{n - 1}} (β)_{m_{n - 1} + 1}} \sum_{k = 0}^{m_{0}} \frac{(α)_{k} (β)_{k}}{k! (γ)_{k}} + \frac{m_{n - 1}! (γ)_{m_{n - 1}}}{(α)_{m_{n - 1}} (β)_{m_{n - 1} + 1}} \frac{(α)_{m_{0}} (β)_{m_{0}}}{m_{0}! (γ)_{m_{0}}} \\ - \frac{m_{n - 1}! (γ)_{m_{n - 1}}}{(α)_{m_{n - 1} + 1} (β)_{m_{n - 1} + 1}} \sum_{k = 0}^{m_{0}} \frac{(α)_{k} (β)_{k + 1}}{k! (γ)_{k}} + \frac{m_{n - 1}! (γ)_{m_{n - 1}}}{(α)_{m_{n - 1} + 1} (β)_{m_{n - 1} + 1}} \frac{(α)_{m_{0}} (β)_{m_{0} + 1}}{m_{0}! (γ)_{m_{0}}} \\ + \sum_{k = 0}^{m_{0} - 1} \frac{(α)_{k + 1} (β)_{k + 1}}{k! (γ)_{k + 1}} \sum_{m_{n - 1} \leq m_{n}} \frac{m_{n}! (γ)_{m_{n}}}{(α)_{m_{n} + 1} (β)_{m_{n} + 1}} + \sum_{k = 0}^{m_{0}} \frac{(α)_{k} (β)_{k}}{k! (γ)_{k}} \sum_{m_{n - 1} \leq m_{n}} \frac{m_{n}! (γ)_{m_{n} + 1}}{(α)_{m_{n} + 1} (β)_{m_{n} + 1}} \\ - \sum_{k = 0}^{m_{0}} \frac{(α)_{k} (β)_{k}}{k! (γ)_{k}} \sum_{m_{n - 1} < m_{n}} \frac{m_{n}! (γ)_{m_{n}}}{(α)_{m_{n}} (β)_{m_{n} + 1}} - \sum_{k = 0}^{m_{0}} \frac{(α)_{k} (β)_{k + 1}}{k! (γ)_{k}} \sum_{m_{n - 1} < m_{n}} \frac{m_{n}! (γ)_{m_{n}}}{(α)_{m_{n} + 1} (β)_{m_{n} + 1}} \end{matrix} \\ = \begin{matrix} \sum_{k = 0}^{m_{0} - 1} \frac{(α)_{k} (β)_{k}}{k! (γ)_{k}} \frac{m_{n - 1}! (γ)_{m_{n - 1}}}{(α)_{m_{n - 1}} (β)_{m_{n - 1}} (m_{n - 1} - k)} \\ + \frac{m_{n - 1}! (γ)_{m_{n - 1}}}{(α)_{m_{n - 1}} (β)_{m_{n - 1} + 1}} \frac{(α)_{m_{0}} (β)_{m_{0}}}{m_{0}! (γ)_{m_{0}}} \\ + \frac{m_{n - 1}! (γ)_{m_{n - 1}}}{(α)_{m_{n - 1} + 1} (β)_{m_{n - 1} + 1}} \frac{(α)_{m_{0}} (β)_{m_{0} + 1}}{m_{0}! (γ)_{m_{0}}} \\ + \sum_{k = 0}^{m_{0}} \frac{(α)_{k} (β)_{k}}{(k - 1)! (γ)_{k}} \sum_{m_{n - 1} \leq m_{n}} \frac{m_{n}! (γ)_{m_{n}}}{(α)_{m_{n} + 1} (β)_{m_{n} + 1}} + \sum_{k = 0}^{m_{0}} \frac{(α)_{k} (β)_{k}}{k! (γ)_{k}} \sum_{m_{n - 1} \leq m_{n}} \frac{m_{n}! (γ)_{m_{n} + 1}}{(α)_{m_{n} + 1} (β)_{m_{n} + 1}} \\ - \sum_{k = 0}^{m_{0}} \frac{(α)_{k} (β)_{k}}{k! (γ)_{k}} \sum_{m_{n - 1} \leq m_{n}} \frac{m_{n}! (γ)_{m_{n}}}{(α)_{m_{n}} (β)_{m_{n} + 1}} - \sum_{k = 0}^{m_{0}} \frac{(α)_{k} (β)_{k + 1}}{k! (γ)_{k}} \sum_{m_{n - 1} \leq m_{n}} \frac{m_{n}! (γ)_{m_{n}}}{(α)_{m_{n} + 1} (β)_{m_{n} + 1}} \end{matrix} \end{aligned}$
$\begin{aligned} = \begin{matrix} \sum_{k = 0}^{m_{0} - 1} \frac{(α)_{k} (β)_{k}}{k! (γ)_{k}} \frac{m_{n - 1}! (γ)_{m_{n - 1}}}{(α)_{m_{n - 1}} (β)_{m_{n - 1}} (m_{n - 1} - k)} \\ + \frac{m_{n - 1}! (γ)_{m_{n - 1}}}{(α)_{m_{n - 1}} (β)_{m_{n - 1} + 1}} \frac{(α)_{m_{0}} (β)_{m_{0}}}{m_{0}! (γ)_{m_{0}}} \\ + \frac{m_{n - 1}! (γ)_{m_{n - 1}}}{(α)_{m_{n - 1} + 1} (β)_{m_{n - 1} + 1}} \frac{(α)_{m_{0}} (β)_{m_{0} + 1}}{m_{0}! (γ)_{m_{0}}} \\ + (γ - α - β) \sum_{k = 0}^{m_{0}} \frac{(α)_{k} (β)_{k}}{k! (γ)_{k}} \sum_{m_{n - 1} \leq m_{n}} \frac{m_{n}! (γ)_{m_{n}}}{(α)_{m_{n} + 1} (β)_{m_{n} + 1}} \end{matrix} \end{aligned}$
以上より示せた.

$\begin{aligned} T^{⋆} (k_{1}, \dots, k_{n}; (α, β, γ)) \\ = \begin{matrix} T^{⋆} (k_{n}, k_{1}, \dots, k_{n - 1}; (α, β, γ)) \\ + (γ - α - β) Z_{I I}^{⋆} (k_{n}, k_{1}, \dots, k_{n - 1}, 2; (α, β, γ)) \\ + Z (n | n - 1 | k_{1} + \dots + k_{n} - 2 n) + Z (n - 1 | n | k_{1} + \dots + k_{n} - 2 n) \\ - \sum_{j = 0}^{k_{n} - 3} Z_{I}^{⋆} (j + 1, k_{1}, \dots, k_{n - 1}, k_{n} - j; (α, β, γ)) \\ + (k_{n} - 2) Z (n | n | k_{1} + \dots + k_{n} - 2 n + 1; (α, β, γ)) \end{matrix} \end{aligned}$

$\begin{aligned} \sum_{\begin{array}{c} 0 \leq m_{0} \leq m_{1} \leq \dots \leq m_{n} \\ m_{0} \neq m_{n} \end{array}} \frac{(α)_{m_{0}} (β)_{m_{0}}}{m_{0}! (γ)_{m_{0}}} \frac{m_{n}! (γ)_{m_{n}}}{(α)_{m_{n}} (β)_{m_{n}}} \frac{1}{(m_{0} + γ)^{j}} \\ \times {\prod_{i = 1}^{n - 1} \frac{1}{(m_{i} + α) (m_{i} + β) (m_{i} + γ)^{k_{i} - 2}}} \frac{1}{(m_{n} + α) (m_{n} + β) (m_{n} + γ)^{k_{n} - j - 2}} \frac{1}{m_{n} - m_{0}} \\ = \sum_{\begin{array}{c} 0 \leq m_{0} \leq m_{1} \leq \dots \leq m_{n} \\ m_{0} \neq m_{n} \end{array}} \frac{(α)_{m_{0}} (β)_{m_{0}}}{m_{0}! (γ)_{m_{0}}} \frac{m_{n}! (γ)_{m_{n}}}{(α)_{m_{n}} (β)_{m_{n}}} \frac{1}{(m_{0} + γ)^{j + 1}} \\ \times {\prod_{i = 1}^{n - 1} \frac{1}{(m_{i} + α) (m_{i} + β) (m_{i} + γ)^{k_{i} - 2}}} \frac{1}{(m_{n} + α) (m_{n} + β) (m_{n} + γ)^{k_{n} - j - 3}} (\frac{1}{m_{n} - m_{0}} - \frac{1}{m_{n} + γ}) \end{aligned}$
$\begin{aligned} = \sum_{\begin{array}{c} 0 \leq m_{0} \leq m_{1} \leq \dots \leq m_{n} \\ m_{0} \neq m_{n} \end{array}} \frac{(α)_{m_{0}} (β)_{m_{0}}}{m_{0}! (γ)_{m_{0}}} \frac{m_{n}! (γ)_{m_{n}}}{(α)_{m_{n}} (β)_{m_{n}}} \frac{1}{(m_{0} + γ)^{j + 1}} \\ \times {\prod_{i = 1}^{n - 1} \frac{1}{(m_{i} + α) (m_{i} + β) (m_{i} + γ)^{k_{i} - 2}}} \frac{1}{(m_{n} + α) (m_{n} + β) (m_{n} + γ)^{k_{n} - j - 3}} \frac{1}{m_{n} - m_{0}} \\ - Z_{I}^{⋆} (j + 1, k_{1}, \dots, k_{n - 1}, k_{n} - j; (α, β, γ)) \\ + Z (n | n | k_{1} + \dots + k_{n} - 2 n + 1; (α, β, γ)) \end{aligned}$
$j = 0 \dots k_{n} - 3$ で和をとると,
$\begin{aligned} T^{⋆} (k_{1}, \dots, k_{n}; (α, β, γ)) \\ = \sum_{\begin{array}{c} m_{0} \leq m_{1} \leq \dots \leq m_{n} \\ m_{0} \neq m_{n} \end{array}} \frac{(α)_{m_{0}} (β)_{m_{0}}}{m_{0}! (γ)_{m_{0}}} \frac{m_{n}! (γ)_{m_{n}}}{(α)_{m_{n}} (β)_{m_{n}}} \frac{1}{(m_{0} + γ)^{k_{n} - 2}} \prod_{i = 1}^{n - 1} \frac{1}{(m_{i} + α) (m_{i} + β) (m_{i} + γ)^{k_{i} - 2}} \\ \times \frac{1}{(m_{n} + α) (m_{n} + β) (m_{n} - m_{0})} \\ - \sum_{j = 0}^{k_{n} - 3} Z_{I}^{⋆} (j + 1, k_{1}, \dots, k_{n - 1}, k_{n} - j; (α, β, γ)) \\ + (k_{n} - 2) Z (n | n | k_{1} + \dots + k_{n} - 2 n + 1; (α, β, γ)) \end{aligned}$
右辺の第一項を補題を用いて,
$\begin{aligned} \begin{matrix} \sum_{\begin{array}{c} 0 \leq m_{0} \leq m_{1} \leq \dots \leq m_{n} \\ m_{0} \neq m_{n} \end{array}} \frac{(α)_{m_{0}} (β)_{m_{0}}}{m_{0}! (γ)_{m_{0}}} \frac{1}{(m_{0} + γ)^{k_{n} - 2}} \frac{m_{n}! (γ)_{m_{n}}}{(α)_{m_{n}} (β)_{m_{n}}} \prod_{i = 1}^{n - 1} \frac{1}{(m_{i} + α) (m_{i} + β) (m_{i} + γ)^{k_{i} - 2}} \\ \times \frac{1}{(m_{n} + α) (m_{n} + β) (m_{n} - m_{0})} \end{matrix} \\ = \begin{matrix} \sum_{\begin{array}{c} m_{0} \leq m_{1} \leq \dots \leq m_{n - 1} \\ m_{0} \neq m_{n - 1} \end{array}} \frac{1}{(m_{0} + α) (m_{0} + β) (m_{0} + γ)^{k_{n} - 2}} \prod_{i = 1}^{n - 1} \frac{1}{(m_{i} + α) (m_{i} + β) (m_{i} + γ)^{k_{i} - 2}} \\ \times (\frac{(α)_{m_{0} + 1} (β)_{m_{0} + 1}}{m_{0}! (γ)_{m_{0}}} \sum_{m_{n - 1} \leq m_{n}} \frac{m_{n}! (γ)_{m_{n}}}{(α)_{m_{n} + 1} (β)_{m_{n} + 1} (m_{n} - m_{0})}) \\ + \sum_{\begin{array}{c} 0 \leq m_{0} = m_{1} = \dots = m_{n - 1} \end{array}} \frac{1}{(m_{0} + α) (m_{0} + β) (m_{0} + γ)^{k_{n} - 2}} \prod_{i = 1}^{n - 1} \frac{1}{(m_{i} + α) (m_{i} + β) (m_{i} + γ)^{k_{i} - 2}} \\ \times (\frac{(α)_{m_{0} + 1} (β)_{m_{0} + 1}}{m_{0}! (γ)_{m_{0}}} \sum_{m_{n - 1} < m_{n}} \frac{m_{n}! (γ)_{m_{n}}}{(α)_{m_{n} + 1} (β)_{m_{n} + 1} (m_{n} - m_{0})}) \end{matrix} \end{aligned}$
$\begin{aligned} = \begin{matrix} \sum_{\begin{array}{c} 0 \leq m_{0} \leq m_{1} \leq \dots \leq m_{n - 1} \\ m_{0} \neq m_{n - 1} \end{array}} \frac{1}{(m_{0} + α) (m_{0} + β) (m_{0} + γ)^{k_{n} - 2}} \prod_{i = 1}^{n - 1} \frac{1}{(m_{i} + α) (m_{i} + β) (m_{i} + γ)^{k_{i} - 2}} \\ \times (\frac{(γ)_{m_{n - 1}} m_{n - 1}!}{(α)_{m_{n - 1}} (β)_{m_{n - 1}}} \sum_{k = 0}^{m_{0}} \frac{(α)_{k} (β)_{k}}{k! (γ)_{k} (m_{n - 1} - m_{0})} \\ + (γ - α - β) (\sum_{k = 0}^{m_{0}} \frac{(α)_{k} (β)_{k}}{k! (γ)_{k}}) (\sum_{m_{n - 1} \leq m_{n}} \frac{(γ)_{m_{n}} m_{n}!}{(α)_{m_{n} + 1} (β)_{m_{n} + 1}})) \\ + \sum_{\begin{array}{c} 0 \leq m_{0} = m_{1} = \dots = m_{n - 1} \end{array}} \frac{1}{(m_{0} + α) (m_{0} + β) (m_{0} + γ)^{k_{n} - 2}} \prod_{i = 1}^{n - 1} \frac{1}{(m_{i} + α) (m_{i} + β) (m_{i} + γ)^{k_{i} - 2}} \\ \times (\frac{m_{n - 1}! (γ)_{m_{n - 1}}}{(α)_{m_{n - 1}} (β)_{m_{n - 1}}} \sum_{k = 0}^{m_{0} - 1} \frac{(α)_{k} (β)_{k}}{k! (γ)_{k} (m_{n - 1} - k)} \\ + \frac{m_{n - 1}! (γ)_{m_{n - 1}}}{(α)_{m_{n - 1}} (β)_{m_{n - 1} + 1}} \frac{(α)_{m_{0}} (β)_{m_{0}}}{m_{0}! (γ)_{m_{0}}} \\ + \frac{m_{n - 1}! (γ)_{m_{n - 1}}}{(α)_{m_{n - 1} + 1} (β)_{m_{n - 1} + 1}} \frac{(α)_{m_{0}} (β)_{m_{0} + 1}}{m_{0}! (γ)_{m_{0}}} \\ + (γ - α - β) (\sum_{k = 0}^{m_{0}} \frac{(α)_{k} (β)_{k}}{k! (γ)_{k}}) (\sum_{m_{n - 1} \leq m_{n}} \frac{m_{n}! (γ)_{m_{n}}}{(α)_{m_{n} + 1} (β)_{m_{n} + 1}})) \end{matrix} \end{aligned}$
$\begin{aligned} = \begin{matrix} T^{⋆} (k_{n}, k_{1}, \dots, k_{n - 1}; (α, β, γ)) \\ + (γ - α - β) \sum_{\begin{array}{c} 0 \leq k \leq m_{0} \leq \dots \leq m_{n} \\ m_{0} \neq m_{n - 1} \end{array}} \frac{(α)_{k} (β)_{k}}{k! (γ)_{k}} \frac{m_{n}! (γ)_{m_{n}}}{(α)_{m_{n}} (β)_{m_{n}}} \frac{1}{(m_{0} + α) (m_{0} + β) (m_{0} + γ)^{k_{n} - 2}} \\ \times \prod_{i = 0}^{n - 1} \frac{1}{(m_{i} + α) (m_{i} + β) (m_{i} + γ)^{k_{i} - 2}} \frac{1}{(m_{n} + α) (m_{n} + β)} \\ + Z (n | n - 1 | k_{1} + \dots + k_{n} - 2 n) + Z (n - 1 | n | k_{1} + \dots + k_{n} - 2 n) \\ + (γ - α - β) \sum_{\begin{array}{c} 0 \leq k \leq m_{0} = \dots = m_{n - 1} \leq m_{n} \end{array}} \frac{(α)_{k} (β)_{k}}{k! (γ)_{k}} \frac{m_{n}! (γ)_{m_{n}}}{(α)_{m_{n}} (β)_{m_{n}}} \frac{1}{(m_{0} + α) (m_{0} + β) (m_{0} + γ)^{k_{n} - 2}} \\ \times \prod_{i = 0}^{n - 1} \frac{1}{(m_{i} + α) (m_{i} + β) (m_{i} + γ)^{k_{i} - 2}} \frac{1}{(m_{n} + α) (m_{n} + β)} \end{matrix} \\ = \begin{matrix} T^{⋆} (k_{n}, k_{1}, \dots, k_{n - 1}; (α, β, γ)) \\ + (γ - α - β) \sum_{\begin{array}{c} 0 \leq k \leq m_{0} \leq \dots \leq m_{n} \end{array}} \frac{(α)_{k} (β)_{k}}{k! (γ)_{k}} \frac{m_{n}! (γ)_{m_{n}}}{(α)_{m_{n}} (β)_{m_{n}}} \frac{1}{(m_{0} + α) (m_{0} + β) (m_{0} + γ)^{k_{n} - 2}} \\ \times \prod_{i = 0}^{n - 1} \frac{1}{(m_{i} + α) (m_{i} + β) (m_{i} + γ)^{k_{i} - 2}} \frac{1}{(m_{n} + α) (m_{n} + β)} \\ + Z (n | n - 1 | k_{1} + \dots + k_{n} - 2 n) + Z (n - 1 | n | k_{1} + \dots + k_{n} - 2 n) \end{matrix} \end{aligned}$
となり示せた.