３変数付き多重級数の巡回和公式の発見について
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.
\begin{equation}
\zeta(s)\coloneqq\sum_{n=1}^\infty\frac{1}{n^s}.
\end{equation}
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.
$r$ を正整数とする.
$r$ 個の正整数の組 $(k_1,\ldots,k_r)$ 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.
\begin{equation}
\zeta(k_1,\ldots,k_r)\coloneqq\sum_{0< n_1<\cdots< n_r}\frac{1}{n_1^{k_1}\cdots n_r^{k_r}}.
\end{equation}
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.
$\mathbf{k}=(\{1\}^{a_1-1},b_1+1,\ldots,\{1\}^{a_r-1},b_r+1)$
$\mathbf{k}^\dagger=(\{1\}^{b_r-1},a_r+1,\ldots,\{1\}^{b_1-1},a_1+1)$としたとき,
\begin{equation}
\zeta(\mathbf{k})=\zeta(\mathbf{k}^\dagger)
\end{equation}
が成り立つ.
具体例として $\mathbf{k}=(3)$とすると, $\mathbf{k}^\dagger=(1,2)$ となるため $\zeta(3)=\zeta(1,2)$ 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.
\begin{align} \sum_{\substack{\mathbf{k}\in\alpha\\0\le i\le k_a-2}}\zeta(i+1,k_1,\ldots,k_{a-1},k_a-i)&=\sum_{\mathbf{k}\in\alpha}\zeta(k_1,\ldots,k_{a-1},k_a+1). \end{align}
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
\begin{equation}
Z(\mathbf{k};\alpha)\coloneqq\sum_{0\le m_1<\cdots< m_n}\frac{(\alpha)_{m_1}}{m_1!}\frac{m_n!}{(\alpha)_{m_n}}\prod_{i=1}^n\frac{1}{(m_i+\alpha)^{k_i}}
\end{equation}
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 $\alpha,\beta$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 $\alpha,\beta,\gamma$ 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
$\displaystyle Z_I^\star(\mathbf{k};\alpha,\beta,\gamma)\coloneqq\sum_{0\le m_1\le \cdots\le m_d}\frac{(\alpha)_{m_1}(\beta)_{m_1}}{(\gamma)_{m_1}m_1!}\frac{(\gamma)_{m_d}m_d!}{(\alpha)_{m_d}(\beta)_{m_d}}\frac{1}{(m_1+\gamma)^{k_1}}\prod_{i=2}^d\frac{1}{(m_i+\alpha)(m_i+\beta)(m_i+\gamma)^{k_i-2}}$
$\displaystyle Z_{I\hspace{-1.2pt}I}^\star(\mathbf{k};\alpha,\beta,\gamma)\coloneqq\sum_{0\le m_1\le \cdots\le m_d}\frac{(\alpha)_{m_1}(\beta)_{m_1}}{(\gamma)_{m_1}m_1!}\frac{(\gamma)_{m_d}m_d!}{(\alpha)_{m_d}(\beta)_{m_d}}\prod_{i=1}^d\frac{1}{(m_i+\alpha)(m_i+\beta)(m_i+\gamma)^{k_i-2}}$
$\displaystyle Z(a|b|c;\alpha,\beta,\gamma)\coloneqq\sum_{n=0}^\infty\frac{1}{(n+\alpha)^a(n+\beta)^b(n+\gamma)^c}$
とすると
\begin{align}
\sum_{i=1}^d\sum_{j=0}^{k_i-3}Z_I^\star(j+1,k_{i+1},\ldots,k_d,k_1,\ldots,k_{i-1},k_i-j;\alpha,\beta,\gamma)+(\alpha+\beta-\gamma)\sum_{i=1}^dZ_{I\hspace{-1.2pt}I}^\star(k_i,\ldots,k_d,k_1,\ldots,k_i,2;\alpha,\beta,\gamma)&=dZ(d|d+1|k-2d;\alpha,\beta,\gamma)+dZ(d+1|d|k-2d;\alpha,\beta,\gamma)+(k-2d)Z(d|d|k-2d+1;\alpha,\beta,\gamma)
\end{align}
が成り立つ.
証明
\begin{align*} \frac{(\alpha)_{m+1}(\beta)_{m+1}}{m!(\gamma)_{m}}\sum_{n\le l}\frac{l!(\gamma)_{l}}{(\alpha)_{l+1}(\beta)_{l+1}(l-m)}&= \begin{multlined}[t] \frac{n!(\gamma)_{n}}{(\alpha)_{n}(\beta)_{n}}\sum_{k=0}^{m}\frac{(\alpha)_{k}(\beta)_{k}}{k!(\gamma)_{k}(n-k)}\\+(\gamma-\alpha-\beta)\sum_{k=0}^{m}\frac{(\alpha)_{k}(\beta)_{k}}{k!(\gamma)_{k}}\sum_{n\le l}\frac{l!(\gamma)_{l}}{(\alpha)_{l+1}(\beta)_{l+1}} \end{multlined} \end{align*}
$\displaystyle S_m(\alpha,\beta,\gamma)
\coloneqq\sum_{n< l}\frac{(l-1)!(\gamma)_{l-1}}{(\alpha)_l(\beta)_l(l-m)}$とおくと, 次のように計算できる.
\begin{align*}
S_m(\alpha,\beta,\gamma)
&=\sum_{n< l}\frac{(l-1)!(\gamma)_{l-1}}{(\alpha)_l(\beta)_l(l-m)}\\
&=\frac{1}{\alpha+m-1}\sum_{n< l}\frac{(l-1)!(\gamma)_{l-1}}{(\alpha)_{l-1}(\beta)_l}\left(\frac{1}{l-m}-\frac{1}{\alpha+l-1}\right)\\
&=\begin{multlined}[t]\frac{1}{(\alpha+m-1)(\beta+m-1)}\sum_{n< l}\frac{(l-1)!(\gamma)_{l-1}}{(\alpha)_{l-1}(\beta)_{l-1}}\left(\frac{1}{l-m}-\frac{1}{\beta+l-1}\right)\\-\frac{1}{\alpha+m-1}\sum_{n< l}\frac{(l-1)!(\gamma)_{l-1}}{(\alpha)_{l}(\beta)_l}\end{multlined}\\
&=\begin{multlined}[t]\frac{1}{(\alpha+m-1)(\beta+m-1)}\sum_{n< l}\frac{(l-1)!(\gamma)_{l-1}}{(\alpha)_{l-1}(\beta)_{l-1}(l-m)}\\-\frac{1}{(\alpha+m-1)(\beta+m-1)}\sum_{n< l}\frac{(l-1)!(\gamma)_{l-1}}{(\alpha)_{l-1}(\beta)_{l}}\\-\frac{1}{(\alpha+m-1)}\sum_{n< l}\frac{(l-1)!(\gamma)_{l-1}}{(\alpha)_{l}(\beta)_l}\end{multlined}
\end{align*}
(横に長くなってしまったので折り曲げる)
\begin{align*}
&=\begin{multlined}[t]\frac{1}{(\alpha+m-1)(\beta+m-1)}\frac{n!(\gamma)_{n}}{(\alpha)_{n}(\beta)_{n}(n+1-m)}+\frac{1}{(\alpha+m-1)(\beta+m-1)}\sum_{n< l}\frac{l!(\gamma)_{l}}{(\alpha)_{l}(\beta)_{l}(l+1-m)}\\-\frac{1}{(\alpha+m-1)(\beta+m-1)}\sum_{n< l}\frac{(l-1)!(\gamma)_{l-1}}{(\alpha)_{l-1}(\beta)_{l}}-\frac{1}{(\alpha+m-1)}\sum_{n< l}\frac{(l-1)!(\gamma)_{l-1}}{(\alpha)_{l}(\beta)_l}\end{multlined}\\
&=\begin{multlined}[t]\frac{1}{(\alpha+m-1)(\beta+m-1)}\frac{n!(\gamma)_{n}}{(\alpha)_{n}(\beta)_{n}(n+1-m)}\\+\frac{m-1}{(\alpha+m-1)(\beta+m-1)}\sum_{n< l}\frac{(l-1)!(\gamma)_{l}}{(\alpha)_{l}(\beta)_{l}}\left(\frac{1}{l+1-m}+\frac{1}{m-1}\right)\\-\frac{1}{(\alpha+m-1)(\beta+m-1)}\sum_{n< l}\frac{(l-1)!(\gamma)_{l-1}}{(\alpha)_{l-1}(\beta)_{l}}-\frac{1}{(\alpha+m-1)}\sum_{n< l}\frac{(l-1)!(\gamma)_{l-1}}{(\alpha)_{l}(\beta)_l}\end{multlined}
\end{align*}
\begin{align*}
&=\begin{multlined}[t]\frac{1}{(\alpha+m-1)(\beta+m-1)}\frac{n!(\gamma)_{n}}{(\alpha)_{n}(\beta)_{n}(n+1-m)}+\frac{m-1}{(\alpha+m-1)(\beta+m-1)}\sum_{n< l}\frac{(l-1)!(\gamma)_{l}}{(\alpha)_{l}(\beta)_{l}(l+1-m)}+\\\frac{1}{(\alpha+m-1)(\beta+m-1)}\sum_{n< l}\frac{(l-1)!(\gamma)_{l}}{(\alpha)_{l}(\beta)_{l}}-\frac{1}{(\alpha+m-1)(\beta+m-1)}\sum_{n< l}\frac{(l-1)!(\gamma)_{l-1}}{(\alpha)_{l-1}(\beta)_{l}}\\-\frac{1}{(\alpha+m-1)}\sum_{n< l}\frac{(l-1)!(\gamma)_{l-1}}{(\alpha)_{l}(\beta)_l}\end{multlined}\\
&=\begin{multlined}[t]\frac{1}{(\alpha+m-1)(\beta+m-1)}\frac{n!(\gamma)_{n}}{(\alpha)_{n}(\beta)_{n}(n+1-m)}\\+\frac{(m-1)(\gamma+m-2)}{(\alpha+m-1)(\beta+m-1)}\sum_{n< l}\frac{(l-1)!(\gamma)_{l-1}}{(\alpha)_{l}(\beta)_{l}}\left(\frac{1}{l+1-m}+\frac{1}{\gamma+m-2}\right)\\+\frac{1}{(\alpha+m-1)(\beta+m-1)}\sum_{n< l}\frac{(l-1)!(\gamma)_{l}}{(\alpha)_{l}(\beta)_{l}}-\frac{1}{(\alpha+m-1)(\beta+m-1)}\sum_{n< l}\frac{(l-1)!(\gamma)_{l-1}}{(\alpha)_{l-1}(\beta)_{l}}\\-\frac{1}{(\alpha+m-1)}\sum_{n< l}\frac{(l-1)!(\gamma)_{l-1}}{(\alpha)_{l}(\beta)_l}\end{multlined}\\
&=\begin{multlined}[t]
\frac{1}{(\alpha+m-1)(\beta+m-1)}\frac{n!(\gamma)_{n}}{(\alpha)_{n}(\beta)_{n}(n+1-m)}+\frac{(m-1)(\gamma+m-2)}{(\alpha+m-1)(\beta+m-1)}S_{m-1}(\alpha,\beta,\gamma)\\+\frac{(m-1)}{(\alpha+m-1)(\beta+m-1)}\sum_{n< l}\frac{(l-1)!(\gamma)_{l-1}}{(\alpha)_{l}(\beta)_{l}}+\frac{1}{(\alpha+m-1)(\beta+m-1)}\sum_{n< l}\frac{(l-1)!(\gamma)_{l}}{(\alpha)_{l}(\beta)_{l}}\\-\frac{1}{(\alpha+m-1)(\beta+m-1)}\sum_{n< l}\frac{(l-1)!(\gamma)_{l-1}}{(\alpha)_{l-1}(\beta)_{l}}-\frac{1}{(\alpha+m-1)}\sum_{n< l}\frac{(l-1)!(\gamma)_{l-1}}{(\alpha)_{l}(\beta)_l}\end{multlined}.
\end{align*}
両辺に $\frac{(\alpha)_{m}(\beta)_{m}}{(m-1)!(\gamma)_{m-1}}$ を掛け $S_0(\alpha,\beta,\gamma)$ の項が $0$ になるため補題を得る.
\begin{align*} &\frac{(\alpha)_{m_0+1}(\beta)_{m_0+1}}{m_0!(\gamma)_{m_0}}\sum_{m_{n-1}< m_n}\frac{m_n!(\gamma)_{m_n}}{(\alpha)_{m_n+1}(\beta)_{m_n+1}(m_n-m_0)}\\ &=\begin{multlined}[t] \sum_{k=0}^{m_0-1}\frac{(\alpha)_{k}(\beta)_{k}}{k!(\gamma)_{k}}\frac{m_{n-1}!(\gamma)_{m_{n-1}}}{(\alpha)_{m_{n-1}}(\beta)_{m_{n-1}}(m_{n-1}-k)}\\+\frac{m_{n-1}!(\gamma)_{m_{n-1}}}{(\alpha)_{m_{n-1}}(\beta)_{m_{n-1}+1}}\frac{(\alpha)_{m_0}(\beta)_{m_0}}{m_0!(\gamma)_{m_0}}\\+\frac{m_{n-1}!(\gamma)_{m_{n-1}}}{(\alpha)_{m_{n-1}+1}(\beta)_{m_{n-1}+1}}\frac{(\alpha)_{m_0}(\beta)_{m_0+1}}{m_0!(\gamma)_{m_0}}\\+(\gamma-\alpha-\beta)\sum_{k=0}^{m_0}\frac{(\alpha)_{k}(\beta)_{k}}{k!(\gamma)_{k}}\sum_{m_{n-1}\le m_n}\frac{m_n!(\gamma)_{m_n}}{(\alpha)_{m_n+1}(\beta)_{m_n+1}} \end{multlined} \end{align*}
\begin{align*}
F_{m_0}(\alpha,\beta,\gamma)&=\frac{(\alpha)_{m_0+1}(\beta)_{m_0+1}}{m_0!(\gamma)_{m_0}}\sum_{m_{n-1}< m_n}\frac{m_n!(\gamma)_{m_n}}{(\alpha)_{m_n+1}(\beta)_{m_n+1}(m_n-m_0)}\\
&=\frac{(\alpha)_{m_0+1}(\beta)_{m_0+1}}{m_0!(\gamma)_{m_0}}\sum_{m_{n-1}\le m_n}\frac{(m_n+1)!(\gamma)_{m_n+1}}{(\alpha)_{m_n+2}(\beta)_{m_n+2}(m_n+1-m_0)}\\
&=\frac{(\alpha)_{m_0}(\beta)_{m_0+1}}{m_0!(\gamma)_{m_0}}\sum_{m_{
n-1}\le m_n}\frac{(m_n+1)!(\gamma)_{m_n+1}}{(\alpha)_{m_n+1}(\beta)_{m_n+2}}\left(\frac{1}{m_n+1-m_0}-\frac{1}{\alpha+m_n+1}\right)\\
&=\frac{(\alpha)_{m_0}(\beta)_{m_0+1}}{m_0!(\gamma)_{m_0}}\sum_{m_{n-1}\le m_n}\frac{(m_n+1)!(\gamma)_{m_n+1}}{(\alpha)_{m_n+1}(\beta)_{m_n+2}(m_n+1-m_0)}-\frac{(\alpha)_{m_0}(\beta)_{m_0+1}}{m_0!(\gamma)_{m_0}}\sum_{m_{n-1}\le m_n}\frac{(m_n+1)!(\gamma)_{m_n+1}}{(\alpha)_{m_n+2}(\beta)_{m_n+2}}\\
&=\begin{multlined}[t]\frac{(\alpha)_{m_0}(\beta)_{m_0}}{m_0!(\gamma)_{m_0}}\sum_{m_{n-1}\le m_n}\frac{(m_n+1)!(\gamma)_{m_n+1}}{(\alpha)_{m_n+1}(\beta)_{m_n+1}}\left(\frac{1}{m_n+1-m_0}-\frac{1}{\beta+m_n+1}\right)\\-\frac{(\alpha)_{m_0}(\beta)_{m_0+1}}{m_0!(\gamma)_{m_0}}\sum_{m_{n-1}\le m_n}\frac{(m_n+1)!(\gamma)_{m_n+1}}{(\alpha)_{m_n+2}(\beta)_{m_n+2}}\end{multlined}\\
&=\begin{multlined}[t]\frac{(\alpha)_{m_0}(\beta)_{m_0}}{m_0!(\gamma)_{m_0}}\sum_{m_{n-1}\le m_n}\frac{(m_n+1)!(\gamma)_{m_n+1}}{(\alpha)_{m_n+1}(\beta)_{m_n+1}(m_n+1-m_0)}-\frac{(\alpha)_{m_0}(\beta)_{m_0}}{m_0!(\gamma)_{m_0}}\sum_{m_{n-1}\le m_n}\frac{(m_n+1)!(\gamma)_{m_n+1}}{(\alpha)_{m_n+1}(\beta)_{m_n+2}}\\-\frac{(\alpha)_{m_0}(\beta)_{m_0+1}}{m_0!(\gamma)_{m_0}}\sum_{m_{n-1}\le m_n}\frac{(m_n+1)!(\gamma)_{m_n+1}}{(\alpha)_{m_n+2}(\beta)_{m_n+2}}\end{multlined}\\
&=\begin{multlined}[t]\frac{(\alpha)_{m_0}(\beta)_{m_0}}{(m_0-1)!(\gamma)_{m_0}}\sum_{m_{n-1}\le m_n}\frac{m_n!(\gamma)_{m_n+1}}{(\alpha)_{m_n+1}(\beta)_{m_n+1}}\left(\frac{1}{m_n+1-m_0}+\frac{1}{m_0}\right)\\-\frac{(\alpha)_{m_0}(\beta)_{m_0}}{m_0!(\gamma)_{m_0}}\sum_{m_{n-1}\le m_n}\frac{(m_n+1)!(\gamma)_{m_n+1}}{(\alpha)_{m_n+1}(\beta)_{m_n+2}}-\frac{(\alpha)_{m_0}(\beta)_{m_0+1}}{m_0!(\gamma)_{m_0}}\sum_{m_{n-1}\le m_n}\frac{(m_n+1)!(\gamma)_{m_n+1}}{(\alpha)_{m_n+2}(\beta)_{m_n+2}}\end{multlined}\\
&=\begin{multlined}[t]\frac{(\alpha)_{m_0}(\beta)_{m_0}}{(m_0-1)!(\gamma)_{m_0}}\sum_{m_{n-1}\le m_n}\frac{m_n!(\gamma)_{m_n+1}}{(\alpha)_{m_n+1}(\beta)_{m_n+1}(m_n+1-m_0)}\\+\frac{(\alpha)_{m_0}(\beta)_{m_0}}{m_0!(\gamma)_{m_0}}\sum_{m_{n-1}\le m_n}\frac{m_n!(\gamma)_{m_n+1}}{(\alpha)_{m_n+1}(\beta)_{m_n+1}}
-\frac{(\alpha)_{m_0}(\beta)_{m_0}}{m_0!(\gamma)_{m_0}}\sum_{m_{n-1}\le m_n}\frac{(m_n+1)!(\gamma)_{m_n+1}}{(\alpha)_{m_n+1}(\beta)_{m_n+2}}\\-\frac{(\alpha)_{m_0}(\beta)_{m_0+1}}{m_0!(\gamma)_{m_0}}\sum_{m_{n-1}\le m_n}\frac{(m_n+1)!(\gamma)_{m_n+1}}{(\alpha)_{m_n+2}(\beta)_{m_n+2}}\end{multlined}\\
&=\begin{multlined}[t]\frac{(\alpha)_{m_0}(\beta)_{m_0}}{(m_0-1)!(\gamma)_{m_0-1}}\sum_{m_{n-1}\le m_n}\frac{m_n!(\gamma)_{m_n}}{(\alpha)_{m_n+1}(\beta)_{m_n+1}}\left(\frac{1}{m_n+1-m_0}+\frac{1}{\gamma+m_0-1}\right)\\+\frac{(\alpha)_{m_0}(\beta)_{m_0}}{m_0!(\gamma)_{m_0}}\sum_{m_{n-1}\le m_n}\frac{m_n!(\gamma)_{m_n+1}}{(\alpha)_{m_n+1}(\beta)_{m_n+1}}
-\frac{(\alpha)_{m_0}(\beta)_{m_0}}{m_0!(\gamma)_{m_0}}\sum_{m_{n-1}\le m_n}\frac{(m_n+1)!(\gamma)_{m_n+1}}{(\alpha)_{m_n+1}(\beta)_{m_n+2}}\\-\frac{(\alpha)_{m_0}(\beta)_{m_0+1}}{m_0!(\gamma)_{m_0}}\sum_{m_{n-1}\le m_n}\frac{(m_n+1)!(\gamma)_{m_n+1}}{(\alpha)_{m_n+2}(\beta)_{m_n+2}}\end{multlined}\\
&=\begin{multlined}[t]\frac{(\alpha)_{m_0}(\beta)_{m_0}}{(m_0-1)!(\gamma)_{m_0-1}}\sum_{m_{n-1}\le m_n}\frac{m_n!(\gamma)_{m_n}}{(\alpha)_{m_n+1}(\beta)_{m_n+1}(m_n+1-m_0)}\\+\frac{(\alpha)_{m_0}(\beta)_{m_0}}{(m_0-1)!(\gamma)_{m_0}}\sum_{m_{n-1}\le m_n}\frac{m_n!(\gamma)_{m_n}}{(\alpha)_{m_n+1}(\beta)_{m_n+1}}+\frac{(\alpha)_{m_0}(\beta)_{m_0}}{m_0!(\gamma)_{m_0}}\sum_{m_{n-1}\le m_n}\frac{m_n!(\gamma)_{m_n+1}}{(\alpha)_{m_n+1}(\beta)_{m_n+1}}
\\-\frac{(\alpha)_{m_0}(\beta)_{m_0}}{m_0!(\gamma)_{m_0}}\sum_{m_{n-1}\le m_n}\frac{(m_n+1)!(\gamma)_{m_n+1}}{(\alpha)_{m_n+1}(\beta)_{m_n+2}}-\frac{(\alpha)_{m_0}(\beta)_{m_0+1}}{m_0!(\gamma)_{m_0}}\sum_{m_{n-1}\le m_n}\frac{(m_n+1)!(\gamma)_{m_n+1}}{(\alpha)_{m_n+2}(\beta)_{m_n+2}}\end{multlined}\\
&=\begin{multlined}[t]F_{m_0-1}(\alpha,\beta,\gamma)+\frac{(\alpha)_{m_0}(\beta)_{m_0}}{(m_0-1)!(\gamma)_{m_0-1}}\frac{m_{n-1}!(\gamma)_{m_{n-1}}}{(\alpha)_{m_{n-1}+1}(\beta)_{m_{n-1}+1}(m_{n-1}+1-m_0)}\\+\frac{(\alpha)_{m_0}(\beta)_{m_0}}{(m_0-1)!(\gamma)_{m_0}}\sum_{m_{n-1}\le m_n}\frac{m_n!(\gamma)_{m_n}}{(\alpha)_{m_n+1}(\beta)_{m_n+1}}+\frac{(\alpha)_{m_0}(\beta)_{m_0}}{m_0!(\gamma)_{m_0}}\sum_{m_{n-1}\le m_n}\frac{m_n!(\gamma)_{m_n+1}}{(\alpha)_{m_n+1}(\beta)_{m_n+1}}\\
-\frac{(\alpha)_{m_0}(\beta)_{m_0}}{m_0!(\gamma)_{m_0}}\sum_{m_{n-1}< m_n}\frac{m_n!(\gamma)_{m_n}}{(\alpha)_{m_n}(\beta)_{m_n+1}}-\frac{(\alpha)_{m_0}(\beta)_{m_0+1}}{m_0!(\gamma)_{m_0}}\sum_{m_{n-1}< m_n}\frac{m_n!(\gamma)_{m_n}}{(\alpha)_{m_n+1}(\beta)_{m_n+1}}\end{multlined}
\end{align*}
今度は$m_0=0$から和をとると,
\begin{align*}
&\frac{(\alpha)_{m_0+1}(\beta)_{m_0+1}}{m_0!(\gamma)_{m_0}}\sum_{m_{n-1}< m_n}\frac{m_n!(\gamma)_{m_n}}{(\alpha)_{m_n+1}(\beta)_{m_n+1}(m_n-m_0)}\\&=
\begin{multlined}[t]
\sum_{k=0}^{m_0-1}\frac{(\alpha)_{k+1}(\beta)_{k+1}}{k!(\gamma)_{k}}\frac{m_{n-1}!(\gamma)_{m_{n-1}}}{(\alpha)_{m_{n-1}+1}(\beta)_{m_{n-1}+1}(m_{n-1}-k)}\\+\sum_{k=0}^{m_0-1}\frac{(\alpha)_{k+1}(\beta)_{k+1}}{k!(\gamma)_{k+1}}\sum_{m_{n-1}\le m_n}\frac{m_n!(\gamma)_{m_n}}{(\alpha)_{m_n+1}(\beta)_{m_n+1}}+\sum_{k=0}^{m_0}\frac{(\alpha)_{k}(\beta)_{k}}{k!(\gamma)_{k}}\sum_{m_{n-1}\le m_n}\frac{m_n!(\gamma)_{m_n+1}}{(\alpha)_{m_n+1}(\beta)_{m_n+1}}\\
-\sum_{k=0}^{m_0}\frac{(\alpha)_{k}(\beta)_{k}}{k!(\gamma)_{k}}\sum_{m_{n-1}< m_n}\frac{m_n!(\gamma)_{m_n}}{(\alpha)_{m_n}(\beta)_{m_n+1}}-\sum_{k=0}^{m_0}\frac{(\alpha)_{k}(\beta)_{k+1}}{k!(\gamma)_{k}}\sum_{m_{n-1}< m_n}\frac{m_n!(\gamma)_{m_n}}{(\alpha)_{m_n+1}(\beta)_{m_n+1}}\\
\end{multlined}\\
&=\begin{multlined}[t]
\sum_{k=0}^{m_0-1}\frac{(\alpha)_{k}(\beta)_{k+1}}{k!(\gamma)_{k}}\frac{m_{n-1}!(\gamma)_{m_{n-1}}}{(\alpha)_{m_{n-1}}(\beta)_{m_{n-1}+1}}\left(\frac{1}{m_{n-1}-k}-\frac{1}{\alpha+m_{n-1}}\right)\\+\sum_{k=0}^{m_0-1}\frac{(\alpha)_{k+1}(\beta)_{k+1}}{k!(\gamma)_{k+1}}\sum_{m_{n-1}\le m_n}\frac{m_n!(\gamma)_{m_n}}{(\alpha)_{m_n+1}(\beta)_{m_n+1}}+\sum_{k=0}^{m_0}\frac{(\alpha)_{k}(\beta)_{k}}{k!(\gamma)_{k}}\sum_{m_{n-1}\le m_n}\frac{m_n!(\gamma)_{m_n+1}}{(\alpha)_{m_n+1}(\beta)_{m_n+1}}\\
-\sum_{k=0}^{m_0}\frac{(\alpha)_{k}(\beta)_{k}}{k!(\gamma)_{k}}\sum_{m_{n-1}< m_n}\frac{m_n!(\gamma)_{m_n}}{(\alpha)_{m_n}(\beta)_{m_n+1}}-\sum_{k=0}^{m_0}\frac{(\alpha)_{k}(\beta)_{k+1}}{k!(\gamma)_{k}}\sum_{m_{n-1}< m_n}\frac{m_n!(\gamma)_{m_n}}{(\alpha)_{m_n+1}(\beta)_{m_n+1}}\\
\end{multlined}\\
&=\begin{multlined}[t]
\sum_{k=0}^{m_0-1}\frac{(\alpha)_{k}(\beta)_{k}}{k!(\gamma)_{k}}\frac{m_{n-1}!(\gamma)_{m_{n-1}}}{(\alpha)_{m_{n-1}}(\beta)_{m_{n-1}}}\left(\frac{1}{m_{n-1}-k}-\frac{1}{\beta+m_{n-1}}\right)\\-\frac{m_{n-1}!(\gamma)_{m_{n-1}}}{(\alpha)_{m_{n-1}+1}(\beta)_{m_{n-1}+1}}\sum_{k=0}^{m_0-1}\frac{(\alpha)_k(\beta)_{k+1}}{k!(\gamma)_k}\\+\sum_{k=0}^{m_0-1}\frac{(\alpha)_{k+1}(\beta)_{k+1}}{k!(\gamma)_{k+1}}\sum_{m_{n-1}\le m_n}\frac{m_n!(\gamma)_{m_n}}{(\alpha)_{m_n+1}(\beta)_{m_n+1}}+\sum_{k=0}^{m_0}\frac{(\alpha)_{k}(\beta)_{k}}{k!(\gamma)_{k}}\sum_{m_{n-1}\le m_n}\frac{m_n!(\gamma)_{m_n+1}}{(\alpha)_{m_n+1}(\beta)_{m_n+1}}\\
-\sum_{k=0}^{m_0}\frac{(\alpha)_{k}(\beta)_{k}}{k!(\gamma)_{k}}\sum_{m_{n-1}< m_n}\frac{m_n!(\gamma)_{m_n}}{(\alpha)_{m_n}(\beta)_{m_n+1}}-\sum_{k=0}^{m_0}\frac{(\alpha)_{k}(\beta)_{k+1}}{k!(\gamma)_{k}}\sum_{m_{n-1}< m_n}\frac{m_n!(\gamma)_{m_n}}{(\alpha)_{m_n+1}(\beta)_{m_n+1}}
\end{multlined}
\end{align*}
\begin{align*}
&=\begin{multlined}[t]
\sum_{k=0}^{m_0-1}\frac{(\alpha)_{k}(\beta)_{k}}{k!(\gamma)_{k}}\frac{m_{n-1}!(\gamma)_{m_{n-1}}}{(\alpha)_{m_{n-1}}(\beta)_{m_{n-1}}(m_{n-1}-k)}\\-\frac{m_{n-1}!(\gamma)_{m_{n-1}}}{(\alpha)_{m_{n-1}}(\beta)_{m_{n-1}+1}}\sum_{k=0}^{m_0-1}\frac{(\alpha)_k(\beta)_{k}}{k!(\gamma)_k}\\-\frac{m_{n-1}!(\gamma)_{m_{n-1}}}{(\alpha)_{m_{n-1}+1}(\beta)_{m_{n-1}+1}}\sum_{k=0}^{m_0-1}\frac{(\alpha)_k(\beta)_{k+1}}{k!(\gamma)_k}\\+\sum_{k=0}^{m_0-1}\frac{(\alpha)_{k+1}(\beta)_{k+1}}{k!(\gamma)_{k+1}}\sum_{m_{n-1}\le m_n}\frac{m_n!(\gamma)_{m_n}}{(\alpha)_{m_n+1}(\beta)_{m_n+1}}+\sum_{k=0}^{m_0}\frac{(\alpha)_{k}(\beta)_{k}}{k!(\gamma)_{k}}\sum_{m_{n-1}\le m_n}\frac{m_n!(\gamma)_{m_n+1}}{(\alpha)_{m_n+1}(\beta)_{m_n+1}}\\
-\sum_{k=0}^{m_0}\frac{(\alpha)_{k}(\beta)_{k}}{k!(\gamma)_{k}}\sum_{m_{n-1}< m_n}\frac{m_n!(\gamma)_{m_n}}{(\alpha)_{m_n}(\beta)_{m_n+1}}-\sum_{k=0}^{m_0}\frac{(\alpha)_{k}(\beta)_{k+1}}{k!(\gamma)_{k}}\sum_{m_{n-1}< m_n}\frac{m_n!(\gamma)_{m_n}}{(\alpha)_{m_n+1}(\beta)_{m_n+1}}\end{multlined}\\
&=\begin{multlined}[t]
\sum_{k=0}^{m_0-1}\frac{(\alpha)_{k}(\beta)_{k}}{k!(\gamma)_{k}}\frac{m_{n-1}!(\gamma)_{m_{n-1}}}{(\alpha)_{m_{n-1}}(\beta)_{m_{n-1}}(m_{n-1}-k)}\\-\frac{m_{n-1}!(\gamma)_{m_{n-1}}}{(\alpha)_{m_{n-1}}(\beta)_{m_{n-1}+1}}\sum_{k=0}^{m_0}\frac{(\alpha)_k(\beta)_{k}}{k!(\gamma)_k}+\frac{m_{n-1}!(\gamma)_{m_{n-1}}}{(\alpha)_{m_{n-1}}(\beta)_{m_{n-1}+1}}\frac{(\alpha)_{m_0}(\beta)_{m_0}}{m_0!(\gamma)_{m_0}}\\-\frac{m_{n-1}!(\gamma)_{m_{n-1}}}{(\alpha)_{m_{n-1}+1}(\beta)_{m_{n-1}+1}}\sum_{k=0}^{m_0}\frac{(\alpha)_k(\beta)_{k+1}}{k!(\gamma)_k}+\frac{m_{n-1}!(\gamma)_{m_{n-1}}}{(\alpha)_{m_{n-1}+1}(\beta)_{m_{n-1}+1}}\frac{(\alpha)_{m_0}(\beta)_{m_0+1}}{m_0!(\gamma)_{m_0}}\\+\sum_{k=0}^{m_0-1}\frac{(\alpha)_{k+1}(\beta)_{k+1}}{k!(\gamma)_{k+1}}\sum_{m_{n-1}\le m_n}\frac{m_n!(\gamma)_{m_n}}{(\alpha)_{m_n+1}(\beta)_{m_n+1}}+\sum_{k=0}^{m_0}\frac{(\alpha)_{k}(\beta)_{k}}{k!(\gamma)_{k}}\sum_{m_{n-1}\le m_n}\frac{m_n!(\gamma)_{m_n+1}}{(\alpha)_{m_n+1}(\beta)_{m_n+1}}\\
-\sum_{k=0}^{m_0}\frac{(\alpha)_{k}(\beta)_{k}}{k!(\gamma)_{k}}\sum_{m_{n-1}< m_n}\frac{m_n!(\gamma)_{m_n}}{(\alpha)_{m_n}(\beta)_{m_n+1}}-\sum_{k=0}^{m_0}\frac{(\alpha)_{k}(\beta)_{k+1}}{k!(\gamma)_{k}}\sum_{m_{n-1}< m_n}\frac{m_n!(\gamma)_{m_n}}{(\alpha)_{m_n+1}(\beta)_{m_n+1}}
\end{multlined}
\\&=\begin{multlined}[t]
\sum_{k=0}^{m_0-1}\frac{(\alpha)_{k}(\beta)_{k}}{k!(\gamma)_{k}}\frac{m_{n-1}!(\gamma)_{m_{n-1}}}{(\alpha)_{m_{n-1}}(\beta)_{m_{n-1}}(m_{n-1}-k)}\\+\frac{m_{n-1}!(\gamma)_{m_{n-1}}}{(\alpha)_{m_{n-1}}(\beta)_{m_{n-1}+1}}\frac{(\alpha)_{m_0}(\beta)_{m_0}}{m_0!(\gamma)_{m_0}}\\+\frac{m_{n-1}!(\gamma)_{m_{n-1}}}{(\alpha)_{m_{n-1}+1}(\beta)_{m_{n-1}+1}}\frac{(\alpha)_{m_0}(\beta)_{m_0+1}}{m_0!(\gamma)_{m_0}}\\+\sum_{k=0}^{m_0}\frac{(\alpha)_{k}(\beta)_{k}}{(k-1)!(\gamma)_{k}}\sum_{m_{n-1}\le m_n}\frac{m_n!(\gamma)_{m_n}}{(\alpha)_{m_n+1}(\beta)_{m_n+1}}+\sum_{k=0}^{m_0}\frac{(\alpha)_{k}(\beta)_{k}}{k!(\gamma)_{k}}\sum_{m_{n-1}\le m_n}\frac{m_n!(\gamma)_{m_n+1}}{(\alpha)_{m_n+1}(\beta)_{m_n+1}}\\
-\sum_{k=0}^{m_0}\frac{(\alpha)_{k}(\beta)_{k}}{k!(\gamma)_{k}}\sum_{m_{n-1}\le m_n}\frac{m_n!(\gamma)_{m_n}}{(\alpha)_{m_n}(\beta)_{m_n+1}}-\sum_{k=0}^{m_0}\frac{(\alpha)_{k}(\beta)_{k+1}}{k!(\gamma)_{k}}\sum_{m_{n-1}\le m_n}\frac{m_n!(\gamma)_{m_n}}{(\alpha)_{m_n+1}(\beta)_{m_n+1}}\\
\end{multlined}
\end{align*}
\begin{align*}
&=\begin{multlined}[t]
\sum_{k=0}^{m_0-1}\frac{(\alpha)_{k}(\beta)_{k}}{k!(\gamma)_{k}}\frac{m_{n-1}!(\gamma)_{m_{n-1}}}{(\alpha)_{m_{n-1}}(\beta)_{m_{n-1}}(m_{n-1}-k)}\\+\frac{m_{n-1}!(\gamma)_{m_{n-1}}}{(\alpha)_{m_{n-1}}(\beta)_{m_{n-1}+1}}\frac{(\alpha)_{m_0}(\beta)_{m_0}}{m_0!(\gamma)_{m_0}}\\+\frac{m_{n-1}!(\gamma)_{m_{n-1}}}{(\alpha)_{m_{n-1}+1}(\beta)_{m_{n-1}+1}}\frac{(\alpha)_{m_0}(\beta)_{m_0+1}}{m_0!(\gamma)_{m_0}}\\+(\gamma-\alpha-\beta)\sum_{k=0}^{m_0}\frac{(\alpha)_{k}(\beta)_{k}}{k!(\gamma)_{k}}\sum_{m_{n-1}\le m_n}\frac{m_n!(\gamma)_{m_n}}{(\alpha)_{m_n+1}(\beta)_{m_n+1}}
\end{multlined}
\end{align*}
以上より示せた.
\begin{align*} &T^\star(k_1,\ldots,k_n;(\alpha,\beta,\gamma)) \\&=\begin{multlined}[t] T^\star(k_n,k_1,\ldots,k_{n-1};(\alpha,\beta,\gamma))\\ +(\gamma-\alpha-\beta)Z_{I\hspace{-1.2pt}I}^\star(k_n,k_1,\ldots,k_{n-1},2;(\alpha,\beta,\gamma))\\ +Z(n|n-1|k_1+\cdots+k_n-2n)+Z(n-1|n|k_1+\cdots+k_n-2n) \\-\sum_{j=0}^{k_n-3}Z_I^\star(j+1,k_1,\ldots,k_{n-1},k_n-j;(\alpha,\beta,\gamma)) \\+(k_n-2)Z(n|n|k_1+\cdots+k_n-2n+1;(\alpha,\beta,\gamma)) \end{multlined} \end{align*}
\begin{align*}
&\sum_{\substack{0\le m_0\le m_1\le \cdots\le m_n\\m_0\neq m_n}}\frac{(\alpha)_{m_0}(\beta)_{m_0}}{m_0!(\gamma)_{m_0}}\frac{m_n!(\gamma)_{m_n}}{(\alpha)_{m_n}(\beta)_{m_n}}\frac{1}{(m_0+\gamma)^j}\\&\times\left\{\prod_{i=1}^{n-1}\frac{1}{(m_i+\alpha)(m_i+\beta)(m_i+\gamma)^{k_i-2}}\right\}\frac{1}{(m_n+\alpha)(m_n+\beta)(m_n+\gamma)^{k_n-j-2}}\frac{1}{m_n-m_0}\\
&=\sum_{\substack{0\le m_0\le m_1\le \cdots\le m_n\\m_0\neq m_n}}\frac{(\alpha)_{m_0}(\beta)_{m_0}}{m_0!(\gamma)_{m_0}}\frac{m_n!(\gamma)_{m_n}}{(\alpha)_{m_n}(\beta)_{m_n}}\frac{1}{(m_0+\gamma)^{j+1}}\\&\times\left\{\prod_{i=1}^{n-1}\frac{1}{(m_i+\alpha)(m_i+\beta)(m_i+\gamma)^{k_i-2}}\right\}\frac{1}{(m_n+\alpha)(m_n+\beta)(m_n+\gamma)^{k_n-j-3}}\left(\frac{1}{m_n-m_0}-\frac{1}{m_n+\gamma}\right)
\end{align*}
\begin{align*}
&=\sum_{\substack{0\le m_0\le m_1\le \cdots\le m_n\\m_0\neq m_n}}\frac{(\alpha)_{m_0}(\beta)_{m_0}}{m_0!(\gamma)_{m_0}}\frac{m_n!(\gamma)_{m_n}}{(\alpha)_{m_n}(\beta)_{m_n}}\frac{1}{(m_0+\gamma)^{j+1}}\\&\times\left\{\prod_{i=1}^{n-1}\frac{1}{(m_i+\alpha)(m_i+\beta)(m_i+\gamma)^{k_i-2}}\right\}\frac{1}{(m_n+\alpha)(m_n+\beta)(m_n+\gamma)^{k_n-j-3}}\frac{1}{m_n-m_0}\\
&-Z_I^\star(j+1,k_1,\ldots,k_{n-1},k_n-j;(\alpha,\beta,\gamma))\\
&+Z(n|n|k_1+\cdots+k_n-2n+1;(\alpha,\beta,\gamma))
\end{align*}
$j=0\ldots k_n-3$ で和をとると,
\begin{align*}
&T^\star(k_1,\ldots,k_n;(\alpha,\beta,\gamma))\\
&=\sum_{\substack{m_0\le m_1\le \cdots\le m_n\\m_0\neq m_n}}\frac{(\alpha)_{m_0}(\beta)_{m_0}}{m_0!(\gamma)_{m_0}}\frac{m_n!(\gamma)_{m_n}}{(\alpha)_{m_n}(\beta)_{m_n}}\frac{1}{(m_0+\gamma)^{k_n-2}}\prod_{i=1}^{n-1}\frac{1}{(m_i+\alpha)(m_i+\beta)(m_i+\gamma)^{k_i-2}}\\
&\times\frac{1}{(m_n+\alpha)(m_n+\beta)(m_n-m_0)}\\
&-\sum_{j=0}^{k_n-3}Z_I^\star(j+1,k_1,\ldots,k_{n-1},k_n-j;(\alpha,\beta,\gamma))\\
&+(k_n-2)Z(n|n|k_1+\cdots+k_n-2n+1;(\alpha,\beta,\gamma))
\end{align*}
右辺の第一項を補題を用いて,
\begin{align*}
&\begin{multlined}[t]\sum_{\substack{0\le m_0\le m_1\le \cdots\le m_n\\m_0\neq m_n}}\frac{(\alpha)_{m_0}(\beta)_{m_0}}{m_0!(\gamma)_{m_0}}\frac{1}{(m_0+\gamma)^{k_n-2}}\frac{m_n!(\gamma)_{m_n}}{(\alpha)_{m_n}(\beta)_{m_n}}\prod_{i=1}^{n-1}\frac{1}{(m_i+\alpha)(m_i+\beta)(m_i+\gamma)^{k_i-2}}\\\times\frac{1}{(m_n+\alpha)(m_n+\beta)(m_n-m_0)}\end{multlined}\\
&=\begin{multlined}[t]\sum_{\substack{m_0\le m_1\le \cdots\le m_{n-1}\\m_0\neq m_{n-1}}}\frac{1}{(m_0+\alpha)(m_0+\beta)(m_0+\gamma)^{k_n-2}}\prod_{i=1}^{n-1}\frac{1}{(m_i+\alpha)(m_i+\beta)(m_i+\gamma)^{k_i-2}}\\\times\left(\frac{(\alpha)_{m_0+1}(\beta)_{m_0+1}}{m_0!(\gamma)_{m_0}}\sum_{m_{n-1}\le m_n}\frac{m_n!(\gamma)_{m_n}}{(\alpha)_{m_n+1}(\beta)_{m_n+1}(m_n-m_0)}\right)\\
\\+\sum_{\substack{0\le m_0= m_1=\cdots= m_{n-1}}}\frac{1}{(m_0+\alpha)(m_0+\beta)(m_0+\gamma)^{k_n-2}}\prod_{i=1}^{n-1}\frac{1}{(m_i+\alpha)(m_i+\beta)(m_i+\gamma)^{k_i-2}}\\\times\left(\frac{(\alpha)_{m_0+1}(\beta)_{m_0+1}}{m_0!(\gamma)_{m_0}}\sum_{m_{n-1}< m_n}\frac{m_n!(\gamma)_{m_n}}{(\alpha)_{m_n+1}(\beta)_{m_n+1}(m_n-m_0)}\right)\end{multlined}
\end{align*}
\begin{align*}
&=\begin{multlined}[t]\sum_{\substack{0\le m_0\le m_1\le \cdots\le m_{n-1}\\m_0\neq m_{n-1}}}\frac{1}{(m_0+\alpha)(m_0+\beta)(m_0+\gamma)^{k_n-2}}\prod_{i=1}^{n-1}\frac{1}{(m_i+\alpha)(m_i+\beta)(m_i+\gamma)^{k_i-2}}\\\times\left(\frac{(\gamma)_{m_{n-1}}m_{n-1}!}{(\alpha)_{m_{n-1}}(\beta)_{m_{n-1}}}\sum_{k=0}^{m_0}\frac{(\alpha)_k(\beta)_k}{k!(\gamma)_k(m_{n-1}-m_0)}\right.\\\left.+(\gamma-\alpha-\beta)\left(\sum_{k=0}^{m_0}\frac{(\alpha)_k(\beta)_k}{k!(\gamma)_k}\right)\left(\sum_{m_{n-1}\le m_n}\frac{(\gamma)_{m_n}m_n!}{(\alpha)_{m_n+1}(\beta)_{m_n+1}}\right)\right)\\+\sum_{\substack{0\le m_0= m_1=\cdots= m_{n-1}}}\frac{1}{(m_0+\alpha)(m_0+\beta)(m_0+\gamma)^{k_n-2}}\prod_{i=1}^{n-1}\frac{1}{(m_i+\alpha)(m_i+\beta)(m_i+\gamma)^{k_i-2}}\\\times\left(\frac{m_{n-1}!(\gamma)_{m_{n-1}}}{(\alpha)_{m_{n-1}}(\beta)_{m_{n-1}}}
\sum_{k=0}^{m_0-1}\frac{(\alpha)_{k}(\beta)_{k}}{k!(\gamma)_{k}(m_{n-1}-k)}\right.\\\left.+\frac{m_{n-1}!(\gamma)_{m_{n-1}}}{(\alpha)_{m_{n-1}}(\beta)_{m_{n-1}+1}}\frac{(\alpha)_{m_0}(\beta)_{m_0}}{m_0!(\gamma)_{m_0}}\right.\\\left.+\frac{m_{n-1}!(\gamma)_{m_{n-1}}}{(\alpha)_{m_{n-1}+1}(\beta)_{m_{n-1}+1}}\frac{(\alpha)_{m_0}(\beta)_{m_0+1}}{m_0!(\gamma)_{m_0}}\right.\\\left.+(\gamma-\alpha-\beta)\left(\sum_{k=0}^{m_0}\frac{(\alpha)_{k}(\beta)_{k}}{k!(\gamma)_{k}}\right)\left(\sum_{m_{n-1}\le m_n}\frac{m_n!(\gamma)_{m_n}}{(\alpha)_{m_n+1}(\beta)_{m_n+1}}\right)\right)\end{multlined}\\
\end{align*}
\begin{align*}
&=\begin{multlined}[t]T^\star(k_n,k_1,\ldots,k_{n-1};(\alpha,\beta,\gamma))\\+(\gamma-\alpha-\beta)\sum_{\substack{0\le k\le m_0\le \cdots\le m_n\\m_0\neq m_{n-1}}}\frac{(\alpha)_{k}(\beta)_{k}}{k!(\gamma)_{k}}\frac{m_n!(\gamma)_{m_n}}{(\alpha)_{m_n}(\beta)_{m_n}}\frac{1}{(m_0+\alpha)(m_0+\beta)(m_0+\gamma)^{k_n-2}}\\\times\prod_{i=0}^{n-1}\frac{
1}{(m_i+\alpha)(m_i+\beta)(m_i+\gamma)^{k_i-2}}\frac{1}{(m_n+\alpha)(m_n+\beta)}\\
+Z(n|n-1|k_1+\cdots+k_n-2n)+Z(n-1|n|k_1+\cdots+k_n-2n)\\+
(\gamma-\alpha-\beta)\sum_{\substack{0\le k\le m_0= \cdots= m_{n-1}\le m_n\\}}\frac{(\alpha)_{k}(\beta)_{k}}{k!(\gamma)_{k}}\frac{m_n!(\gamma)_{m_n}}{(\alpha)_{m_n}(\beta)_{m_n}}\frac{1}{(m_0+\alpha)(m_0+\beta)(m_0+\gamma)^{k_n-2}}\\\times\prod_{i=0}^{n-1}\frac{
1}{(m_i+\alpha)(m_i+\beta)(m_i+\gamma)^{k_i-2}}\frac{1}{(m_n+\alpha)(m_n+\beta)}
\end{multlined}\\
&=\begin{multlined}[t]T^\star(k_n,k_1,\ldots,k_{n-1};(\alpha,\beta,\gamma))\\+(\gamma-\alpha-\beta)\sum_{\substack{0\le k\le m_0\le \cdots\le m_n\\}}\frac{(\alpha)_{k}(\beta)_{k}}{k!(\gamma)_{k}}\frac{m_n!(\gamma)_{m_n}}{(\alpha)_{m_n}(\beta)_{m_n}}\frac{1}{(m_0+\alpha)(m_0+\beta)(m_0+\gamma)^{k_n-2}}\\\times\prod_{i=0}^{n-1}\frac{
1}{(m_i+\alpha)(m_i+\beta)(m_i+\gamma)^{k_i-2}}\frac{1}{(m_n+\alpha)(m_n+\beta)}\\
+Z(n|n-1|k_1+\cdots+k_n-2n)+Z(n-1|n|k_1+\cdots+k_n-2n)
\end{multlined}
\end{align*}
となり示せた.
インデックスを $(k_{i+1},\ldots,k_n,k_1,\ldots,k_i)$ とおき, $i=1,\ldots n$ で和をとると主結果を得る.
最後に
多重ゼータ値そのものを研究するというのは非常に難しいこと(だと僕は思っています)ですが, 今回のように既出の結果にパラメータをつけて拡張していく研究というのはまだまだ未発展だと思うので今後も取り組んでいく価値があると思います.
とりあえず, 今までのことが形になって世に出たことは嬉しい限りです.
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