多重ゼータ値とその類似　Part３

1744

　
$Definition .$

$R_{F} (k) = \sum_{0 < n_{1} < \dots < n_{r}} C_{n_{1}}^{- 1} n^{- k}$

$R_{L} (k) = \sum_{0 < n_{1} < \dots < n_{r}} n^{- k} C_{n_{r}}^{- 1}$

$ζ (k_{1}, \dots, [k_{s}], \dots, k_{r}) = \sum_{\begin{matrix} 0 < n_{1} < \dots < n_{r} \\ n_{s} \in Z - \frac{1}{2} \end{matrix}} n^{- k}$

$T (k) = \sum_{0 \leq n_{1} < \dots < n_{r}} {(n + \frac{1}{2})}^{- k}$

$R_{L} (k) = \sum_{0 \leq n_{1} < \dots < n_{r}} {(n + \frac{1}{2})}^{- k} C_{n_{r}}^{- 1}$

$R^{F L} (k) = \sum_{0 \leq n_{1} < \dots < n_{r}} C_{n_{1}} {(n + \frac{1}{2})}^{- k} C_{n_{r}}$

$[\underset{r}{\underset{⏟}{k, \dots, k}}] = [k]_{r}, ([k_{1}], \dots, [k_{r}]) = ([k_{1}, \dots, k_{r}])$

　
$T h e o r e m 3.1 .$

$[3.1] R_{F}^{⋆} (k) = ζ ([1]_{k_{r} - 1}, 1, \dots, [1]_{k_{2} - 1}, 1, [1]_{k_{1} - 2}, 2)$

　
$Proof .$

$\begin{aligned} \int_{0}^{1} \frac{1}{1 - t} {(\circ \frac{1}{t})}^{k_{1} - 2} {(\circ \frac{1}{t \sqrt{1 - t}})}^{2} \dots {(\circ \frac{1}{t})}^{k_{r - 1} - 2} {(\circ \frac{1}{t \sqrt{1 - t}})}^{2} {(\circ \frac{1}{t})}^{k_{r} - 2} \circ \frac{1}{t \sqrt{1 - t}} \\ = \sum_{0 < n_{1}} \int_{0}^{1} t^{n_{1} - 1} {(\circ \frac{1}{t})}^{k_{1} - 2} {(\circ \frac{1}{t \sqrt{1 - t}})}^{2} \dots {(\circ \frac{1}{t})}^{k_{r - 1} - 2} {(\circ \frac{1}{t \sqrt{1 - t}})}^{2} {(\circ \frac{1}{t})}^{k_{r} - 2} \circ \frac{1}{t \sqrt{1 - t}} \\ = \sum_{0 < n_{1}} \frac{1}{n_{1}} \int_{0}^{1} t^{n_{1} - 1} {(\circ \frac{1}{t})}^{k_{1} - 3} {(\circ \frac{1}{t \sqrt{1 - t}})}^{2} \dots {(\circ \frac{1}{t})}^{k_{r - 1} - 2} {(\circ \frac{1}{t \sqrt{1 - t}})}^{2} {(\circ \frac{1}{t})}^{k_{r} - 2} \circ \frac{1}{t \sqrt{1 - t}} \\ = \dots \\ = R_{F}^{⋆} (k_{1}, \dots, k_{r}) \\ \int_{0}^{1} \frac{1}{1 - t} {(\circ \frac{1}{t})}^{k_{1} - 2} {(\circ \frac{1}{t \sqrt{1 - t}})}^{2} \dots {(\circ \frac{1}{t})}^{k_{r - 1} - 2} {(\circ \frac{1}{t \sqrt{1 - t}})}^{2} {(\circ \frac{1}{t})}^{k_{r} - 2} \circ \frac{1}{t \sqrt{1 - t}} \\ = \int_{0}^{1} \frac{1}{(1 - t) \sqrt{t}} {(\circ \frac{1}{1 - t})}^{k_{r} - 2} {(\circ \frac{1}{(1 - t) \sqrt{t}})}^{2} \dots {(\circ \frac{1}{1 - t})}^{k_{2} - 2} {(\circ \frac{1}{(1 - t) \sqrt{t}})}^{2} {(\circ \frac{1}{1 - t})}^{k_{1} - 2} \circ \frac{1}{t} \\ = \sum_{0 < n_{1}} \int_{0}^{1} \frac{t^{n_{1} - 1}}{\sqrt{t}} {(\circ \frac{1}{1 - t})}^{k_{r} - 2} {(\circ \frac{1}{(1 - t) \sqrt{t}})}^{2} \dots {(\circ \frac{1}{1 - t})}^{k_{2} - 2} {(\circ \frac{1}{(1 - t) \sqrt{t}})}^{2} {(\circ \frac{1}{1 - t})}^{k_{1} - 2} \circ \frac{1}{t} \\ = \sum_{0 < n_{1}} \frac{1}{n_{1} - \frac{1}{2}} \int_{0}^{1} \frac{t^{n_{1} - \frac{1}{2}}}{1 - t} {(\circ \frac{1}{1 - t})}^{k_{r} - 3} {(\circ \frac{1}{(1 - t) \sqrt{t}})}^{2} \dots {(\circ \frac{1}{1 - t})}^{k_{2} - 2} {(\circ \frac{1}{(1 - t) \sqrt{t}})}^{2} {(\circ \frac{1}{1 - t})}^{k_{1} - 2} \circ \frac{1}{t} \\ = \dots \\ = ζ ([1]_{k_{r} - 1}, 1, \dots, [1]_{k_{2} - 1}, 1, [1]_{k_{1} - 2}, 2) \end{aligned}$
$◻$

　
$T h e o r e m 3.2 .$

$[3.2] ζ^{⋆} ({1}_{a}, b + 1) = \sum_{0 < n_{1} < \dots < n_{b} \leq m_{1} \leq \dots \leq m_{a}} \frac{1}{n_{1} \dots m_{a - 1} m_{a}^{2}}$

　
$Proof .$

$\int_{0}^{1} \frac{1}{1 - t} {(\circ \frac{1}{t (1 - t)})}^{a} {(\circ \frac{1}{t})}^{b}$

と， $t \to 1 - t$ と置換したものをそれぞれ計算する。
$◻$

　
$T h e o r e m 3.3 .$

$[3.3] T^{⋆} (k) = \sum_{j = 0}^{r - 1} \sum_{i = 0}^{j} R_{L} ({1}_{k_{r} - 2}, 2, \dots, {1}_{d_{i, j} - 2}, 2, \dots, {1}_{k_{1} - 2}, 2) (d_{i, j} = \sum_{h = j - i + 1}^{r - i} k_{h})$

　
$Proof .$

$\int_{0}^{1} \frac{1}{(1 - t) \sqrt{t}} {(\circ \frac{1}{t})}^{k_{1} - 1} \circ \frac{1}{t (1 - t)} \dots {(\circ \frac{1}{t})}^{k_{r - 1} - 1} \circ \frac{1}{t (1 - t)} {(\circ \frac{1}{t})}^{k_{r} - 1}$

と， $t \to 1 - t$ と置換したものをそれぞれ計算する。シグマの等号部分をすべて分離することで右辺のようになる。
$◻$

　
$T h e o r e m 3.4 .$

$[3.4] R_{L} (k) = T (k^{†})$

　
$Proof .$

$\int_{0}^{1} \frac{1}{1 - t} {(\circ \frac{1}{1 - t})}^{a_{1} - 1} {(\circ \frac{1}{t})}^{b_{1}} {(\circ \frac{1}{1 - t})}^{a_{2}} {(\circ \frac{1}{t})}^{b_{2}} \dots {(\circ \frac{1}{1 - t})}^{a_{r}} {(\circ \frac{1}{t})}^{b_{r} - 1} \circ \frac{1}{t \sqrt{1 - t}}$

と， $t \to 1 - t$ と置換したものをそれぞれ計算する。
$◻$

　
$T h e o r e m 3.5 .$

$[3.5] R_{L} ({1}_{a - 1}, 2) = π \sum_{0 \leq n} {(n + \frac{1}{2})}^{- a} C_{n}^{2}$

　
$Proof .$

$\int_{0}^{1} \frac{1}{(1 - t) \sqrt{t}} {(\circ \frac{1}{1 - t})}^{a - 1} \circ \frac{1}{\sqrt{t (1 - t)}}$

と， $t \to 1 - t$ と置換したものをそれぞれ計算する。
$◻$

　
$T h e o r e m 3.6 .$

$[3.6] R_{L} (i n v (k), 2) = π R^{F L} (i n v (k^{†}), 1)$

　
$Proof .$

$\int_{0}^{1} \frac{1}{(1 - t) \sqrt{t}} {(\circ \frac{1}{t})}^{a_{1}} {(\circ \frac{1}{1 - t})}^{b_{1}} \dots {(\circ \frac{1}{t})}^{a_{r}} {(\circ \frac{1}{1 - t})}^{b_{r}} \circ \frac{1}{\sqrt{t (1 - t)}}$

と， $t \to 1 - t$ と置換したものをそれぞれ計算する。
$◻$

　
$Definition .$

$Z [α, β, γ, δ] (k) = \frac{Γ (α + γ) Γ (δ)}{Γ (α + γ + δ)} \sum_{0 \leq n_{1} < \dots < n_{r}} \frac{(1 - β)_{n_{1}}}{n_{1}!} (n + α)^{- k} \frac{(α + γ)_{n_{r}}}{(α + γ + δ)_{n_{r}}}$

　
$T h e o r e m 3.7 .$

$[3.7] Z [α, β, γ, δ] (i n v (k), 1) = Z [δ, γ, β, α] (i n v (k^{†}), 1)$

　
$Proof .$

$\int_{0}^{1} t^{- 1 + α} (1 - t)^{- 1 + β} {(\circ \frac{1}{t})}^{a_{1}} {(\circ \frac{1}{1 - t})}^{b_{1}} \dots {(\circ \frac{1}{t})}^{a_{r}} {(\circ \frac{1}{1 - t})}^{b_{r}} \circ t^{- 1 + γ} (1 - t)^{- 1 + δ}$