论文标题
Euler的转换，Zeta的功能和Wallis公式的概括
Euler's transformation, zeta functions and generalizations of Wallis' formula
论文作者
论文摘要
在本说明中，我们将Euler的转换公式从交替系列扩展到更一般的系列。然后，我们通过广义差异操作员$δ_{C} $为Riemann Zeta函数$ζ$提供新的表达式，该差异提供了$ζ（S）$的分析延续和新方法，以评估$ M = 0,1,1,2，\ ldots $的特殊值$ζ（-m）$。在应用这些结果时，我们进一步扩展了Huylebrouck对Wallis的著名公式的概括，分别为$π$，分别为$π$。他们暗示了几个有趣的特殊情况，包括$ \ frac {2π} {3^{\ frac {\ frac {3} {2}}}} = \ frac {3^{\ frac {4} {3} {3} {3}}}}}}}}}} {2^{2^{\ frac {\ frac {\ frac {4} {4} {4} {4} {4} {4} {4} {4} {4} \ frac {2^{\ frac {1} {3}}} \ cdot3^{\ frac {1} {3}} {3}} \ cdot3^^{\ frac {1} {1} {3} {3}}}}} \ cdot4^{\ frac {1} {3}}} \ cdot6^{\ frac {2} {3} {3}} \ cdot6^{\ frac {\ frac {2} {3} {3}}}}}} { 4^{\ frac {1} {3}} \ cdot4^{\ frac {1} {3}} {3}} \ cdot5^{\ frac {1} {3}}}}} \ cdot5 ^{\ frac {1} {3}} \ cdot4^{\ frac {2} {3}} {3}} \ cdot5^{\ frac {2} {2} {3}}}}}}}} \ cdots， $$ $$ 3^{γ-\ frac {\ log 3} {2}} = \ frac {3^{\ frac {1} {3}}} \ cdot3^^{\ frac {\ frac {1} {3}}}}} {2^{\ frac {1}} \ frac {6^{\ frac {1} {6}} \ cdot6^{\ frac {\ frac {1} {6}}}}} {5^{\ frac {1} {5}} {5}}} \ cdot7^{\ cdot7^{\ frac {\ frac {1} {1}}}} c {9^{\ frac {1} {9}} \ cdot9^{\ frac {1} {1} {9}}}} {8^{\ frac {1} {8}} {8}}}}}}}}}}}}}} \ cdot10^{\ cdot10^{\ frac {\ frac {1} {1} {1}} {10}} {10}} $$和$$ \ left（3 \ left（\ frac {2πe^γ} {a^{12}}} \ right） \ frac {1} {3^2}}} \ cdot3^{\ frac {\ frac {1} {3^2}}}}}} {2^{\ frac {1} {2^2}}}}}}} \ cdot4^{\ cdot4^{\ frac {\ frac {\ frac {1} \ frac {6^{\ frac {1} {6^2}} \ cdot6^{\ frac {\ frac {1} {6^2}}}} {5^{\ frac {1} {1} {5^2}} {5^2}}}}} \ cdot7^{\ cdot7^{\ frac {\ frac}}}}}}}}}}}}}} 2}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}} 2 {9^{\ frac {1} {9^2}} \ cdot9^{\ frac {\ frac {1} {9^2}}}}} {8^{\ frac {1} {1} {8^2}}} {8^2}}} \ cdot10^{\ frac {\ frac {\ frac {\ frac {\ frac {\ frac {1}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}} $γ$是Euler-Mascheroni常数，而$ a $是Glaisher-kinkelin常数。
In this note, we extend Euler's transformation formula from the alternating series to more general series. Then we give new expressions for the Riemann zeta function $ζ(s)$ by the generalized difference operator $Δ_{c}$, which provide analytic continuation of $ζ(s)$ and new ways to evaluate the special values of $ζ(-m)$ for $m=0,1,2,\ldots$. Applying these results, we further extend Huylebrouck's generalization of Wallis' well-known formula for $π$ in the half planes Re$(s)>0$ and Re$(s)>-1$, respectively. They imply several interesting special cases including $$ \frac{2π}{3^{\frac{3}{2}}}=\frac{3^{\frac{4}{3}}}{2^{\frac{4}{3}}} \frac{2^{\frac{1}{3}}\cdot3^{\frac{1}{3}}\cdot3^{\frac{1}{3}}\cdot4^{\frac{1}{3}}\cdot6^{\frac{2}{3}}\cdot6^{\frac{2}{3}}}{4^{\frac{1}{3}}\cdot4^{\frac{1}{3}}\cdot5^{\frac{1}{3}}\cdot5^{\frac{1}{3}}\cdot4^{\frac{2}{3}}\cdot5^{\frac{2}{3}}}\cdots, $$ $$ 3^{γ-\frac{\log 3}{2}}=\frac{3^{\frac{1}{3}}\cdot3^{\frac{1}{3}}}{2^{\frac{1}{2}}\cdot4^{\frac{1}{4}}} \frac{6^{\frac{1}{6}}\cdot6^{\frac{1}{6}}}{5^{\frac{1}{5}}\cdot7^{\frac{1}{7}}}\frac{9^{\frac{1}{9}}\cdot9^{\frac{1}{9}}}{8^{\frac{1}{8}}\cdot10^{\frac{1}{10}}}\cdots, $$ and $$ \left(3\left(\frac{2πe^γ}{A^{12}}\right)^{2}\right)^{\frac{π^2}{18}}=\frac{3^{\frac{1}{3^2}}\cdot3^{\frac{1}{3^2}}}{2^{\frac{1}{2^2}}\cdot4^{\frac{1}{4^2}}} \frac{6^{\frac{1}{6^2}}\cdot6^{\frac{1}{6^2}}}{5^{\frac{1}{5^2}}\cdot7^{\frac{1}{7^2}}}\frac{9^{\frac{1}{9^2}}\cdot9^{\frac{1}{9^2}}}{8^{\frac{1}{8^2}}\cdot10^{\frac{1}{10^2}}}\cdots,$$ where $γ$ is the Euler-Mascheroni constant and $A$ is the Glaisher-Kinkelin constant.
