Euler's transformation, zeta functions and generalizations of Wallis' formula

Abstract

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 $\zeta(s)$ by the generalized difference operator $\Delta_{c}$, which provide analytic continuation of $\zeta(s)$ and new ways to evaluate the special values of $\zeta(-m)$ for $m=0,1,2,\ldots$. Applying these results, we further extend Huylebrouck's generalization of Wallis' well-known formula for $\pi$ in the half planes Re$(s)>0$ and Re$(s)>-1$, respectively. They imply several interesting special cases including $$ \frac{2\pi}{3^{\frac{3}{2}}}=\frac{3^{\frac{4}{3}}}{2^{\frac{4}{3}}} \frac{2^{\frac{1}{3}}\cdot3^{\frac{1}{3}}\cdot3^{\frac{1}{3}}\cdot4^{\frac{1}{3}}\cdot6^{\frac{2}{3}}\cdot6^{\frac{2}{3}}}{4^{\frac{1}{3}}\cdot4^{\frac{1}{3}}\cdot5^{\frac{1}{3}}\cdot5^{\frac{1}{3}}\cdot4^{\frac{2}{3}}\cdot5^{\frac{2}{3}}}\cdots, $$ $$ 3^{\gamma-\frac{\log 3}{2}}=\frac{3^{\frac{1}{3}}\cdot3^{\frac{1}{3}}}{2^{\frac{1}{2}}\cdot4^{\frac{1}{4}}} \frac{6^{\frac{1}{6}}\cdot6^{\frac{1}{6}}}{5^{\frac{1}{5}}\cdot7^{\frac{1}{7}}}\frac{9^{\frac{1}{9}}\cdot9^{\frac{1}{9}}}{8^{\frac{1}{8}}\cdot10^{\frac{1}{10}}}\cdots, $$ and $$ \left(3\left(\frac{2\pi e^{\gamma}}{A^{12}}\right)^{2}\right)^{\frac{\pi^2}{18}}=\frac{3^{\frac{1}{3^2}}\cdot3^{\frac{1}{3^2}}}{2^{\frac{1}{2^2}}\cdot4^{\frac{1}{4^2}}} \frac{6^{\frac{1}{6^2}}\cdot6^{\frac{1}{6^2}}}{5^{\frac{1}{5^2}}\cdot7^{\frac{1}{7^2}}}\frac{9^{\frac{1}{9^2}}\cdot9^{\frac{1}{9^2}}}{8^{\frac{1}{8^2}}\cdot10^{\frac{1}{10^2}}}\cdots,$$ where $\gamma$ is the Euler-Mascheroni constant and $A$ is the Glaisher-Kinkelin constant.