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

TL;DR

This paper extends Euler's transformation to general series, provides new expressions for the Riemann zeta function, and generalizes Wallis' formula for pi, revealing new identities and evaluation methods for special values.

Contribution

It introduces generalized difference operators for zeta function analysis and extends Wallis' formula to broader complex planes, offering novel analytic and computational tools.

Findings

New expressions for $ta(s)$ using generalized difference operators

Analytic continuation of the Riemann zeta function

Extended Wallis' formulas for pi in wider half-planes

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.