Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Euler's transformation, zeta functions and generalizations of Wallis'
  formula   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.

23
The categories of this paper are: solv-int
21
The categories of this paper are: adap-org
12
The categories of this paper are: nucl-th
26
The categories of this paper are: bayes-an
37
The categories of this paper are: supr-con
36
The categories of this paper are: cmp-lg
5
The categories of this paper are: gr-qc
17
The categories of this paper are: q-fin
35
The categories of this paper are: mtrl-th
29
The categories of this paper are: funct-an
19
The categories of this paper are: eess
7
The categories of this paper are: hep-th
33
The categories of this paper are: plasm-ph
30
The categories of this paper are: q-alg
6
The categories of this paper are: astro-ph
10
The categories of this paper are: q-bio