Two Conjectured Bernoulli Number Identities (Open)
Summary: 1) Prove or disprove that for all $m > 4$,
\[
\sum\limits_{k=1}^{\left[ \frac{m-1}{2}\right] }\left( \frac{1}{m-2k}%
+H_{2k}\left(
\begin{array}{c}
m \\
2k
\end{array}
\right) \right) B_{2k}=\frac{m}{2}-\frac{1}{m}-\frac{1}{2}\frac{1}{m-1},
\]
2) Prove or disprove that for all $m>4$, and $k=1,2,\dots ,m-3$,
\[
\begin{array}{c}
\sum\limits_{j=1}^{\left[ \frac{m-k-1}{2}\right] }\left( \frac{1}{\left(
m-2j-k\right) \left( k+2j\right) }\left(
\begin{array}{c}
k+2j \\
k
\end{array}%
\right) +\frac{1}{m}\left(
\begin{array}{c}
m \\
k
\end{array}
\right) \left(
\begin{array}{c}
m-k \\
2j
\end{array}
\right) H_{2j+k-1}\right) B_{2j} \\
=\frac{1}{k^{2}}\left(
\begin{array}{c}
m-1 \\
k-1
\end{array}
\right) -\frac{1}{\left( m-k\right) k}-\frac{1}{2\left( m-1-k\right) }+\frac{%
m-k-2}{2(m-k)}\left(
\begin{array}{c}
m-1 \\
k
\end{array}
\right) H_{k}.
\end{array}
\]
Classification: Primary, Discrete Mathematics; Secondary, Combinatorial Identities
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Huizeng Qin
College of Science
ShanDong University of Technology
12 ZhangZhou Road, Zibo, ShanDong 255049
China
Email: [email protected]
Phone: +(86)533-2781376