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

Huizeng Qin
College of Science
ShanDong University of Technology
12 ZhangZhou Road, Zibo, ShanDong 255049
Phone: +(86)533-2781376

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