论文标题
涉及谐波数字的汇总系列
Series with summands involving harmonic numbers
论文作者
论文摘要
对于每个正整数$ m $，$ m $ th订单谐波数字由$$ h_n^{（m）} = \ sum_ {0 <k \ le n} \ frac1 {k^m} \ \（n = 0,1,1,2，\ ldots）。例如，我们猜测$$ \ sum_ {k = 0}^\ infty（6k+1）\ frac {\ binom {\ binom {2k} k^3} {256^k} \ left（h_ {2k}^{2K}^{（3）}} - } - \ frac {7} - {7} {64} {64} {64} {64} {64} {{64} {{64} {{{64} {{{64} {{64}} = \ frac {25ζ（3）} {8π} -g，$ g $ the $ g $表示加泰罗尼亚常数$ \ sum_ {k = 0}^\ infty（-1）^k/（2k+1）^2 $。本文包含$ 70 $的猜想，由作者在2022--2023期间提出。
For each positive integer $m$, the $m$th order harmonic numbers are given by $$H_n^{(m)}=\sum_{0<k\le n}\frac1{k^m}\ \ (n=0,1,2,\ldots).$$ We discover exact values of some series involving harmonic numbers of order not exceeding four. For example, we conjecture that $$\sum_{k=0}^\infty(6k+1)\frac{\binom{2k}k^3}{256^k}\left(H_{2k}^{(3)}-\frac{7}{64}H_{k}^{(3)}\right) =\frac{25ζ(3)}{8π}-G,$$ where $G$ denotes the Catalan constant $\sum_{k=0}^\infty(-1)^k/(2k+1)^2$. This paper contains $70$ conjectures posed by the author during 2022--2023.
