We present explicit formulas for the following family of parametric binomial sums involving harmonic numbers for \(p=0,1,2\) and \(|t|\le 1\).

$$\begin{aligned} \sum _{k=1}^{\infty }\frac{H_{k-1}t^k}{k^p\left( {\begin{array}{c}n+k\\ k\end{array}}\right) } \quad \text{ and }\quad \sum _{k=1}^{\infty }\frac{t^k}{k^p\left( {\begin{array}{c}n+k\\ k\end{array}}\right) }. \end{aligned}$$

We also generalize the following relation between the Stirling numbers of the first kind and the Riemann zeta function to polygamma function and give some applications.

$$\begin{aligned} \zeta (n+1)=\sum _{k=n}^{\infty }\frac{s(k,n)}{kk!}, \quad n=1,2,3,\ldots . \end{aligned}$$

As examples,

$$\begin{aligned} \zeta (3)=\frac{1}{7}\sum _{k=1}^{\infty }\frac{H_{k-1}4^k}{k^2\left( {\begin{array}{c}2k\\ k\end{array}}\right) }, \quad \text{ and }\quad \zeta (3)=\frac{8}{7}+\frac{1}{7}\sum _{k=1}^{\infty } \frac{H_{k-1}4^k}{k^2(2k+1)\left( {\begin{array}{c}2k\\ k\end{array}}\right) }, \end{aligned}$$

which are new series representations for the Apéry constant \(\zeta (3)\).

中文翻译：

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涉及谐波数的参数二项式和

我们为以下参数二项和式系列提供了明确的公式，其中涉及\（p = 0,1,2 \）和\（| t | \ le 1 \）的调和数。

$$ \ begin {aligned} \ sum _ {k = 1} ^ {\ infty} \ frac {H_ {k-1} t ^ k} {k ^ p \ left（{\ begin {array} {c} n + k \\ k \ end {array}} \ right）} \ quad \ text {和} \ quad \ sum _ {k = 1} ^ {\ infty} \ frac {t ^ k} {k ^ p \ left （{\ begin {array} {c} n + k \\ k \ end {array}} \ right）}。\ end {aligned} $$

我们还将第一类斯特林数与黎曼Zeta函数之间的以下关系推广到多伽玛函数，并给出了一些应用。

$$ \ begin {aligned} \ zeta（n + 1）= \ sum _ {k = n} ^ {\ infty} \ frac {s（k，n）} {kk！}，\ quad n = 1,2 ，3，\ ldots。\ end {aligned} $$

举例来说，

$$ \ begin {aligned} \ zeta（3）= \ frac {1} {7} \ sum _ {k = 1} ^ {\ infty} \ frac {H_ {k-1} 4 ^ k} {k ^ 2 \ left（{\ begin {array} {c} 2k \\ k \ end {array}} \ right）}，\ quad \ text {和} \ quad \ zeta（3）= \ frac {8} {7 } + \ frac {1} {7} \ sum _ {k = 1} ^ {\ infty} \ frac {H_ {k-1} 4 ^ k} {k ^ 2（2k + 1）\ left（{\ begin {array} {c} 2k \\ k \ end {array}} \ right）}，\ end {aligned} $$

这是Apéry常数\（\ zeta（3）\）的新系列表示。