Consider this family of series: $$\sum _{n=0} ^{\infty} \frac{1}{\left(n+\frac{m}{2}\right)^{\beta}} \frac{d^{\alpha}}{{dx}^{\alpha}} \left. \binom{x-m}{n} \right|_{x=0}, \quad \text{where}~ m \geq0$$
Here's how this paper by Borwein and Straub considers the case of $m=0$ and finds a closed form. First we need to calculate the derivatives of the binomial coefficient. To this end, let's define the multiple harmonic number for $n\geq 1$ as: $$H^{[\alpha]}_{n-1} = \sum _{n < i_1< \cdots <i_{\alpha}}\frac{1}{i_1 i_2 \cdots i_{\alpha}} =\sum _{i_1=1} ^{n-1} \frac{1}{i_1}\sum _{i_2=1} ^{i_1} \frac{1}{i_2} \cdots\sum _{i_{\alpha}=1} ^{i_{\alpha-1}} \frac{1}{i_{\alpha}}$$
The claim is that the following identity holds for $n \geq 1$: $$\frac{(-1)^{\alpha}}{\alpha!} \frac{d^{\alpha}}{{dx}^{\alpha}} \left. \binom{x}{n} \right|_{x=0} = \frac{(-1)^n}{n} H^{[\alpha-1]} _{n-1} \tag{☼}$$
Then, it is stated that the series reduces to the Nielsen polylogarithm function like so: $$\sum _{n=1} ^{\infty} \frac{(\pm1)^n}{n^{\beta}} H^{[\alpha]}_{n-1} = S_{\beta,\alpha}(\pm1) \tag{☾}$$ How can we show that formulas $(☼)$ and $(☾)$ are true? And could they be generalized to the case of $m = 1$?
