Remarks on Ramanujan problem by using fractional order derivatives (Q1887390)

The equality \[ \begin{multlined} \frac 1k+\frac 1{k(k+m)}+\frac 1{k(k+m)(k+2m)}+\frac 1{k(k+m)(k+2m)(k+3m)}+\cdots\\ =\Gamma\left(\frac km\right)\left(\frac 1m\right)^{1-\frac km}\left(\frac{m\,e^{\frac 1m}}{(\frac km-n)!}\right. \int_0^{(\frac 1m)^{\frac 1m}} e^{-\eta^m}\eta^{k-m(n-1)-1}d\eta+\frac{(\frac 1m)^{\frac km-n}}{(\frac km-n)!}\\ -\frac{(\frac 1m)^{\frac km-n}}{(\frac km-n)!} \sum_{p=0}^{n-1}\left(\frac 1m\right)^p \sum_{q=0}^p\frac{(-1)^q}{q!(p-q)!}\cdot \left.\frac{\frac km-n+p+1}{\frac km-n+q+1} \right) \end{multlined} \] is proved for \(k,m\in\mathbb N\).