Some identities for generalized harmonic numbers


S. Koparal, N. Ömür, K. N. Südemen




In this paper, we derive some nonlinear differential equations from generating function of generalized harmonic numbers and give some identities involving generalized harmonic numbers and special numbers by using these differential equations. For example, for any positive integers $N,$ $n,$ $r,$ $\alpha $ and any integer $m\geq 2,$ \begin{align*} \dfrac{S_{1}(n+N,r+1)}{n!} &=umimits_{j=0}^{n}umimits_{i=0}^{n}umimits_{l=0}^{i}um% imits_{z=0}^{l}umimits_{k=0}^{r}eft( -1\right) ^{l-z-i}ḅinom{m}{% l-z}ḅinom{i-l+m-2}{i-l}\dfrac{N^{j}lpha ^{i}}{j!eft( n-i\right) !} & \quadimes S_{1}(N,r-k+1)S_{1}eft( n-i,k\right) H(z,j-1,lpha ) \end{align*} where $S_{1}\left( n,k\right) $ is Stirling number of the first kind.