Abstract

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!} &=\sum\limits_{j=0}^{n}\sum\limits_{i=0}^{n}\sum\limits_{l=0}^{i}\sum% \limits_{z=0}^{l}\sum\limits_{k=0}^{r}\left( -1\right) ^{l-z-i}\dbinom{m}{% l-z}\dbinom{i-l+m-2}{i-l}\dfrac{N^{j}\alpha ^{i}}{j!\left( n-i\right) !}\\ & \quad\times S_{1}(N,r-k+1)S_{1}\left( n-i,k\right) H(z,j-1,\alpha ) \end{align*} where $S_{1}\left( n,k\right) $ is Stirling number of the first kind.

Keywords: Generalized hyperharmonic numbers of order $r$; Daehee numbers; Stirling numbers of the first kind and the second kind; generating function.

MSC: 05A15, 05A19, 11B73

DOI: 10.57016/MV-BLXV3088

Pages:  288--302     

Volume  76 ,  Issue  4 ,  2024