Some identities for generalized harmonic numbers

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 >= 2, S1(n + N, r + 1) n ! n X = j=0 n X i=0 i X l=0 l X z=0 r X k=0 ! ! N j alpha i (1)l-z-i m i l + m 2 l z i l j! ( n i)! x S1(N, r k + 1) S 1 ( n i, k) H(z, j 1 , alpha) where S 1 ( n, k ) is Stirling number of the first kind.