Abstract
In this paper, we present some identities relating the hyperharmonic, the Daehee and the derangement numbers, and we derive some nonlinear differential equations from the generating function of a hyperharmonic number. In addition, we use this differential equation to obtain some identities in which the hyperharmonic numbers and the Daehee numbers are involved.
Highlights
For any n, we denote by (x)n the falling factorial (x)0 = 1, (x)n = x(x – 1)(x – 2) · · · (x – n + 1) and x n for rising factorial x 0 = 1, x n = x(x + 1)(x + 2) · · · (x + n – 1)
The unsigned Stirling numbers of the first kind |S1(n, k)| count the number of permutations of n elements with k disjoint cycles and the definition is given by n x n = S1(n, k) xk
We present some identities between the Daehee and hyperharmonic numbers
Summary
The Cauchy numbers of order r, denoted by Cn(r), are defined by the generating function to be t log(1 + t) From (1) and (2), we note that Hn(1) is the ordinary harmonic number Hn. Many authors have studied the hyperharmonic numbers [1,2,3, 5, 10, 21]. The Daehee numbers, denoted by Dn, are defined by the generating function to be log(1 + t) ∞ tn t The higher-order Daehee numbers, denoted by D(nr), are defined by the generating function, log(1 + t) t r
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