# Identities between harmonic, hyperharmonic and Daehee numbers

Open Access
Research

## 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.

## Keywords

Hyperharmonic numbers Daehee numbers Differential equation

## MSC

05A19 11B37 34A30

## 1 Introduction

For any n, we denote by $$(x)_{n}$$ the falling factorial $$(x)_{0} =1, (x)_{n} = x(x-1)(x-2) \cdots (x-n+1)$$ and $$\langle x\rangle _{n}$$ for rising factorial $$\langle x\rangle _{0} =1, \langle x\rangle _{n}= x(x+1)(x+2) \cdots (x+n-1)$$. Formally, $$(x)_{n} = \langle x\rangle _{n} =0$$ if $$n < 0$$.

The Stirling numbers are defined by $$x^{n}$$ and $$(x)_{n}$$ as
\begin{aligned}& x^{n}= \sum_{k=0}^{n} S_{2} (n, k) (x)_{k}, \\ & (x)_{n}= \sum_{k=0}^{n} S_{1} (n, k) x^{k}, \end{aligned}
where $$S_{1} (n,k)$$ and $$S_{2} (n,k)$$ are called the Stirling numbers of the first kind and the second kind, respectively.
As is well known, the unsigned Stirling numbers of the first kind, denoted by $$\vert S_{1}(n,k) \vert$$, are $$(-1)^{n+k}S_{1} (n,k)$$. The unsigned Stirling numbers of the first kind $$\vert S_{1}(n,k) \vert$$ count the number of permutations of n elements with k disjoint cycles and the definition is given by
$$\langle x\rangle _{n} = \sum _{k=0}^{n} \bigl\vert S_{1} (n, k) \bigr\vert x^{k}.$$
A derangement is a permutation of the elements of a set, such that no element appears in its original position. In other words, derangement is a permutation that has no fixed points. The number of derangements of a set of size n, denoted by $$d_{n}$$, is called the nth derangement number. The generating function of derangement numbers is given by
$$\frac{e^{-t}}{1-t} = \sum_{n=0}^{\infty} d_{n} \frac{t^{n}}{n!}.$$
The Cauchy numbers of orderr, denoted by $$C_{n}^{(r)}$$, are defined by the generating function to be
$$\biggl( \frac{t}{\log(1+t)} \biggr)^{r} = \sum _{n=0}^{\infty} C_{n}^{(r)} \frac{t^{n}}{n!}.$$
It is well known that the nth harmonic numbers, denoted by $$H_{n}$$, are defined by
\begin{aligned} H_{n} = \frac{1}{1} + \frac{1}{2} + \frac{1}{3} + \cdots + \frac{1}{n} \end{aligned}
(1)
with $$H_{0} = 0$$.

The harmonic numbers have many applications in combinatorics and other areas. Several interesting properties of harmonic numbers can be found in .

In , the nth hyperharmonic numbers of order r, denoted by $$H_{n}^{(r)}$$, are defined by
\begin{aligned} H_{n}^{(r)} = \textstyle\begin{cases} 0 & \text{if } n \le 0 \text{ or } r< 0, \\ \frac{1}{n} & \text{if } n > 0 \text{ and } r = 0, \\ \sum_{i=1}^{n}H_{i}^{(r-1)} & \text{if } r,n \ge 1. \end{cases}\displaystyle \end{aligned}
(2)

From (1) and (2), we note that $$H_{n}^{(1)}$$ is the ordinary harmonic number $$H_{n}$$. Many authors have studied the hyperharmonic numbers [1, 2, 3, 5, 10, 21].

The Daehee numbers, denoted by $$D_{n}$$, are defined by the generating function to be
$$\frac{\log(1+t)}{t}= \sum_{n=0}^{\infty} D_{n} \frac{t^{n}}{n!}.$$
(3)
It is clear that
$$D_{0} = 1,\qquad D_{1} = - \frac{1}{2}, \ldots, D_{n} = (-1)^{n} \frac{n!}{n+1}.$$
(4)

The Daehee numbers serve as an intermediate medium connecting between several special numbers [6, 7, 11, 13, 31]. The higher-order Daehee numbers led to many combinatorial identities [8, 9, 13, 19, 22, 24, 29, 30]. In addition, the degenerate Daehee numbers have been defined and studied [13, 29, 30]. Recently many interesting results have been published regarding the degenerate Daehee numbers.

The higher-order Daehee numbers, denoted by $$D_{n}^{(r)}$$, are defined by the generating function,
$$\biggl( \frac{\log(1+t)}{t} \biggr)^{r} = \sum _{n=0}^{\infty} D_{n}^{(r)} (x) \frac{t^{n}}{n!}\quad \text{(see [8, 9, 13, 19, 22, 24, 29, 30])}.$$
(5)

Recently, a group of mathematicians used a differential equations to study special numbers. In [11, 13], the Daehee and degenerate Daehee numbers are considered by using differential equations arising from the generating function. The identities for ordered Bell numbers  and Bernoulli numbers of the second kind  were derived arising from the differential equations of the generating functions, and the other identities of special polynomials can be found in [15, 16, 17, 18, 20, 23, 25, 26, 27]. In this paper, we present some identities between the Daehee and hyperharmonic numbers. In addition, we derive some nonlinear differential equations from the generating function of the hyperharmonic number. In addition, we use this differential equations to obtain some identities in which the hyperharmonic numbers and the Daehee numbers are involved.

## 2 Harmonic numbers and hyperharmonic numbers

Since $$-\log(1-t) = t+ \frac{t^{2}}{2} + \frac{t^{3}}{3}+ \cdots$$ , the generating function of the harmonic numbers $$H_{n}$$ is as follows:
$$- \frac{ \log(1-t)}{1-t} = \sum_{n=0}^{\infty} H_{n} t^{n}.$$
(6)
From the definition of hyperharmonic numbers (2) and the generating function of harmonic numbers (6), we get the generating function of the hyperharmonic numbers:
$$- \frac{ \log(1-t)}{(1-t)^{r}} = \sum_{n=0}^{\infty} H_{n}^{(r)} t^{n}.$$
(7)

The generating functions of harmonic and hyperharmonic numbers can be found in [1, 5] and .

A recurrence relation of the hyperharmonic numbers can be obtained by the generating function as follows:
\begin{aligned} &{-} \frac{ \log(1-t)}{(1-t)^{r}} (1-t) = \sum_{n=0}^{\infty} H_{n}^{(r)} t^{n} - \sum _{n=0}^{\infty} H_{n}^{(r)} t^{n+1} \\ &\phantom{{-} \frac{ \log(1-t)}{(1-t)^{r}} (1-t) }= \sum_{n=0}^{\infty} H_{n}^{(r)} t^{n} - \sum_{n=1}^{\infty} H_{n-1}^{(r)} t^{n} \\ &\phantom{{-} \frac{ \log(1-t)}{(1-t)^{r}} (1-t) }= \sum_{n=0}^{\infty} \bigl( H_{n}^{(r)} - H_{n-1}^{(r)} \bigr) t^{n}, \\ &{- }\frac{ \log(1-t)}{(1-t)^{r}} (1-t) = - \frac{ \log(1-t)}{(1-t)^{r-1}} \\ &\phantom{{- }\frac{ \log(1-t)}{(1-t)^{r}} (1-t) }= \sum_{n=0}^{\infty} H_{n}^{(r-1)} t^{n}. \end{aligned}
Therefore we get
$$H_{n}^{(r)}= H_{n-1}^{(r)} + H_{n}^{(r-1)}\quad \text{for } n \ge 1.$$
(8)

This recurrence relation (8) is shown in , which we obtained in another way.

We note that, for $$1 \le s \le r$$,
\begin{aligned} - \frac{\log(1-t)}{(1-t)^{r}} &= - \frac{\log(1-t)}{(1-t)^{r-s}} \frac{1}{(1-t)^{s}} \\ &= \sum_{l=0}^{\infty} H_{l}^{(r-s)} t^{l} \sum_{k=0}^{\infty} \binom{-s}{k} (-1)^{k} t^{k} \\ &= \sum_{l=0}^{\infty} H_{l}^{(r-s)} t^{l} \sum_{k=0}^{\infty} \binom{s+k-1}{s-1} t^{k} \\ &= \sum_{n=0}^{\infty} \sum _{l=0}^{n} H_{l}^{(r-s)} \binom{s+n-l-1}{s-1} t^{n}. \end{aligned}
(9)
Equation (9) yields some identities that are presented in  as follows:
\begin{aligned} H_{n}^{(r)} &= \sum _{m=1}^{n} \binom{n+r-m-1}{ r-1} \frac{1}{m}, \\ H_{n}^{(r)} &= \sum_{m=1}^{n} \binom{n+r-m-s-1}{ r-s-1} H_{m}^{(s)},\quad 0 \le s \le n-1. \end{aligned}
(10)

## 3 Relations between hyperharmonic numbers and Daehee numbers

From the definition of Daehee numbers, we obtain
\begin{aligned} \frac{\log(1+t)}{t} &= \frac{- \log(1+t)}{(1+t)} \frac{1+t}{ -t} \\ &= \sum_{n=1}^{\infty} (-1)^{n+1} H_{n} t^{n} \biggl( 1 + \frac{1}{t} \biggr) \\ &= \sum_{n=1}^{\infty} (-1)^{n+1} H_{n} t^{n} + \sum_{n=0}^{\infty} (-1)^{n} H_{n+1} t^{n} \\ &= \sum_{n=1}^{\infty} \bigl( (-1)^{n+1} H_{n} + (-1)^{n} H_{n+1} \bigr) t^{n} + H_{1}. \end{aligned}
(11)
Since $$H_{n+1} -H_{n} = \frac{1}{n+1}$$, from (4) and (11), we get
$$D_{0} = H_{1},\qquad D_{n} = (-1)^{n} n! ( H_{n+1} - H_{n} ) \quad\text{for } n \ge 1.$$
Let us investigate the relationship between the Daehee and hyperharmonic numbers.
\begin{aligned} \frac{\log(1+t)}{t} &= \frac{- \log(1+t)}{(1+t)^{r}} \frac{(1+t)^{r}}{ -t} \\ &= \sum_{i=1}^{\infty} (-1)^{i+1} H_{i}^{(r)} t^{i-1} \sum _{j=0}^{\infty} \binom{r}{j} t^{j} \\ &= \sum_{i=0}^{\infty} (-1)^{i} H_{i+1}^{(r)} t^{i} \sum _{j=0}^{\infty} \binom{r}{j} t^{j} \\ &= \sum_{n=0}^{\infty} \sum _{i=0}^{n} (-1)^{i} \binom{r}{n-i} H_{i+1}^{(r)} t^{n}. \end{aligned}
(12)

From (12) and the definition of the Daehee numbers (3), we get the following identity.

### Theorem 1

For any non-negative integern,
$$D_{n} = n! \sum_{i=0}^{n} (-1)^{i} \binom{r}{n-i} H_{i+1}^{(r)}.$$
From (4), we note that $$\sum_{i=0}^{n-1} \frac{ (-1)^{i} D_{i}}{i!} = H_{n}$$. Theorem 1 yields the following identity:
$$H_{n} = \sum_{i=0}^{n-1} \sum_{k=0}^{i}(-1)^{k+i} H_{k+1}^{(r)} \binom{r}{i-k}, \quad\text{for } n \ge 1.$$
Let us consider higher-order Daehee numbers:
\begin{aligned} \biggl( \frac{\log(1+t)}{t} \biggr)^{r} &= - \frac{\log(1+t)}{(1+t)^{k}} \biggl( \frac{\log(1+t)}{t} \biggr)^{r-1} \frac{(1+t)^{k}}{-t} \\ &= \Biggl( \sum_{i=0}^{\infty} (-1)^{i} H_{i+1}^{(k)} t^{i} \Biggr) \Biggl( \sum_{j=0}^{\infty} {D_{j}^{(r-1)}} \frac{t^{j}}{j!} \Biggr) \Biggl( \sum_{l=0}^{\infty} (k)_{l} \frac{t^{l}}{l!} \Biggr) \\ &= \Biggl( \sum_{i=0}^{\infty} (-1)^{i} H_{i+1}^{(k)} t^{i} \Biggr) \Biggl( \sum_{m=0}^{\infty} \sum _{j=0}^{m} \binom{m}{j}{D_{j}^{(r-1)}} (k)_{m-j} \frac{t^{m}}{m!} \Biggr) \\ &= \sum_{n=0}^{\infty}\sum _{i=0}^{n} \sum_{j=0}^{n-i} (-1)^{i} \binom{n-i}{j} \frac { (k)_{n-i-j} D_{j}^{(r-1)} H_{i+1}^{(k)}}{(n-i)!} {t^{n}}. \end{aligned}
(13)

The following is found in Eq. (13) along with the definition of higher-order Daehee numbers.

### Theorem 2

For any non-negative integernand$$k \ge 1$$,
$$D_{n}^{(r)} = n! \sum _{i=0}^{n} \sum_{j=0}^{n-i} (-1)^{i} \binom{n-i}{j} \frac { (k)_{n-i-j} D_{j}^{(r-1)} H_{i+1}^{(k)}}{(n-i)!}.$$
Now, we want to express $$H_{n}$$ as a summation of $$D_{k}$$. We have
\begin{aligned} -\frac{\log(1-t)}{1-t} &= \sum _{n=0}^{\infty} H_{n} t^{n} \\ &= \frac{ \log(1-t)}{-t} \frac{t}{ 1-t} \\ &= \sum_{i=0}^{\infty} (-1)^{i} \frac{D_{i}}{i!} t^{i} \sum_{j=1}^{\infty} t^{j} \\ &= \sum_{n=1}^{\infty} \sum _{j=0}^{n-1} (-1)^{j} \frac{D_{j}}{j!} t^{n}. \end{aligned}
(14)
By comparing coefficients of the first line and fourth line in (14), we get an obvious identity:
$$H_{n} = \sum_{j=0}^{n-1} (-1)^{j} \frac{D_{j}}{j!}.$$
Let us observe the definition of hyperharmonic and Daehee numbers. We have
\begin{aligned} - \frac{\log(1-t)}{(1-t)^{r}} &= \frac{\log(1-t)}{-t} \frac{t}{(1-t)^{r}} \\ &= \sum_{k=0}^{\infty}(-1)^{k} D_{k} \frac{t^{k}}{k!} \sum_{l=0}^{\infty} (-r)_{l} (-1)^{l} \frac{t^{l+1}}{l!} \\ &= \sum_{k=0}^{\infty} (-1)^{k} D_{k} \frac{t^{k}}{k!} \sum_{l=1}^{\infty} (-r)_{l-1} (-1)^{l-1} l \frac{t^{l}}{l!} \\ &= \sum_{n=1}^{\infty} \sum _{k=0}^{n-1} \binom{n}{k} (-1)^{n-1} D_{k} (-r)_{k-1} (n-k) \frac{t^{n}}{n!}. \end{aligned}
(15)

Equation (15) yields Theorem 3.

### Theorem 3

For any positive integern,
$$n! H_{n}^{(r)} = \sum _{k=0}^{n-1} \binom{n}{k} (-1)^{n-1}(n-k) (-r)_{k-1} D_{k}.$$
(16)

Theorem 3 shows that hyperharmonic numbers, $$H_{n}^{(r)}$$, can be expressed as a kind of sum of Daehee numbers. Naturally we can think of whether it is possible to express the Daehee number $$D_{n}$$ in terms of the hyperharmonic numbers $$H_{n}^{(r)}$$.

Let us observe Eq. (15) from a different point of view:
\begin{aligned} - \frac{\log(1-t)}{(1-t)^{r}} &= \biggl( \frac{\log(1-t)}{-t} \biggr)^{r} \biggl( \frac{-t}{ \log(1-t)} \biggr)^{r-1} \frac{t}{(1-t)^{r}} \\ &= \sum_{k=0}^{\infty}(-1)^{k} D_{k}^{(r)} \frac{t^{k}}{k!} \sum _{l=0}^{\infty} (-1)^{l} C_{l}^{(r-1)} \frac{t^{l}}{l!} \sum_{m=0}^{\infty} (-r)_{m} (-1)^{m} \frac{t^{m+1}}{m!} \\ &= \sum_{k=0}^{\infty}(-1)^{k} D_{k}^{(r)} \frac{t^{k}}{k!} \sum _{l=0}^{\infty} (-1)^{l} C_{l}^{(r-1)} \frac{t^{l}}{l!} \sum_{m=1}^{\infty} (-r)_{m-1} m (-1)^{m-1} \frac{t^{m}}{m!} \\ &= \sum_{k=0}^{\infty}(-1)^{k} D_{k}^{(r)} \frac{t^{k}}{k!} \sum _{n=1}^{\infty} \sum_{l=0}^{n-1} \binom{n}{l} (-1)^{n-1} C_{l}^{(r-1)} (-r)_{n-l-1} (n-l) \frac{t^{n}}{n!} \\ &= \sum_{n=1}^{\infty} \sum _{k=0}^{n-1} \sum_{l=0}^{n-1} \binom{n}{k} \binom{n}{l} (-1)^{k+1} D_{n-k}^{(r)} C_{l}^{(r-1)} (-r)_{n-l-1} (n-l) \frac{t^{n}}{n!}. \end{aligned}
(17)

### Theorem 4

For any positive integern,
$$n! H_{n}^{(r)} = \sum _{k=0}^{n-1} \sum_{l=0}^{n-1} \binom{n}{k} \binom{n}{l} (-1)^{k+1} D_{n-k}^{(r)} C_{l}^{(r-1)} (-r)_{n-l-1} (n-l).$$
(18)
By multiplying the generating function of the hyperharmonic numbers by $$e^{-1}$$, the following can be observed:
\begin{aligned} - \frac{\log(1-t)}{(1-t)^{r}} e^{-t} &= \sum _{l=0}^{\infty} H_{l}^{(r)} t^{l} \sum_{k=0}^{\infty} \frac{(-1)^{k}}{k!} t^{k} \\ &= \sum_{n=0}^{\infty} \sum _{l=0}^{n} H_{l}^{(r)} \frac{(-1)^{n-l}}{(n-l)!} t^{n}. \end{aligned}
(19)
From (19), we get the following identity:
\begin{aligned} - \frac{\log(1-t)}{(1-t)^{r}} e^{-t} &= - \frac{\log(1-t)}{(1-t)^{r-1}} \frac{ e^{-t}}{1-t} \\ &= \sum_{l=0}^{\infty} H_{l}^{(r-1)} t^{l} \sum_{k=0}^{\infty} \frac{d_{k}}{k!} t^{k} \\ &= \sum_{n=0}^{\infty} \sum _{l=0}^{n} H_{l}^{(r-1)} \frac{d_{n-l}}{(n-l)!} t^{n}, \end{aligned}
(20)
where $$d_{k}$$ denotes the kth derangement number. From (19) and (20), we get the following identity.

### Theorem 5

For any positive integern,
$$\sum_{l=0}^{n} H_{l}^{(r)} \frac{(-1)^{n-l}}{(n-l)!} = \sum _{l=0}^{n} H_{l}^{(r-1)} \frac{d_{n-l}}{(n-l)!},$$
(21)
where$$d_{k}$$denotes thekth derangement number.

## 4 Some identities of hyperharmonic numbers and Daehee numbers arising from differential equations

From now on, throughout this article, we set
\begin{aligned} G & = G(t) =- \log(1-t), \\ F & = F(t) = \log(1+t), \end{aligned}
and
\begin{aligned}& F^{N} =\underbrace{F \times\cdots \times F}_{N\text{-times}}, \\ &F^{(0)}= F,\qquad F^{(N)} = \frac{d}{d t} F^{(N-1)}. \end{aligned}
In , Kwon et al. showed that $$F= F(t) = \log(1+t)$$ is a solution of the following differential equation:
$$F^{(N)}= (-1)^{N-1} (N-1)! \sum _{n=0}^{\infty} (-1)^{n} N^{n} \frac{F^{n}}{n!}.$$
(22)

From Eq. (22), some relationships between the Daehee numbers and other special numbers have been found .

In , the authors presented two identities,
\begin{aligned} F^{(N)} &= (-1)^{N-1} (N-1)!\sum _{n=0}^{\infty} \Biggl( \sum _{m=0}^{n}(-1)^{m} N^{m} S_{1} (n,m) \Biggr) \frac{t^{n}}{n!} \\ &= \sum_{n=0}^{\infty} (n+N) D_{n+N-1} \frac{t^{n}}{n!}. \end{aligned}
(23)
From the definition of G, we get
$$G' = \frac{1}{1-t} = e^{- \log(1-t)} = e^{G}.$$
(24)
By differentiation of both sides of Eq. (24), we get
\begin{aligned}& G''= e^{G} G' = e^{G} e^{G} = e^{2G}, \\ &G^{(3)}= e^{2G} (2 G)' = 2 e^{3G}. \end{aligned}
By repeating this process, we can easily get
$$G^{(N)} = (N-1)! e^{NG},\quad \text{for } N \ge 1.$$
(25)
From Eq. (12), we obtain
\begin{aligned} G^{(N)} &= (N-1)! e^{NG} \\ &= N! \sum_{m=0}^{\infty} N^{m-1} \frac{ G^{m}}{m!} \\ &= N! \sum_{m=0}^{\infty} N^{m-1} \frac{ (- \log(1-t) )^{m}}{m!} \\ &= N! \sum_{m=0}^{\infty} N^{m-1} (-1)^{m} \sum_{n=m}^{\infty} (-1)^{n} S_{1} (n,m) \frac{t^{n}}{n!} \\ &= N! \sum_{n=0}^{\infty} \sum _{m=0}^{n} N^{m-1} (-1)^{n+m} S_{1} (n,m) \frac{t^{n}}{n!}. \end{aligned}
(26)
From the definition of G and the hyperharmonic numbers,
\begin{aligned} G^{(N)} &= \biggl( \frac{d}{dt} \biggr)^{N} \biggl( \frac{-\log(1-t)}{(1-t)^{r}} \cdot (1-t)^{r} \biggr) \\ &= \biggl( \frac{d}{dt} \biggr)^{N} \Biggl( \sum _{m=0}^{\infty}H_{m}^{(r)} t^{m} \sum_{k=0}^{\infty} \binom{r}{k}(-1)^{k} t^{k} \Biggr) \\ &= \biggl( \frac{d}{dt} \biggr)^{N} \Biggl( \sum _{n=0}^{\infty} \sum_{k=0}^{n} \binom{r}{n-k} (-1)^{n-k} H_{k}^{(r)} t^{n} \Biggr) \\ &= \sum_{n=0}^{\infty} \sum _{k=0}^{n} \biggl( \frac{d}{dt} \biggr)^{N} \binom{r}{n-k} (-1)^{n-k} H_{k}^{(r)} t^{n} \\ &= \sum_{n=N}^{\infty} \sum _{k=0}^{n} \binom{r}{n-k} (-1)^{n-k} H_{k}^{(r)} (n)_{N} t^{n-N} \\ &= \sum_{n=0}^{\infty} \sum _{k=0}^{n+N} \binom{r}{n+N-k} (-1)^{n+N-k} H_{k}^{(r)} (n+N)_{N} t^{n}. \end{aligned}
(27)

Equations (26) and (27) yield the following theorem.

### Theorem 6

For any positive integerNand non-negative integern,
$$\frac{1}{n!} \sum_{k=0}^{n} N^{k-1} (-1)^{k} S_{1} (n,k) = \binom{n+N}{N} \sum_{k=0}^{n+N} \binom{r}{n+N-k} (-1)^{N-k} H_{k}^{(r)}.$$
We note that
$$F(t) = - G (-t).$$
(28)
From (28), we have
$$F^{(N)}(t) = (-1)^{N+1} G^{(N)} (-t).$$
(29)
Apply (29) to (27), then
\begin{aligned} (-1)^{N+1} G^{(N)} (-t) &= (-1)^{N+1} \sum_{n=0}^{\infty} \sum _{k=0}^{n+N} \binom{r}{n+N-k} (-1)^{N-k} H_{k}^{(r)} (n+N)_{N} t^{n} \\ &= \sum_{n=0}^{\infty} \sum _{k=0}^{n+N} \binom{r}{n+N-k} (-1)^{k+1} H_{k}^{(r)} (n+N)_{N} t^{n}. \end{aligned}
(30)

The definition of the Daehee numbers (3), (29) and (30) yields the following identity. This is a kind of inversion formula associated with Theorem 3.

### Theorem 7

For any positive integerN,
$$D_{n+N-1} = (n+N-1)_{N-1} \sum_{k=0}^{n+N} \binom{r}{n+N-k} (-1)^{k+1} H_{k}^{(r)}.$$
From the definition of higher-order Daehee numbers (5),
\begin{aligned} G^{m} &= \bigl( - \log(1-t) \bigr)^{m} \\ &= \biggl( \frac{ \log(1-t)}{-t} \biggr)^{m} t^{m} \\ &= \sum_{l=0}^{\infty} (-1)^{l} D_{l}^{(m)} \frac{t^{l+m}}{l!}. \end{aligned}
(31)
Let us observe Eq. (27) in a different way:
\begin{aligned} G^{(N)} &= (N-1)! e^{NG} \\ &= (N-1)! \sum_{m=0}^{\infty} N^{m} \frac{ G^{m}}{m!} \\ &= (N-1)! \sum_{m=0}^{\infty} \frac{N^{m}}{m!} \sum_{k=0}^{\infty} (-1)^{k} D_{k}^{(m)} (k+m)_{m} \frac{t^{k+m}}{(k+m)!} \\ &= (N-1)! \sum_{n=0}^{\infty} \sum _{k=0}^{n}(-1)^{k} \binom{n}{k} N^{n-k} D_{k}^{(n-k)} \frac{t^{n}}{n!}. \end{aligned}
(32)

From (27) and (32), we have a relation between hyperharmonic and higher-order Daehee numbers.

### Theorem 8

For any positive integerN,
\begin{aligned} \sum_{k=0}^{n+N} & \binom{r}{n+N-k} (-1)^{n+N-k} H_{k}^{(r)} (n+N)_{N} n! \\ & = (N-1)! \sum_{k=0}^{n}(-1)^{k} \binom{n}{k} N^{n-k} D_{k}^{(n-k)}. \end{aligned}
Substituting $$1-e^{t}$$ instead of t at (14) and (15), we have
\begin{aligned} G^{(N)} \bigl( 1 - e^{t} \bigr) &= (N-1)! \sum_{n=0}^{\infty} (-1)^{n} N^{n} \frac{t^{n}}{n!} \end{aligned}
(33)
and
\begin{aligned} & G^{(N)} \bigl( 1 - e^{t} \bigr) \\ &\quad= \sum_{m=0}^{\infty} \sum _{k=0}^{m+N} \binom{r}{m+N-k} (-1)^{N-k} H_{k}^{(r)} (m+N)_{N} \bigl(e^{t}-1 \bigr)^{m} \\ &\quad= \sum_{m=0}^{\infty} \sum _{k=0}^{m+N} \sum_{n=m}^{\infty} \binom{r}{m+N-k} (-1)^{N-k} H_{k}^{(r)} (m+N)_{N} m! S_{2} (n,m) \frac{t^{n}}{n!} \\ &\quad= \sum_{n=0}^{\infty} \sum _{m=0}^{n} \sum_{k=0}^{m+N} \binom{r}{m+N-k} (-1)^{N-k} H_{k}^{(r)} (m+N)_{N} m! S_{2} (n,m) \frac{t^{n}}{n!}. \end{aligned}
(34)

From (33) and (34), we have the following theorem.

### Theorem 9

For any positive integerNand non-negative integern,
$$(-1)^{n} (N-1)! N^{n} = \sum_{m=0}^{n} \sum_{k=0}^{m+N} \binom{r}{m+N-k} (-1)^{N-k} H_{k}^{(r)} (m+N)_{N} m! S_{2} (n,m).$$

## 5 Results and discussion

In this paper, we have studied the harmonic, the hyperharmonic, the Daehee and the higher-order Daehee numbers which are different from the previous research articles. In Sect. 2, we present some elementary identities between the harmonic and the hyperharmonic numbers. In Sect. 3, we study some relations and properties for the harmonic and the hyperharmonic numbers, the Daehee and the higher-order Daehee numbers. Additionally, the derangement numbers and the Cauchy numbers are also studied in Sect. 3. In Sect. 4, we study a nonlinear differential equation arising from the generating function of the harmonic numbers and we give some identities of harmonic and hyperharmonic numbers, the Daehee and higher-order Daehee numbers which are derived from this nonlinear differential equation.

## 6 Conclusion

For a long time, research on the harmonic numbers was mainly focused on the study of inequalities. In this paper, we tried to study of the inequalities of the harmonic numbers by showing the relationship between harmonic numbers with other special numbers.

## References

1. 1.
Benjamin, A.T., Gaebler, D., Gaebler, R.: A combinatorial approach to hyperharmonic numbers. Integers 3(A15), 1–9 (2003)
2. 2.
Cheon, G.S., Mikkawy, M.E.A.: Generalized harmonic number identities and a related matrix representation. J. Korean Math. Soc. 44, 487–498 (2007)
3. 3.
Conway, J.H., Guy, R.K.: The Book of Numbers. Springer, New York (1996)
4. 4.
Dil, A., Kurt, V.: Polynomials related to harmonic numbers and evaluation of harmonic number series. Integers 12, 38 (2012)
5. 5.
Dil, A., Mezo, I.: A symmetric algorithm for hyperharmonic and Fibonacci numbers. Appl. Math. Comput. 206, 942–951 (2008)
6. 6.
Do, Y., Lim, D.: On $$(h, q)$$-Daehee numbers and polynomials. Adv. Differ. Equ. 2015, 107 (2015)
7. 7.
Dolgy, D.V., Kim, D.S., Kim, T., Mansour, T.: Barnes-type Daehee with λ-parameter and degenerate Euler mixed-type polynomials. J. Inequal. Appl. 2015, 154 (2015).
8. 8.
El-Desouky, B.S., Mustafa, A.: New results on higher-order Daehee and Bernoulli numbers and polynomials. Adv. Differ. Equ. 2016, 32 (2016).
9. 9.
El-Desouky, B.S., Mustafa, A., Abdel-Moneim, F.M.: Multiparameter higher order Daehee and Bernoulli numbers and polynomials. Appl. Math. 2017, 775–785 (2017).
10. 10.
Graham, R., Knuth, D., Patashnik, K.: Concrete Mathematics. Addison-Wesley, Reading (1989)
11. 11.
Jang, G.W., Kim, T.: Revisit of identities of Daehee numbers arising from nonlinear differential equations. Proc. Jangjeon Math. Soc. 20, 163–177 (2017)
12. 12.
Jang, G.W., Kim, T.: Some identities of ordered Bell numbers arising from differential equation. Adv. Stud. Contemp. Math. 27, 385–397 (2017)
13. 13.
Jang, G.W., Kwon, J., Lee, J.G.: Some identities of degenerate Daehee numbers arising from nonlinear differential equation. Adv. Differ. Equ. 2017, 206 (2017)
14. 14.
Kim, D.S., Kim, T.: Some identities for Bernoulli numbers of the second kind arising from a non-linear differential equation. Bull. Korean Math. Soc. 52, 2001–2010 (2015)
15. 15.
Kim, D.S., Kim, T.: A note on nonlinear Changhee differential equations. Russ. J. Math. Phys. 23, 88–92 (2016)
16. 16.
Kim, D.S., Kim, T.: Identities involving degenerate Euler numbers and polynomials arising from non-linear differential equations. J. Nonlinear Sci. Appl. 9, 2086–2098 (2016)
17. 17.
Kim, D.S., Kim, T.: On degenerate Bell numbers and polynomials. Rev. R. Acad. Cienc. Exactas Fís. Nat., Ser. A Mat. 111(2), 435–446 (2017)
18. 18.
Kim, D.S., Kim, T., Mansour, T., Seo, J.J.: Linear differential equations for families of polynomials. J. Inequal. Appl. 2016, Article ID 95 (2016).
19. 19.
Kim, D.S., Kim, T.: Seo, J.J.: Higher-order Daehee polynomials of the first kind with umbral calculs. Adv. Stud. Contemp. Math. (Kyungshang) 24(1), 5–18 (2014)
20. 20.
Kim, T.: Identities involving Frobenius–Euler polynomials arising from non-linear differential equations. J. Number Theory 132, 2854–2865 (2012)
21. 21.
Kim, T., Kim, D.S.: Identities involving harmonic and hyperharmonic numbers. Adv. Differ. Equ. 2013, 235 (2013)
22. 22.
Kim, T., Kim, D.S.: Degenerate Laplace transform and degenerate gamma function. Russ. J. Math. Phys. 24, 241–248 (2017)
23. 23.
Kim, T., Kim, D.S., Hwang, K.W., Seo, J.J.: Some identities of Laguerre polynomials arising from differential equations. Adv. Differ. Equ. 2016, 159 (2016)
24. 24.
Kim, T., Kim, D.S., Komatsu, T., Lee, S.H.: Higher-order Daehee of the second kind and poly-Cauchy of the second kind mixed-type polynomials. J. Nonlinear Convex Anal. 16, 1993–2015 (2015)
25. 25.
Kim, T., Kim, D.S., Kwon, H.I., Seo, J.J.: Revisit nonlinear differential equations associated with Bernoulli numbers of the second kind. Glob. J. Pure Appl. Math. 12, 1893–1901 (2016) Google Scholar
26. 26.
Kim, T., Kim, D.S., Kwon, H.I., Seo, J.J.: Differential equations arising from the generating function of general modified degenerate Euler numbers. Adv. Differ. Equ. 2016, 129 (2016)
27. 27.
Kim, T., Yao, Y., Kim, D.S., Jang, G.W.: Degenerate r-Stirling numbers and r-Bell polynomials. Russ. J. Math. Phys. 25, 44–58 (2018)
28. 28.
Kwon, H.I., Kim, T., Seo, J.J.: A note on Daehee numbers arising from differential equations. Glob. J. Pure Appl. Math. 12(3), 2349–2354 (2016) Google Scholar
29. 29.
Park, J.W., Kwon, J.: A note on the degenerate high order Daehee polynomials. Appl. Math. Sci. 9, 4635–4642 (2015) Google Scholar
30. 30.
Pyo, S.S., Kim, T., Rim, S.H.: Identities of the degenerate Daehee numbers with the Bernoulli numbers of the second kind arising from nonlinear differential equation. J. Nonlinear Sci. Appl. 10, 6219–6228 (2017)
31. 31.
Simsek, Y.: Identities on the Changhee numbers and Apostol-type Daehee polynomials. Adv. Stud. Contemp. Math. (Kyungshang) 27, 199–212 (2017)