The generalized harmonic numbers {H}_{n}^{(s)} of order *s* are defined by (*cf*. [1]; see also [2, 3], [[4], p.156] and [[5], Section 3.5])

{H}_{n}^{(s)}:=\sum _{j=1}^{n}\frac{1}{{j}^{s}}\phantom{\rule{1em}{0ex}}(n\in \mathbb{N};s\in \mathbb{C}),

(1.1)

and

{H}_{n}:={H}_{n}^{(1)}=\sum _{j=1}^{n}\frac{1}{j}\phantom{\rule{1em}{0ex}}(n\in \mathbb{N})

(1.2)

are the harmonic numbers. Here ℕ and ℂ denote the set of positive integers and the set of complex numbers, respectively, and we assume that

{H}_{0}:=0,\phantom{\rule{2em}{0ex}}{H}_{0}^{(s)}:=0\phantom{\rule{1em}{0ex}}(s\in \mathbb{C}\setminus \{0\})\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}{H}_{0}^{(0)}:=1.

The generalized harmonic functions {H}_{n}^{(s)}(z) are defined by (see [2, 6]; see also [7, 8])

{H}_{n}^{(s)}(z):=\sum _{j=1}^{n}\frac{1}{{(j+z)}^{s}}\phantom{\rule{1em}{0ex}}(n\in \mathbb{N};s\in \mathbb{C}\setminus {\mathbb{Z}}^{-};{\mathbb{Z}}^{-}:=\{-1,-2,-3,\dots \})

(1.3)

so that, obviously,

{H}_{n}^{(s)}(0)={H}_{n}^{(s)}.

Equation (1.1) can be written in the following form:

{H}_{n}^{(s)}=\zeta (s)-\zeta (s,n+1)\phantom{\rule{1em}{0ex}}(\mathrm{\Re}(s)>1;n\in \mathbb{N})

(1.4)

by recalling the well-known (easily-derivable) relationship between the Riemann zeta function \zeta (s) and the Hurwitz (or generalized) zeta function \zeta (s,a) (see [[4], Eq. 2.3(9)])

\zeta (s)=\zeta (s,n+1)+\sum _{k=1}^{n}{k}^{-s}\phantom{\rule{1em}{0ex}}(n\in {\mathbb{N}}_{0}:=\mathbb{N}\cup \{0\}).

(1.5)

The polygamma functions {\psi}^{(n)}(s) (n\in \mathbb{N}) are defined by

{\psi}^{(n)}(s):=\frac{{d}^{n+1}}{d{z}^{n+1}}log\mathrm{\Gamma}(s)=\frac{{d}^{n}}{d{s}^{n}}\psi (s)\phantom{\rule{1em}{0ex}}(n\in {\mathbb{N}}_{0};s\in \mathbb{C}\setminus {\mathbb{Z}}_{0}^{-}:={\mathbb{Z}}^{-}\cup \{0\}),

(1.6)

where \mathrm{\Gamma}(s) is the familiar gamma function, and the psi-function *ψ* is defined by

\psi (s):=\frac{d}{ds}log\mathrm{\Gamma}(s)\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}{\psi}^{(0)}(s)=\psi (s)\phantom{\rule{1em}{0ex}}(s\in \mathbb{C}\setminus {\mathbb{Z}}_{0}^{-}).

A well-known (and potentially useful) relationship between the polygamma functions {\psi}^{(n)}(s) and the generalized zeta function \zeta (s,a) is given by

{\psi}^{(n)}(s)={(-1)}^{n+1}n!\sum _{k=0}^{\mathrm{\infty}}\frac{1}{{(k+s)}^{n+1}}={(-1)}^{n+1}n!\zeta (n+1,s)\phantom{\rule{1em}{0ex}}(n\in \mathbb{N};s\in \mathbb{C}\setminus {\mathbb{Z}}_{0}^{-}).

(1.7)

It is also easy to have the following expression (*cf*. [[4], Eq. 1.2(54)]):

{\psi}^{(m)}(s+n)-{\psi}^{(m)}(s)={(-1)}^{m}m!{H}_{n}^{(m+1)}(s-1)\phantom{\rule{1em}{0ex}}(m,n\in {\mathbb{N}}_{0}),

(1.8)

which immediately gives {H}_{n}^{(s)} another expression for {H}_{n}^{(s)} as follows (*cf*. [[9], Eq. (20)]):

{H}_{n}^{(m)}=\frac{{(-1)}^{m-1}}{(m-1)!}[{\psi}^{(m-1)}(n+1)-{\psi}^{(m-1)}(1)]\phantom{\rule{1em}{0ex}}(m\in \mathbb{N};n\in {\mathbb{N}}_{0}).

(1.9)

By using finite differences, Spivey [10] presented many summation formulas involving binomial coefficients, the Stirling numbers of the first and second kind and harmonic numbers, two of which are chosen to be recalled here: [[10], Identity 14]

\sum _{k=0}^{n}\left(\genfrac{}{}{0ex}{}{n}{k}\right){H}_{k}={2}^{n}({H}_{n}-\sum _{k=1}^{n}\frac{1}{k{2}^{k}})\phantom{\rule{1em}{0ex}}(n\in {\mathbb{N}}_{0}),

(1.10)

which was also given by Paule and Schneider [[11], Eq. (39)] by deriving it automatically by means of the *Sigma* package in [12], together with the following identity [[10], Identity 20]:

\sum _{k=0}^{n}{(-1)}^{k}\left(\genfrac{}{}{0ex}{}{n}{k}\right){H}_{k}=-\frac{1}{n}\phantom{\rule{1em}{0ex}}(n\in \mathbb{N}).

(1.11)

Paule and Schneider [11] proved five conjectured harmonic number identities similar to those arising in the context of supercongruences for Apéry numbers, one of which is recalled here as follows [[11], Eq. (5)]:

\sum _{j=0}^{n}(1-5j{H}_{j}+5j{H}_{n-j}){\left(\genfrac{}{}{0ex}{}{n}{j}\right)}^{5}={(-1)}^{n}\sum _{j=0}^{n}{\left(\genfrac{}{}{0ex}{}{n}{j}\right)}^{2}\left(\genfrac{}{}{0ex}{}{n+j}{j}\right).

(1.12)

Greene and Knuth [[13], p.10] recorded six commonly used identities that involve both binomial coefficients and harmonic numbers, two of which are recalled here:

Alzer *et al.* [[14], Eq. (3.62)] proved, by using the principle of mathematical induction, that

\sum _{j=1}^{n}\frac{{H}_{j}}{j}=\frac{1}{2}[{({H}_{n})}^{2}+{H}_{n}^{(2)}]\phantom{\rule{1em}{0ex}}(n\in \mathbb{N}).

(1.15)

By using (1.15) in conjunction with the following elementary identity (see [2]):

{H}_{j+1}={H}_{j}+\frac{1}{j+1},

(1.16)

we obtain

\sum _{j=1}^{n}\frac{{H}_{j}}{j+1}=\frac{1}{2}[{({H}_{n+1})}^{2}-{H}_{n+1}^{(2)}]\phantom{\rule{1em}{0ex}}(n\in \mathbb{N}).

(1.17)

Chu and De Donno [15] made use of the classical hypergeometric summation theorems to derive several striking identities for harmonic numbers other than those discovered recently by Paule and Schneider [11], one of which is recalled below [[15], Thereoem 1].

One interesting special case of (1.18) is when we set \mu =0. We thus find that

\sum _{k=0}^{n}\left(\genfrac{}{}{0ex}{}{n}{k}\right)\left(\genfrac{}{}{0ex}{}{n+\lambda n}{n-k}\right){H}_{\lambda n+k}=\left(\genfrac{}{}{0ex}{}{2n+\lambda n}{n}\right)(2{H}_{\lambda n+n}-{H}_{\lambda n+2n}),

(1.19)

which can be further specialized, with \lambda =0, to the following form:

\sum _{k=0}^{n}{\left(\genfrac{}{}{0ex}{}{n}{k}\right)}^{2}{H}_{k}=\left(\genfrac{}{}{0ex}{}{2n}{n}\right)(2{H}_{n}-{H}_{2n}).

(1.20)

Dattolli and Srivastava [16] proposed several generating functions involving harmonic numbers by making use of an interesting approach based on the umbral calculus. Subsequently, Cvijović [17] showed the truth of the conjectured relations in [16] by using simple analytical arguments.

For a concise and beautiful description of these numbers, we refer also to WolframMathWorld’s website [18].

As we have seen in the above brief eclectic review, harmonic and generalized harmonic numbers are involved in a variety of useful identities. Of course, certain interesting properties of harmonic and generalized harmonic numbers have been studied (see, *e.g.*, [19]). Here we aim at presenting further interesting identities about certain interesting finite series associated with binomial coefficients, harmonic numbers and generalized harmonic numbers.