The generalized harmonic numbers of order s are defined by (cf. ; see also [2, 3], [, p.156] and [, Section 3.5])
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
The generalized harmonic functions are defined by (see [2, 6]; see also [7, 8])
so that, obviously,
Equation (1.1) can be written in the following form:
by recalling the well-known (easily-derivable) relationship between the Riemann zeta function and the Hurwitz (or generalized) zeta function (see [, Eq. 2.3(9)])
The polygamma functions () are defined by
where is the familiar gamma function, and the psi-function ψ is defined by
A well-known (and potentially useful) relationship between the polygamma functions and the generalized zeta function is given by
It is also easy to have the following expression (cf. [, Eq. 1.2(54)]):
which immediately gives another expression for as follows (cf. [, Eq. (20)]):
By using finite differences, Spivey  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: [, Identity 14]
which was also given by Paule and Schneider [, Eq. (39)] by deriving it automatically by means of the Sigma package in , together with the following identity [, Identity 20]:
Paule and Schneider  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 [, Eq. (5)]:
Greene and Knuth [, p.10] recorded six commonly used identities that involve both binomial coefficients and harmonic numbers, two of which are recalled here:
Alzer et al. [, Eq. (3.62)] proved, by using the principle of mathematical induction, that
By using (1.15) in conjunction with the following elementary identity (see ):
Chu and De Donno  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 , one of which is recalled below [, Thereoem 1].
One interesting special case of (1.18) is when we set . We thus find that
which can be further specialized, with , to the following form:
Dattolli and Srivastava  proposed several generating functions involving harmonic numbers by making use of an interesting approach based on the umbral calculus. Subsequently, Cvijović  showed the truth of the conjectured relations in  by using simple analytical arguments.
For a concise and beautiful description of these numbers, we refer also to WolframMathWorld’s website .
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., ). Here we aim at presenting further interesting identities about certain interesting finite series associated with binomial coefficients, harmonic numbers and generalized harmonic numbers.