The generalized harmonic numbers of order s are defined by (cf. [1]; see also [2, 3], [[4], p.156] and [[5], Section 3.5])
(1.1)
and
(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
The generalized harmonic functions are defined by (see [2, 6]; see also [7, 8])
(1.3)
so that, obviously,
Equation (1.1) can be written in the following form:
(1.4)
by recalling the well-known (easily-derivable) relationship between the Riemann zeta function and the Hurwitz (or generalized) zeta function (see [[4], Eq. 2.3(9)])
(1.5)
The polygamma functions () are defined by
(1.6)
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
(1.7)
It is also easy to have the following expression (cf. [[4], Eq. 1.2(54)]):
(1.8)
which immediately gives another expression for as follows (cf. [[9], Eq. (20)]):
(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]
(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]:
(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)]:
(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
(1.15)
By using (1.15) in conjunction with the following elementary identity (see [2]):
(1.16)
we obtain
(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 . We thus find that
(1.19)
which can be further specialized, with , to the following form:
(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.