On Evaluations of Euler-type Sums of Hyperharmonic Numbers (original) (raw)

Euler sums of hyperharmonic numbers

Journal of Number Theory, 2015

The hyperharmonic numbers h (r) n are defined by means of the clasical harmonic numbers. We show that the Euler-type sums with hyperharmonic numbers:

Euler sums of generalized hyperharmonic numbers

2021

are the generalized hyperharmonic numbers (see [4, 10]). Furthermore, H (p,1) n = H (p) n = ∑n j=1 1/n p are the generalized harmonic numbers and H (1,r) n = h (r) n are the classical hyperharmonic numbers. In particularH (1,1) n = Hn are the classical harmonic numbers. Many researchers have been studying Euler sums of harmonic and hyperharmonic numbers (see [4, 6, 7, 9] and references therein), since they play

Summation formulas of hyperharmonic numbers with their generalizations II

2021

In 1990, Spieß gave some identities of harmonic numbers including the types of ∑n l=1 l Hl, ∑n l=1 l Hn−l and ∑n l=1 l HlHn−l. In this paper, we derive several formulas of hyperharmonic numbers including ∑n l=0 l h (r) l h (s) n−l and ∑n l=0 l (h (r) l ) . Some more formulas of generalized hyperharmonic numbers are also shown.

Euler sums of generalized alternating hyperharmonic numbers

2021

We define the notion of the generalized alternating hyperharmonic numbers, and show that Euler sums of the generalized alternating hyperharmonic numbers can be expressed in terms of linear combinations of classical (alternating) Euler sums.

Euler sums of generalized harmonic numbers and connected extensions

Applicable Analysis and Discrete Mathematics

This paper presents the evaluation of the Euler sums of generalized hyperharmonic numbers H(p,q)n ?H(p,q)(r) = ?Xn=1 H(p,q)n/nr in terms of the famous Euler sums of generalized harmonic numbers. Moreover, several infinite series, whose terms consist of certain harmonic numbers and reciprocal binomial coefficients, are evaluated in terms of the Riemann zeta values.

Summation formulas of q-hyperharmonic numbers

Afrika Matematika

In this paper, several weighted summation formulas of q-hyperharmonic numbers are derived. As special cases, several formulas of hyperharmonic numbers of type n ℓ=1 ℓ p H (r) ℓ and n ℓ=0 ℓ p H (r) n−ℓ are obtained.

Hypergeometric series and harmonic number identities

Advances in Applied Mathematics, 2005

The classical hypergeometric summation theorems are exploited to derive several striking identities on harmonic numbers including those discovered recently by Paule and Schneider (2003).

Some summation formulas involving harmonic numbers and generalized harmonic numbers

Mathematical and Computer Modelling, 2011

Harmonic numbers and generalized harmonic numbers have been studied since the distant past and are involved in a wide range of diverse fields such as analysis of algorithms in computer science, various branches of number theory, elementary particle physics and theoretical physics. Here we aim at presenting further interesting identities about certain finite or infinite series involving harmonic numbers and generalized harmonic numbers. We also give a brief eclectic review on some known properties associated with the harmonic and generalized harmonic numbers.

Summation formulae involving multiple harmonic numbers

Applicable Analysis and Discrete Mathematics, 2021

By means of the generating function approach, we derive several summation formulae involving multiple harmonic numbers Hn,? (?), as well as other combinatorial numbers named after Bernoulli, Euler, Bell, Genocchi and Stirling.