Euler sums of generalized 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:

On Evaluations of Euler-type Sums of Hyperharmonic Numbers

arXiv (Cornell University), 2021

We give explicit evaluations of the linear and non-linear Euler sums of hyperharmonic numbers h (r) n with reciprocal binomial coefficients. These evaluations enable us to extend closed form formula of Euler sums of hyperharmonic numbers to an arbitrary integer r. Moreover, we reach at explicit formulas for the shifted Euler-type sums of harmonic and hyperharmonic numbers. All the evaluations are provided in terms of the Riemann zeta values, harmonic numbers and linear 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 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.

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.

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.

On connection between values of Riemann zeta function at integers and generalized harmonic numbers

arXiv: Number Theory, 2015

Using Euler transformation of series we relate values of Hurwitz zeta function at integer and rational values of arguments to certain rapidly converging series where some generalized harmonic numbers appear. The form of these generalized harmonic numbers carries information about the values of the arguments of Hurwitz function. In particular we prove: forallkinmathbbN:\forall k\in \mathbb{N}:forallkinmathbbN: zeta(k,1)allowbreak=allowbreakfrac2k−12k−1−1\zeta (k,1)\allowbreak =\allowbreak \frac{2^{k-1}}{2^{k-1}-1}% \sum_{n=1}^{\infty }\frac{H_{n}^{(k-1)}}{n2^{n}},zeta(k,1)allowbreak=allowbreakfrac2k−12k−1−1 where Hn(k)H_{n}^{(k)}Hn(k) are defined below generalized harmonic numbers. Further we find generating function of the numbers hatzeta(k)=sumj=1infty(−1)j−1/jk.\hat{\zeta}(k)=\sum_{j=1}^{\infty }(-1)^{j-1}/j^{k}. hatzeta(k)=sumj=1infty(−1)j−1/jk.

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.

On connection between values of Riemann zeta function at rationals and generalized harmonic numbers

Mathematica Eterna, 2015

Using Euler transformation of series, we relate values of Hurwitz zeta function (s; t) at integer and rational values of arguments to certain rapidly converging series, where some generalized harmonic numbers appear. Most of the results of the paper can be derived from the recent, more advanced results, on the properties of Arakawa-Kaneko zeta functions. We derive our results directly, by solving simple recursions. The form of mentioned above generalized harmonic numbers carries information, about the values of the arguments of Hurwitz function. In particular we prove: 8k 2 N : (k; 1) = (k) = 2 k 1 2 k 1 1