Euler sums of generalized harmonic numbers and connected extensions

Generalized Reciprocity Laws for Sums of Harmonic Numbers

We present summation identities for generalized harmonic numbers, which generalize reciprocity laws discovered when studying the algorithm quickselect. Furthermore, we demonstrate how the computer algebra system Sigma can be used in order to find/prove such identities. We also discuss alternating harmonic sums, as well as limiting relations.

On Stirling Numbers and Euler Sums

In this paper, we propose the another yet generalization of Stirling numbers of the rst kind for non-integer values of their arguments. We discuss the analytic representations of Stirling numbers through harmonic numbers, the generalized hypergeometric function and the logarithmic beta integral. We present then in nite series involving Stirling numbers and demonstrate how they are related to Eu...

Some summation formulas involving harmonic numbers and generalized harmonic numbers

Euler and the Partial Sums of the Prime Harmonic Series

Abstract. In a 1737 paper, Euler gave the first proof that the sum of the reciprocals of the prime numbers diverges. That paper can be considered as the founding document of analytic number theory, and its key innovation — socalled Euler products — are now ubiquitous in the field. In this note, we probe Euler’s claim there that “the sum of the reciprocals of the prime numbers” is “as the logari...

Sums of Products of q-Euler Polynomials and Numbers

The purpose of this paper is to derive formulae for the sums of products of the q-Euler polynomials and numbers, since many identities can be obtained from our sums of products of the q-Euler polynomials and numbers. In 1 , Simsek evaluated the complete sums for the Euler numbers and polynomials and obtained some identities related to Euler numbers and polynomials from his complete sums, and Ja...