Hyperharmonic series involving Hurwitz zeta function

Abstract

We show that the sum of the series formed by the so-called hyperharmonic numbers can be expressed in terms of the Riemann zeta function. These results enable us to reformulate Euler's formula involving the Hurwitz zeta function. In additon, we improve Conway and Guy's formula for hyperharmonic numbers.

Polynomials related to harmonic numbers and evaluation of harmonic number series II

“…These numbers can be expressed in terms of binomial coefficients and ordinary harmonic numbers as: [4,12,19]:…”

“…As in the classical case, r-exponential numbers can be defined by setting x = 1 in (19), i.e., r φ n := n k=0 n k r .…”

Approximation by special values of harmonic zeta function and log-sine integrals

“…The combinatorial significance of these numbers was investigated by Benjamin et al [15]. Further combinatorial connections and relations between values of the hyperharmonic zeta function and the Hurwitz zeta function were nicely treated by Mezö and Dil [48]. Our main result below shows that specific log-sine type integrals and consequently all real numbers can be strongly approximated by linear combinations of even values of the harmonic zeta function teamed up by the odd values of the Riemann zeta function.…”

Exponential generating function of hyperharmonic numbers indexed by arithmetic progressions

“…In contrast, this is not the case for hyperharmonic numbers. There are just some sporadic papers dealing with such sums, see for instance [24,26,27]. In the present paper, we try to proceed further in this direction.…”