[Paper Review] Evaluation of some simple Euler-type series

Khristo N. Boyadzhiev|arXiv (Cornell University)|Dec 16, 2009

Advanced Mathematical Identities5 references3 citations

TL;DR

This paper evaluates five Euler-type series involving harmonic numbers, Stirling numbers of the first kind, and polylogarithms, reducing their evaluation to integrals of the polylogarithm function. The key contribution is the explicit evaluation of these series in terms of Riemann zeta values, establishing new connections between special functions and zeta constants.

ABSTRACT

Five series are evaluated in terms of zeta values. Three of the series involve harmonic numbers and one involves Stirling numbers of the first kind. The evaluation of these series is reduced to the evaluation of certain integrals, including the moments of the polylogarithm.

Motivation & Objective

• To evaluate specific Euler-type series involving harmonic numbers and Stirling numbers of the first kind.
• To reduce the evaluation of these series to integrals of the polylogarithm function.
• To express the results in terms of Riemann zeta values.
• To establish new identities connecting special functions and zeta constants.

Proposed method

• Reduction of the series to definite integrals involving the polylogarithm function.
• Use of integral representations of harmonic numbers and Stirling numbers of the first kind.
• Application of known integral formulas for polylogarithms and their moments.
• Employment of generating functions and series manipulation techniques.
• Leveraging known results from classical analysis and number theory.
• Transformation of series into closed-form expressions via integral evaluation.

Experimental results

Research questions

• RQ1 How can Euler-type series with harmonic numbers be evaluated in closed form?
• RQ2 What is the relationship between Stirling numbers of the first kind and zeta values through series summation?
• RQ3 Can moments of the polylogarithm function be used to evaluate divergent or conditionally convergent series?
• RQ4 What integral representations underlie the evaluation of these Euler-type series?
• RQ5 How do these series connect to known zeta function identities?

Key findings

• Three of the five series evaluated involve harmonic numbers and are expressed in terms of zeta values.
• One series involving Stirling numbers of the first kind is evaluated using integral representations and zeta function identities.
• The evaluation process relies on computing moments of the polylogarithm function.
• The results demonstrate a direct link between Euler-type series and special values of the Riemann zeta function.
• The paper provides explicit closed-form expressions for the series, confirming their convergence to known zeta constants.
• The method establishes a systematic approach to evaluating similar series through integral transforms.