On Some Continued Fractions and Divergent Series Arising From Integral Families

TL;DR

This work presents a general method to derive Eulerian continued fractions from families of integrals, unifying representations for classical constants and special functions. It yields continued fractions for the natural logarithm $\log$, the Riemann zeta function $\zeta(s)$, and polylogarithms $\mathrm{Li}_s(z)$, and derives several new identities. It also constructs a divergent continued fraction that, via a novel C-Summation, assigns a value to the Euler-Mascheroni constant $\gamma$, illustrating the method's potential for summing certain divergent series. Overall, the paper provides a systematic, integrals-based tool for both convergent and certain divergent sums with broad applicability and room for future development.

Abstract

In this paper we present a method to derive Eulerian continued fractions arising from a sequence of integrals. As examples, through a new derivation, we reproduce classical continued fraction expansions for the natural logarithm, the Riemann zeta function $ζ(s)$, and polylogarithms, while also obtaining several new identities. Finally, we apply the method to construct a divergent continued fraction, which provides a natural assignment of the Euler Mascheroni constant $γ$ as the sum of a particular divergent series through a new summation method which we propose.

Theorems & Definitions (29)

• Theorem 2.1: Euler Continued Fraction Theorem
• Theorem 3.1
• proof
• Theorem 4.1
• proof
• Theorem 4.2
• proof
• Lemma 5.1
• proof
• Theorem 5.2