Divergent series in Gauss's diary and their extensions

TL;DR

This work revisits two divergent $q$-series from Gauss's diary and generalizes them by introducing parameterized families $S_{rho}^{(1)}(q)$ and $S_{kappa}^{(2)}(q)$. It employs Rogers-Fine identities, Ramanujan's continued fractions, and Heine transformations to obtain equivalent convergent representations and explicit sums via analytic continuation, enabling meaningful values for $q>1$ as well as $q<1$. The key contributions include explicit formulas and theorems (Theorems 1 and 2) that express the extended sums as alternating $q$-series and as continued fractions, along with special-case analyses (e.g., self-duality at $\rho=4$ and κ-cases). The approach provides a unifying, regularization-based framework for divergent Gauss-type series that connects classical series transformations with modern analytic continuation concepts, with potential implications for zeta-regularization-style arguments in basic hypergeometric contexts.

Abstract

Two divergent series in Entry 7 of Gauss's diary are extended systematically by introducing additional parameters. Rogers-Fine identities, Ramanujan's continued fractions and Heine's transformation relations of basic hypergeometric series are applied to find equivalent alternative series, which enable us to compute the sums of divergent series in question.