Real quadratic double sums

TL;DR

This paper introduces new q-hypergeometric double sums that serve as generating functions for counting ideals of specific norms in rings of integers of real quadratic fields, expanding the understanding of their arithmetic properties.

Contribution

It provides a dozen new q-hypergeometric double sums linked to real quadratic fields and proves related identities, advancing the study of these series and their algebraic significance.

Findings

New q-hypergeometric double sums for real quadratic fields

Identities relating these sums to ideal counts

Enhanced understanding of arithmetic in quadratic fields

Abstract

In 1988, Andrews, Dyson and Hickerson initiated the study of q-hypergeometric series whose coefficients are dictated by the arithmetic in real quadratic fields. In this paper, we provide a dozen q-hypergeometric double sums which are generating functions for the number of ideals of a given norm in rings of integers of real quadratic fields and prove some related identities.