Reciprocity of Skew Hall-Littlewood-Schubert Series

TL;DR

The paper introduces a new generalized rational function that unifies and refines previous functions, proving its self-reciprocity without using p-adic integration, thus answering an open problem.

Contribution

It presents a simultaneous generalization and refinement of Igusa functions and Hall-Littlewood-Schubert series, establishing their self-reciprocity through elementary methods.

Findings

Established self-reciprocity of the new generalized function.

Unified previous results on Igusa functions and Hall-Littlewood-Schubert series.

Provided an elementary proof avoiding p-adic integration.

Abstract

Carnevale, Schein and Voll proved self-reciprocity of the generalized Igusa functions, and Maglione and Voll did the same for the Hall-Littlewood-Schubert series. We introduce a simultaneous generalization and refinement of these two rational functions, and prove that it satisfies a self-reciprocity property. This answers a problem posed by Maglione and Voll. Our method of proof is elementary, avoiding the use of $p$-adic integration.