Hypergeometric Moments and Hecke Trace Formulas

TL;DR

This paper extends the study of hypergeometric moments over finite fields by generalizing previous results to new cases using Hecke trace formulas, and proposes conjectures for further moments.

Contribution

It introduces new hypergeometric moments over $ ext{Q}$ and primitive data, utilizing recent Hecke trace formulas, and provides algebraic formulas and conjectures for additional moments.

Findings

Generalized hypergeometric moments over $ ext{Q}$ and primitive data.

Established algebraic formulas in finite field setting.

Proposed conjectures for additional $_{2}F_{1}$ and $_{3}F_{2}$ moments.

Abstract

Moments for hypergeometric functions over finite fields were studied in the work of Ono, Pujahari, Saad, and Saikia for several $_{2}F_{1}$ and $_{3}F_{2}$ cases. We generalize their work to prove results for new cases where the hypergeometric data is defined over $\mathbb{Q}$ and primitive. These new moments are established using Hecke trace formulas of hypergeometric origin recently established by Hoffman, Li, Long, and Tu. We also obtain several algebraic formulas in the finite field setting and present conjectures for additional $_{2}F_{1}$ and $_{3}F_{2}$ moments.