Cyclotomic valuation of $q$-Pochhammer symbols and $q$-integrality of basic hypergeometric series

TL;DR

This paper develops a formula for the cyclotomic valuation of $q$-Pochhammer symbols using Dwork maps and establishes a criterion for the $q$-integrality of basic hypergeometric series, extending Christol's results to the $q$-analog setting.

Contribution

It introduces a new formula for cyclotomic valuation of $q$-Pochhammer symbols and a criterion for $q$-integrality of basic hypergeometric series, generalizing Christol's $p$-adic results.

Findings

Derived a formula for cyclotomic valuation using generalized Dwork maps.

Established a criterion for $q$-integrality based on step functions.

Extended Christol's $p$-adic valuation results to the $q$-analog context.

Abstract

We give a formula for the cyclotomic valuation of $q$-Pochhammer symbols in terms of (generalized) Dwork maps. We also obtain a criterion for the $q$-integrality of basic hypergeometric series in terms of certain step functions, which generalize Christol step functions. This provides suitable $q$-analogs of two results proved by Christol: a formula for the $p$-adic valuation of Pochhammer symbols and a criterion for the $N$-integrality of hypergeometric series.