On the $q$-analogue of P\'olya's Theorem

TL;DR

This paper investigates the $q$-analogue of Pólya's theorem, revealing that the algebraicity of diagonals of bivariate rational power series depends on whether $q$ is a root of unity, providing a negative answer in general.

Contribution

It answers a long-standing question about the $q$-analogue of Pólya's theorem, showing the conditions under which the algebraicity property holds or fails.

Findings

Negative answer to Aissen's question in general

Algebraicity holds if and only if $q$ is a root of unity

Clarifies the role of $q$ as a complex parameter in the theorem

Abstract

We answer a question posed by Michael Aissen in 1979 about the $q$-analogue of a classical theorem of George P\'olya (1922) on the algebraicity of (generalized) diagonals of bivariate rational power series. In particular, we prove that the answer to Aissen's question, in which he considers $q$ as a variable, is negative in general. Moreover, we show that when $q$ is a complex number, the answer is positive if and only if $q$ is a root of unity.