Differential transcendence of solutions of difference Riccati equations and Tietze's treatment

TL;DR

This paper revisits Tietze's 1905 work on the differential transcendence of solutions to difference Riccati equations, clarifies its algebraic nature, and applies it to study the q-Airy equation.

Contribution

It provides a purely algebraic reinterpretation of Tietze's original analysis and extends the application to the q-Airy equation.

Findings

Clarified the algebraic basis of Tietze's method

Applied the approach to the q-Airy equation

Enhanced understanding of differential transcendence in difference equations

Abstract

There is the paper by H. Tietze published in 1905 on differential transcendence of solutions of difference Riccati equations. In this paper, we clarify the essence of Tietze's treatment and make it purely algebraic. As an application, the $q$-Airy equation is studied.