$q-$difference equations concerning with $q-$gamma function

TL;DR

This paper investigates solutions to $q$-difference Riccati equations and related second order equations, revealing their connection to the $q$-gamma function and analyzing their growth and value distribution.

Contribution

It establishes the relationship between meromorphic solutions of $q$-difference Riccati equations and the $q$-gamma function, including growth and value distribution properties.

Findings

Meromorphic solutions are related to the $q$-gamma function.

Growth and value distribution of solutions are characterized.

Solutions of second order $q$-difference equations are connected with special functions.

Abstract

We consider a family of solutions of $q-$difference Riccati equation, and prove the meromorphic solutions of $q-$difference Riccati equation and corresponding second order $q-$difference equation are concerning with $q-$gamma function. The growth and value distribution of differences on solutions of $q-$difference Riccati equation are also investigated.