Indefinite $q$-integrals from a method using $q$-Ricatti equations

TL;DR

This paper introduces a reformulated method using $q$-Riccati equations to derive indefinite $q$-integrals for various $q$-special functions, providing a new approach to analyze these functions.

Contribution

The paper develops a novel approach using $q$-Riccati equations to obtain indefinite $q$-integrals, expanding the toolkit for studying $q$-special functions.

Findings

Derived $q$-integrals for multiple $q$-special functions.

Reformulated the method in terms of nonlinear $q$-Riccati equations.

Analyzed specific fragments of $q$-Riccati equations for integral derivation.

Abstract

Earlier work introduced a method for obtaining indefinite $q$-integrals of $q$-special functions from the second-order linear $q$-difference equations that define them. In this paper, we reformulate the method in terms of $q$-Riccati equations, which are nonlinear and first order. We derive $q$-integrals using fragments of these Riccati equations, and here only two specific fragment types are examined in detail. The results presented here are for $q$-Airy function, Ramanujan function, Jackson $q$-Bessel functions, discrete $q$-Hermite polynomials, $q$-Laguerre polynomials, Stieltjes-Wigert polynomial, little $q$-Legendre, and big $q$-Legendre polynomials.