Indefinite q-integrals from a method using q-Riccati equations

Abstract

Abstract In an earlier work, a method was introduced 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 the q -Airy function, the Ramanujan function, the discrete q -Hermite I and II polynomials, the q -hypergeometric functions, the q -Laguerre polynomials, the Stieltjes-Wigert polynomial, the little q -Legendre and the big q -Legendre polynomials.