A Lagrangian method for indefinite q-integrals

TL;DR

This paper extends a Lagrangian method to derive indefinite Jackson $q$-integrals for special functions satisfying second-order linear $q$-difference equations, resulting in many new and known $q$-integrals.

Contribution

It introduces a novel extension of the Lagrangian method to $q$-integrals, enabling derivation of numerous new indefinite and definite $q$-integrals for special functions.

Findings

Derived many new $q$-integrals for special functions.

Extended the Lagrangian method to $q$-difference equations.

Provided examples involving Jackson's $q$-Bessel functions and $q$-hypergeometric functions.

Abstract

A Lagrangian method is introduced recently for deriving indefinite integrals of special functions that satisfy homogeneous (nonhomogeneous) second-order linear differential equations. This paper extends this method to include indefinite Jackson $q$-integrals of special functions satisfying homogeneous (nonhomogeneous) second-order linear $q$-difference equations. Many $q$-integrals, both previously known and completely new, are derived using the method. We introduce samples of indefinite and definite $q$-integrals for Jackson's $q$-Bessel functions, $q$-hypergeometric functions, and some orthogonal polynomials.