Sturm-Liouville Problem in Quantum Calculus

TL;DR

This paper investigates the q-analogue of the Sturm-Liouville problem, providing asymptotic behaviors and expansions for solutions and q-Bessel functions, demonstrating the applicability of classical methods in quantum calculus.

Contribution

It extends classical Sturm-Liouville analysis to the quantum calculus setting, showing that established methods remain valid.

Findings

Asymptotic behavior of solutions at infinity

Asymptotic expansion of q-Bessel functions for alpha > -1/2

Validation of classical methods in quantum calculus context

Abstract

This paper aims to study the q-analogue of the Sturm Liouville problem and to give an asymptotic behaviour at infinity for its solution '. Additionally, we establish an asymptotic expansion of the q-Bessel function $j_\alpha$ for $\alpha >-{1/2}. We are not in situation to claim that our results are new but they have the advantage to show that the method used by Agranovich and Marchenko remain true.