2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/125946Using the notion of quantum integers associated with a complex number $q\neq 0$, we define the quantum Hilbert matrix and various extensions. They are Hankel matrices corresponding to certain little $q$-Jacobi polynomials when $|q|<1$, and for the special value $q=(1-\sqrt{5})/(1+\sqrt{5})$ they are closely related to Hankel matrices of reciprocal Fibonacci numbers called Filbert matrices. We find a formula for the entries of the inverse quantum Hilbert matrix.10 pagesClassical Analysis and ODEs33D45;11B39text