Quantum Hilbert matrices and orthogonal polynomials

Abstract

Using 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.