Quantum Hilbert matrices and orthogonal polynomials

Using the notion of quantum integers associated with a complex number q≠0q≠0, we define the quantum Hilbert matrix and various extensions. They are Hankel matrices corresponding to certain little qq-Jacobi polynomials when |q|<1|q|<1, and for the special value q=(1−5)(1+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.