Quantum Hilbert matrices and orthogonal polynomials

Using the notion of quantum integers associated with a complex number q ≠ 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 − 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.