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Quantum Hilbert matrices and orthogonal polynomials

Jorgen Ellegaard Andersen·Christian Berg
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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.

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Jorgen Ellegaard Andersen, Christian Berg. 2007-03-19. Quantum Hilbert matrices and orthogonal polynomials. https://arxiv.org/abs/math/0703546

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