A family of explicitly diagonalizable weighted Hankel matrices generalizing the Hilbert matrix

Abstract

A three-parameter family of weighted Hankel matrices is introduced with the entries, supposing , , are positive and , , . The famous Hilbert matrix is included as a particular case. The direct sum is shown to commute with a discrete analogue of the dilatation operator. It follows that there exists a three-parameter family of real symmetric Jacobi matrices, , commuting with . The orthogonal polynomials associated with turn out to be the continuous dual Hahn polynomials. Consequently, a unitary mapping diagonalizing can be constructed explicitly. At the same time, diagonalizes and the spectrum of this matrix operator is shown to be purely absolutely continuous and filling the interval where is known explicitly. If the assumption is relaxed while the remaining inequalities on , , are all supposed to be valid, the spectrum contains also a finite discrete part lying above the threshold . Again, all eigenvalues and eigenvectors are described explicitly.

Keywords:

• weighted Hankel matrix
• diagonalization
• spectrum
• Hilbert’s matrix

AMS Subject Classifications:

• 47B35
• 47B37
• 47A10
• 33C45

Acknowledgements

One of the authors (P.Š.) wishes to acknowledge gratefully partial support from grant No. GA13-11058S of the Czech Science Foundation.

Notes

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work was partially supported by the Czech Science Foundation [grant number GA13-11058S].