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.
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.