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A family of explicitly diagonalizable weighted Hankel matrices generalizing the Hilbert matrix. (English) Zbl 1359.47027

The main object under consideration in the present paper is a three-parameter family \(B=B(a,b,c)=\|B_{kj}\|_{k,j\geq0}\) of weighted Hankel-type semi-infinite matrices with entries \[ B_{kj}=\frac{\Gamma(k+j+a)}{\Gamma(k+j+b+c)}\, \sqrt{\frac{\Gamma(k+b)\Gamma(k+c)\Gamma(j+b)\Gamma(j+c)}{\Gamma(k+a)k!\Gamma(j+a)j!}}, \] where \(a,b,c\) are positive parameters subject to \[ a<b+c, \quad b<a+c, \quad c\leq a+b. \] This family contains the celebrated Hilbert matrix \(H(\theta)=B(\theta,\theta,1)\). The authors construct a unitary mapping, diagonalizing \(B\), give a precise spectral analysis of the family. They show that the spectrum is pure absolutely continuous filling the interval \([0,M(a,b,c)]\) with \[ M(a,b,c)=\frac1{\Gamma(b+c-a)}\,\Gamma^2\Bigl(\frac{b+c-a}2\Bigr). \] Under certain relaxed assumptions on \(a,b,c\), there is a finite point spectrum, which is computed explicitly.

MSC:

47B37 Linear operators on special spaces (weighted shifts, operators on sequence spaces, etc.)
47B36 Jacobi (tridiagonal) operators (matrices) and generalizations
47A10 Spectrum, resolvent
33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
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