• Corpus ID: 235790790

Linear differential operators with polynomial coefficients generating generalised Sylvester-Kac matrices

@inproceedings{Dyachenko2021LinearDO,
  title={Linear differential operators with polynomial coefficients generating generalised Sylvester-Kac matrices},
  author={Alexander Dyachenko and Mikhail Tyaglov},
  year={2021}
}
A method of generating differential operators is used to solve the spectral problem for a generalisation of the Sylvester-Kac matrix. As a by-product, we find a linear differential operator with polynomial coefficients of the first order that has a finite sequence of polynomial eigenfunctions generalising the operator considered by M. Kac. In addition, we explain spectral properties of two related tridiagonal matrices whose shape differ from our generalisation. 
1 Citations
A note on tridiagonal matrices with zero main diagonal
We find the spectrum of an arbitrary irreducible complex tridiagonal matrix with two-periodic main diagonal provided that the spectrum of the matrix with the same suband superdiagonals and zero main

References

SHOWING 1-10 OF 39 REFERENCES
Eigenvectors of tridiagonal matrices of Sylvester type
Eigenvectors of the tridiagonal matrices of Sylvester type are explicitly determined. These are closely related to orthogonal polynomials named after Krawtchouk, (dual) Hahn and Racah as well as to
The interesting spectral interlacing property for a certain tridiagonal matrix
In this paper, a new tridiagonal matrix, whose eigenvalues are the same as the Sylvester-Kac matrix of the same order, is provided. The interest of this matrix relies also in that the spectrum of a
A new type of Sylvester–Kac matrix and its spectrum
ABSTRACT The Sylvester–Kac matrix, sometimes known as Clement matrix, has many extensions and applications throughout more than a century of its existence. The computation of the eigenvalues or even
Evaluation of Sylvester Type Determinants Using Orthogonal Polynomials
Random Walk and the Theory of Brownian Motion
(1947). Random Walk and the Theory of Brownian Motion. The American Mathematical Monthly: Vol. 54, No. 7P1, pp. 369-391.
Fibonacci polynomials and Sylvester determinant of tridiagonal matrix
  • W. Chu
  • Mathematics
    Appl. Math. Comput.
  • 2010
A finite quantum oscillator model related to special sets of Racah polynomials
In [R. Oste and J. Van der Jeugt, arXiv: 1507.01821 [math-ph]] we classified all pairs of recurrence relations in which two (dual) Hahn polynomials with different parameters appear. Such pairs are
Doubling (Dual) Hahn Polynomials: Classification and Applications
We classify all pairs of recurrence relations in which two Hahn or dual Hahn polynomials with different parameters appear. Such couples are referred to as (dual) Hahn doubles. The idea and interest
Evaluation of Sylvester type determinants using block-triangularization
It is shown that the values of Sylvester type determinants for various orthogonal polynomials considered by Askey in [R.Askey, Evaluation of some determinants, Proceedings of the 4th ISAAC Congress,
...
...