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A method of generating differential operators is used to solve the spectralproblem for a generalisation of the Sylvester-Kac matrix. As a by-product, wefind a linear differential operator with polynomial coefficients of the firstorder that has a finite sequence of polynomial eigenfunctions generalising theoperator considered by M. Kac. In addition, we explain spectral properties oftwo related tridiagonal matrices whose shape differ from our generalisation.

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