Isospectral matrices and classical polynomials

AbstractThe matrices of order n defined, in terms of the n arbitrary numbers xj, by the formulae X=diag(xj) and Zjk=δjk∑′l=1n(xj−xl)−1+(1−δ jk(xj−xk)−1, are representations of the multiplicative operator ξ and of the differential operator d/dξ in a space spanned by the polynomials in ξ of degree less than n. This elementary fact implies a number of remarkable formulae involving these matrices, including novel representations of the classical polynomials