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Unification of Stieltjes-Calogero type relations for the zeros of classical orthogonal polynomials

Taşeli, Hasan
The classical orthogonal polynomials (COPs) satisfy a second-order differential equation of the form sigma(x)y '' + tau(x)y' + lambda y = 0, which is called the equation of hypergeometric type (EHT). It is shown that two numerical methods provide equivalent schemes for the discrete representation of the EHT. Thus, they lead to the same matrix eigenvalue problem. In both cases, explicit closed-form expressions for the matrix elements have been derived in terms only of the zeros of the COPs. On using the equality of the entries of the resulting matrices in the two discretizations, unified identities related to the zeros of the COPs are then introduced. Hence, most of the formulas in the literature known for the roots of Hermite, Laguerre and Jacobi polynomials are recovered as the particular cases of our more general and unified relationships. Furthermore, we present some novel results that were not reported previously. Copyright (C) 2014 John Wiley & Sons, Ltd.