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Unification of Stieltjes‐Calogero type relations for the zeros of classical orthogonal polynomials
Author(s) -
Alıcı H.,
Taşeli H.
Publication year - 2014
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.3285
Subject(s) - mathematics , laguerre polynomials , hermite polynomials , orthogonal polynomials , type (biology) , pure mathematics , eigenvalues and eigenvectors , hahn polynomials , jacobi polynomials , classical orthogonal polynomials , hypergeometric function , order (exchange) , algebra over a field , gegenbauer polynomials , ecology , physics , finance , quantum mechanics , economics , biology
The classical orthogonal polynomials (COPs) satisfy a second‐order differential equation of the form σ ( x ) y ′′ + τ ( x ) y ′ + λ 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 © 2014 John Wiley & Sons, Ltd.