Discriminants and Functions of the Second Kind of Orthogonal Polynomials
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We consider the problem of evaluating discriminants of general orthogonal polynomials. It is shown that for a general class of weight functions, the functions of the second kind and the orthogonal polynomials are linear independent solutions of the same second order differential equation. We derive a linear fourth-order differential equation satisfied by the numerator polynomials and give two additional linear independent solutions.
1991 Mathematics Subject Classification42C05 33C45
Key words and phrasesDiscriminants orthogonal polynomials zeros functions of the second kind recurrence relations differential equations raising and lowering operators finite dimensional Lie algebras numerator polynomials
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- F. V. Atkinson and W. N. Everitt, Orthogonal polynomials which satisfy second order differential equations. In: E.B. Christoffel, the Influence of His Work on Mathematics and Physical Sciences, edited by Butzer, P. and Fehér, F., pp. 173–181, Burkhäuser Verlag, Basel, 1981.Google Scholar