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\(R_{II}\) type recurrence, generalized eigenvalue problem and orthogonal polynomials on the unit circle. (English) Zbl 1405.33017

Summary: We consider a sequence of polynomials \(\{P_n \}_{n \geq 0}\) satisfying a special \(R_{I I}\) type recurrence relation where the zeros of \(P_n\) are simple and lie on the real line. It turns out that the polynomial \(P_n\), for any \(n \geq 2\), is the characteristic polynomial of a simple \(n \times n\) generalized eigenvalue problem. It is shown that with this \(R_{I I}\) type recurrence relation one can always associate a positive measure on the unit circle. The orthogonality property satisfied by \(P_n\) with respect to this measure is also obtained. Finally, examples are given to justify the results.

MSC:

33C47 Other special orthogonal polynomials and functions
33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
58C40 Spectral theory; eigenvalue problems on manifolds

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References:

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