Applications of new Geronimus type identities for real orthogonal polynomials

Lubinsky, D.

doi:10.1090/S0002-9939-10-10276-7

Applications of new Geronimus type identities for real orthogonal polynomials
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by D. S. Lubinsky PDF

Proc. Amer. Math. Soc. 138 (2010), 2125-2134 Request permission

Abstract:

Let $\mu$ be a positive measure on the real line, with associated orthogonal polynomials $\left \{ p_{n}\right \}$. Let $\mbox {Im}a\neq 0$. Then there is an explicit constant $c_{n}$ such that for all polynomials $P$ of degree at most $2n-2$, \begin{equation*} c_{n}\int _{-\infty }^{\infty }\frac {P\left ( t\right ) }{\left \vert p_{n}\left ( a\right ) p_{n-1}\left ( t\right ) -p_{n-1}\left ( a\right ) p_{n}\left ( t\right ) \right \vert ^{2}}dt=\int P\text { }d\mu . \end{equation*} In this paper, we provide a self-contained proof of this identity. Moreover, we apply the formula to deduce a weak convergence result and a discrepancy estimate, and also to establish a Gauss quadrature associated with $\mu$ with nodes at the zeros of $p_{n}\left ( a\right ) p_{n-1}\left ( t\right ) -p_{n-1}\left ( a\right ) p_{n}\left ( t\right )$.

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