Applications of new Geronimus type identities for real orthogonal polynomials

Let $\mu$ be a positive measure on the real line, with associated orthogonal polynomials $\left\{ p_{n}\right\}$. Let Im$a\neq 0$. Then there is an explicit constant $c_{n}$ such that for all polynomials $P$ of degree at most $2n-2$,

$$c_{n}\int_{-\infty }^{\infty }\frac{P\left( t\right) }{\left\vert... ...}\left( a\right) p_{n}\left( t\right) \right\vert ^{2}}dt=\int P\text{ }d\mu .$$

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 $p_{n}\left( a\right) p_{n-1}\left( t\right) -p_{n-1}\left( a\right) p_{n}\left( t\right)$.