On an Asymptotic Equality for Reproducing Kernels and Sums of Squares of Orthonormal Polynomials

Abstract

In a recent paper, the first author considered orthonormal polynomials $\left \{p_{n}\right \}$ associated with a symmetric measure with unbounded support and with recurrence relation

$$\displaystyle{ xp_{n}\left (x\right ) = A_{n}p_{n+1}\left (x\right ) + A_{n-1}p_{n-1}\left (x\right ),\quad n \geq 0. }$$

Under appropriate restrictions on $\left \{A_{n}\right \}$, the first author established the identity

$$\displaystyle{ \lim _{n\rightarrow \infty }\frac{\sum _{k=0}^{n}p_{k}^{2}\left (x\right )} {\sum _{k=0}^{n}A_{k}^{-1}} =\lim _{n\rightarrow \infty }\frac{p_{2n}^{2}\left (x\right ) + p_{2n+1}^{2}\left (x\right )} {A_{2n}^{-1} + A_{2n+1}^{-1}}, }$$

uniformly for x in compact subsets of the real line. Here, we establish and evaluate this limit for a class of even exponential weights, and also investigate analogues for weights on a finite interval, and for some non-even weights.

This article is dedicated to the memory of Q.I. Rahman