On a specific family of orthogonal polynomials of Bernstein-Szegö type

Abstract: We study a class of weight functions on $[-1,1]$ which are special cases of the broader family studied by Bernstein and Szeg\"o. These weights are parametrized by two positive integers. As these integers tend to infinity, these weights approximate certain weight functions on $\mathbb{R}$ considered by Ismail and Valent. We also study modifications of these weight functions by a continuous parameter $a>0$. These ideas are then used to find finite analogs of some improper integrals first studied by Glaisher and Ramanujan. We also show that some of the functions used in this work are in fact generating functions of certain finite trigonometric sums.