In this paper we consider the one-parameter family of weight functions \(\mu _m(\theta )=1+\cos (m\theta )\) , \(\theta \in [-\pi ,\pi ]\) , \(m \in \mathbb {N}\cup \{ 0 \}\) , and we obtain explicit expressions for the corresponding Szegő and para-orthogonal polynomials of the first and second kind. The associated Szegő quadrature formulas for the estimation of integrals of the form \(\int _{-\pi }^{\pi } F\left( e^{i\theta }\right) \mu _m(\theta )d\theta \) are characterized, their efficient computation based on an eigenvalue problem related to CMV matrices is studied, and error bounds are obtained from the error when approximating the Herglotz-Riesz transform of \(\mu _m\) by certain rational approximants. Most of these results are illustrated with several numerical experiments. By making use of the Joukowsky transformation, we apply the above results to develop a new procedure for the estimation of Christoffel transformations of the Chebyshev weight function on the interval, namely \(\int _{-1}^{1} f(x)\omega (x)dx\) with \(\omega (x)=P(x)\cdot \left( 1-x^2\right) ^{-1/2}\) where P is an arbitrary real polynomial not necessarily positive on \([-1,1]\) , by means of certain related Gauss-type quadrature formulas. Some more numerical experiments are finally carried out along with some conclusions.
