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When a nontrivial measure $\mu$ on the unit circle satisfies the symmetry$d\mu(e^{i(2\pi-\theta)}) = - d\mu(e^{i\theta})$ then the associated OPUC, say$S_n$, are all real. In this case, Delsarte and Genin, in 1986, have shown thatthe two sequences of para-orthogonal polynomials $\{zS_{n}(z) +S_{n}^{\ast}(z)\}$ and $\{zS_{n}(z) - S_{n}^{\ast}(z)\}$ satisfy three termrecurrence formulas and have also explored some further consequences of thesesequences of polynomials such as their connections to sequences of orthogonalpolynomials on the interval $[-1,1]$. The same authors, in (1988), have alsoprovided a means to extend these results to cover any nontrivial measure on theunit circle. However, only recently in Costa, Felix and Sri Ranga (2013) andthen in Castillo, Costa, Sri Ranga and Veronese (2014), the extensionassociated with the para-orthogonal polynomials $zS_{n}(z) - S_{n}^{\ast}(z)$was thoroughly explored, especially from the point of view of the three termrecurrence, and chain sequences play an important part in this exploration. Themain objective of the present manuscript is to provide the theory surroundingthe extension associated with the para-orthogonal polynomials $zS_{n}(z) +S_{n}^{\ast}(z)$ for any nontrivial measure on the unit circle. Like in Costa,Felix and Sri Ranga (2013) and Castillo, Costa, Sri Ranga and Veronese (2014),chain sequences also play an important role in this theory. Examples andapplications are also provided to justify the results obtained.

Para-orthogonal polynomials on the unit circle satisfying three term recurrence formulas