  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 that
the two sequences of para-orthogonal polynomials $\{zS_{n}(z) +
S_{n}^{\ast}(z)\}$ and $\{zS_{n}(z) - S_{n}^{\ast}(z)\}$ satisfy three term
recurrence formulas and have also explored some further consequences of these
sequences of polynomials such as their connections to sequences of orthogonal
polynomials on the interval $[-1,1]$. The same authors, in (1988), have also
provided a means to extend these results to cover any nontrivial measure on the
unit circle. However, only recently in Costa, Felix and Sri Ranga (2013) and
then in Castillo, Costa, Sri Ranga and Veronese (2014), the extension
associated with the para-orthogonal polynomials $zS_{n}(z) - S_{n}^{\ast}(z)$
was thoroughly explored, especially from the point of view of the three term
recurrence, and chain sequences play an important part in this exploration. The
main objective of the present manuscript is to provide the theory surrounding
the 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 and
applications are also provided to justify the results obtained.
