A Favard type theorem for orthogonal polynomials on the unit circle from a three term recurrence formula

The objective of this manuscript is to study directly the Favard type theorem associated with the three term recurrence formula % $$R_{n+1}(z)=[(1+i{c}_{n+1})z+(1-i{c}_{n+1})]R_n(z)-4d_{n+1}z{R}_{n-1}(z),n\ge 1,$$ % with $R_0(z)=1$ and $R_1(z)=(1+i{c}_1)z+(1-i{c}_1)$, where $\{c_n{\}}_{n=1}^{\mathrm{\infty }}$ is a real sequence and $\{d_n{\}}_{n=1}^{\mathrm{\infty }}$ is a positive chain sequence. We establish that there exists an unique nontrivial probability measure $\mu$ on the unit circle for which ${R_n(z) - 2(1-m_n)R_{n-1}(z)}$givesthesequenceoforthogonalpolynomials\mathrm{.}Here,${m_n}_{n=0}^{\infty}$ is the minimal parameter sequence of the positive chain sequence $\{d_n{\}}_{n=1}^{\mathrm{\infty }}$. The element $d_1$ of the chain sequence, which does not effect the polynomials $R_n$, has an influence in the derived probability measure $\mu$ and hence, in the associated orthogonal polynomials on the unit circle. To be precise, if $\{M_n{\}}_{n=0}^{\mathrm{\infty }}$ is the maximal parameter sequence of the chain sequence, then the measure $\mu$ is such that $M_0$ is the size of its mass at $z=1$. An example is also provided to completely illustrates the results obtained.