  We consider a sequence of polynomials $\{P_n\}_{n \geq 0}$ satisfying a
special $R_{II}$ type recurrence relation where the zeros of $P_n$ are simple
and lie on the real line. It turns out that the polynomial $P_n$, for any $n
\geq 2$, is the characteristic polynomial of a simple $n \times n$ generalized
eigenvalue problem. It is shown that with this $R_{II}$ type recurrence
relation one can always associate a positive measure on the unit circle. The
orthogonality property satisfied by $P_n$ with respect to this measure is also
obtained. Finally, examples are given to justify the results.
