  In this work, orthogonal polynomials satisfying $R_I$ type recurrence
relation %$\mathcal{P}_{n+1}(z) = (z-c_n)\mathcal{P}_n(z)-\lambda_n
(z-a_n)\mathcal{P}_{n-1}(z),$ with $\mathcal{P}_{-1}(z) = 0$ and
$\mathcal{P}_0(z) = 1$ are analyzed when the recurrence coefficients are
modified. The structural relationship between the perturbed and the unperturbed
polynomials along with the spectral properties and spectral transformation of
continued fraction are investigated. It is demonstrated that the transfer
matrix method is computationally more efficient than the classical method for
obtaining perturbed $R_I$ polynomials. Further, an interesting consequence of
co-dilation on the Carath\'eodary function is presented. Finally, the study of
co-recursion and co-dilation in connection to the unit circle is carried out
with the help of an illustration. The interlacing and monotonicity of zeros
between L-Jacobi polynomials and their perturbed forms are demonstrated.
