  When the co-recursion and co-dilation in the recurrence relation of certain
sequences of orthogonal polynomials are not at the same level, the behaviour of
the modified orthogonal polynomials is expected to have different properties
compared to the situation of the same level of perturbation. This manuscript
attempts to derive structural relations between the perturbed and original
$R_{II}$ type orthogonal polynomials. The classical result is improved using a
transfer matrix approach. It turns out that the $R_{II}$ fraction with
perturbation is the rational spectral transformation of the unperturbed one.
The derived notions are used to deduce some consequences for the polynomials
orthogonal on the real line. A quadrature rule is formulated from the perturbed
$R_{II}$ type recurrence, leading to observations that are based on numerical
results.

Classical Analysis and ODEs