  Let $\un,$ be a quasi-definite linear functional defined on the space of
polynomials $\mathbb{P}.$ For such a functional we can define a sequence of
monic orthogonal polynomials (SMOP in short) $(P_n)_{n\geq 0},$ which satisfies
a three term recurrence relation. Shifting one unity the recurrence coefficient
indices given the sequence of associated polynomials of the first kind which
are orthogonal with respect to a linear functional denoted by $\un^{(1)}$.
  In the literature two special transformations of the functional $\un$ are
studied, the canonical Christoffel transformation $\widetilde \un=(x-c)\un$ and
the canonical Geronimus transformation $\widehat
\un=\frac{\un}{(x-c)}+M\deltan_c$ , where $c$ is a fixed complex number, $M $
is a free parameter and $\deltan_c$ is the linear functional defined on
$\mathbb{P}$ as $\prodint{\deltan_c,p(x)}=p(c).$
  For the Christoffel transformation with SMOP $(\widetilde P_n)_{n\geq 0}$, we
are interested in analyzing the relation between the linear functionals $
\un^{(1)}$ and $\widetilde{\un}^{(1)}.$ There, the super index denotes the
linear functionals associated with the orthogonal polynomial sequences of the
first kind $(P_n^{(1)})_{n\geq 0}$ and $(\widetilde P_n^{(1)})_{n\geq 0},$
respectively. This problem is also studied for Geronimus transformations. Here
we give close relations between their corresponding monic Jacobi matrices by
using the LU and UL factorizations. To get this result, we first need to study
the relation between $\un^{-1}$ (the inverse functional) and $\un^{(1)}$ which
can be expressed from a quadratic Geronimus transformation.
