  Consider $\{p_n\}_{n=0}^{\infty}$, a sequence of polynomials orthogonal with
respect to $w(x)>0$ on $(a,b)$, and polynomials $\{g_{n,k}\}_{n=0}^{\infty},k
\in \mathbb{N}_0$, orthogonal with respect to $c_k(x)w(x)>0$ on $(a,b)$, where
$c_{k}(x)$ is a polynomial of degree $k$ in $x$. We show how Christoffel's
formula can be used to obtain mixed three-term recurrence equations involving
the polynomials $p_n$, $p_{n-1}$ and $g_{n-m,k},m\in\{2,3,\dots, n-1\}$. In
order for the zeros of $p_n$ and $G_{m-1}g_{n-m,k}$ to interlace (assuming
$p_n$ and $g_{n-m,k}$ are co-prime), the coefficient of $p_{n-1}$, namely
$G_{m-1}$, should be of exact degree $m-1$, in which case restrictions on the
parameter $k$ are necessary. The zeros of $G_{m-1}$ can be considered to be
inner bounds for the extreme zeros of the (classical or $q$-classical)
orthogonal polynomial $p_n$ and we give examples to illustrate the accuracy of
these bounds. Because of the complexity the mixed three-term recurrence
equations in each case, algorithmic tools, mainly Zeilberger's algorithm and
its $q$-analogue, are used to obtain them.
