$t_0$, $t^{s}=(t-t_0)^{s} + s~t_0(t-t_0)^{s-1}+
\frac{t_0^{2}}{2}(s)(s-1)(t-t_0)^{s-2}+{\cal O}([t-t_0]^{s-3})$, one has: $$\begin{aligned}
&~& t^{2p}(z)\cdot{\cal D}^{(q)}(w) = {\cal D}^{(2p+q)}(t-t_0)+ 
\left(2p~t_0 - 2p~\epsilon~{\rm ln}|z-w|\right)~{\cal D}^{(2p+q-1)}+ 
\nonumber \\
&~& + \left(\frac{t_0^{2}}{2}2p(2p-1)+[2pq + 
(2p-4p^2)\epsilon~t_0]~{\rm ln}|z-w|\right)
{\cal D}^{(2p+q-2)}(t-t_0)+ \nonumber \\
&~& + {\cal O}([t-t_0]^{2p+q-3})\end{aligned}$$ where it is worthy of mentioning that inside the subleading terms there are higher logarithms of the form ${\rm ln}^n|z-w|$, where $n =2,3,4, \dots$.

We now come to the OPE between the spatial parts. In view of (\[correl\]), upon expressing $\partial _z$ in normal $\partial_n$ and tangential parts, and imposing Dirichlet boundary conditions on the world-sheet boundary where the operators live on, we observe that such operator products take the form: $$(\partial X^j(z))^2 \cdot \partial_n X^i(w) \sim 
G^{ii}\frac{1}{(z-w)^2}\partial_n X^i \sim \frac{t^{-2p}}{(z-w)^2}
\partial_n X^i, \qquad ({\rm no}~~{\rm sum}~~{\rm over}~~i)$$ Performing the last contraction with the $t^{-2p}$, following the previous general formulæ and collecting appropriate terms, one obtains: $$\begin{aligned}
\label{elalg}
&~& T(z)\cdot{\cal D}^{(q)}[(t-t_0)(w)] 
=\frac{1-\frac{\epsilon^2}{2}}{(z-w)^2}{\cal D}^{(q)} [(t-t_0)(w)] +
\frac{q\epsilon}{(z-w)^2}{\cal D}^{q-1}[(t-t_0)(w)] + \nonumber \\
&~& + \frac{\frac{q(q-1)}{2} -2p^{2}~{\rm ln}|z-w| 
-2p^{2}~\epsilon^2~{\rm ln}^2|z-w|}{(z-w)^2}{\cal D}^{(q-2)} 
[(t-t_0)(w)]
+ {\cal O}([t-t_0]^{q-3})\end{aligned}$$ where again inside the subleading terms there are higher logarithms.

We next notice that, as a consistency check of the formalism, one can calculate the OPE (\[elalg\]) in case one considers matrix 