  In this article, we show that the double scaling limit correlation functions
of a random matrix model when two cuts merge with degeneracy $2m$ (i.e. when
$y\sim x^{2m}$ for arbitrary values of the integer $m$) are the same as the
determinantal formulae defined by conformal $(2m,1)$ models. Our approach
follows the one developed by Berg\`{e}re and Eynard in \cite{BergereEynard} and
uses a Lax pair representation of the conformal $(2m,1)$ models (giving
Painlev\'e II integrable hierarchy) as suggested by Bleher and Eynard in
\cite{BleherEynard}. In particular we define Baker-Akhiezer functions
associated to the Lax pair to construct a kernel which is then used to compute
determinantal formulae giving the correlation functions of the double scaling
limit of a matrix model near the merging of two cuts.
