We continue with the study of the Hankel determinant, $$ D_{ n}(t,\alpha):=\det\left(\int_{ 0}^{ \infty}x^{ j+k}w(x;t,\alpha)dx\right)_{ j,k=0 }^{ n-1}, $$ generated by singularly perturbed Laguerre weight, $$ w(x;t,\alpha):=x^{ \alpha}{ \rm e}^{ -x}\:{ \rm e}^{ -t/x}, \quad 0\leq x<\infty,\;\;\;\alpha>0,\;\;\;\;t>0, $$ obtained through a deformation of the Laguerre weight function, $$ w(x;0,\alpha):=x^{ \alpha}{ \rm e}^{ -x},\quad 0\leq x<\infty,\;\; \alpha>0, $$ via the multiplicative factor ${ \rm e}^{ -t/x}$. \\ An earlier investigation was made on the finite $n$ aspect of the problem, this has appeared in \cite{ ci1}. There, it was found that the logarithm of the Hankel determinant has an integral representation in terms of a particular ${ \rm P_{ III}},$ and its derivative with $t.$ In this paper we show that, under a double scaling, where $n$, the order of the Hankel matrix tends to $\infty,$ and $t$, tends to $0$, the scaled---and therefore, in some sense, infinite dimensional---Hankel determinant, has an integral representation in terms of the $C$ potential, and its derivatives. The second order non-linear differential equation which the $C$ potential satisfies, after a minor change of variables, is another ${ \rm P_{ III}},$ albeit with fewer number of parameters. \\ Expansions of the double scaled determinant for small and large parameter are obtained.
Categories: math.CA