Hankel determinants for a Gaussian weight with Fisher-Hartwig singularities and generalized Painlevé IV equation

We study the Hankel determinant generated by a Gaussian weight with Fisher-Hartwig singularities of root type at tj , j=1,MIDLINE HORIZONTAL ELLIPSIS,N . It characterizes a type of average characteristic polynomial of matrices from Gaussian unitary ensembles. We derive the ladder operators satisfied by the associated monic orthogonal polynomials and three compatibility conditions. By using them and introducing 2N auxiliary quantities {Rn,j,rn,j,j=1,MIDLINE HORIZONTAL ELLIPSIS,N} , we build a series of difference equations. Furthermore, we prove that {Rn,j,rn,j} satisfy Riccati equations. From them we deduce a system of second order PDEs satisfied by {Rn,j,j=1,MIDLINE HORIZONTAL ELLIPSIS,N} , which reduces to a Painleve IV equation for N = 1. We also show that the logarithmic derivative of the Hankel determinant satisfies the generalized sigma-form of a Painleve IV equation.