Hankel Determinants for a Deformed Laguerre Weight with Multiple Variables and Generalized Painlev\'{e} V Equation

Abstract

We study the Hankel determinant generated by the moments of the deformed Laguerre weight function $x^{\alpha}{\rm{e}}^{-x}\prod\limits_{k=1}^{N}(x+t_k)^{\lambda_k}$, where $x\in \left[0,+\infty \right)$, $\alpha,t_k >0, \lambda_k\in\mathbb{R}$ for $k=1,\cdots ,N$. By using the ladder operators for the associated monic orthogonal polynomials and three compatibility conditions, we express the recurrence coefficients in terms of the $2N$ auxiliary quantities which are introduced in the ladder operators and shown to satisfy a system of difference equations. Combining these results with the differential identities obtained from the differentiation of the orthogonality relations, we deduce the Riccati equations for the auxiliary quantities. From them we establish a system of second order PDEs which are reduced to a Painlev\'{e} V equation for $N=1$. Moreover, we derive the second order PDE satisfied by the logarithmic derivative of the Hankel determinant, from which the limiting PDE is obtained under a double scaling. When $N=1$, these two PDEs are reduced to the $\sigma$-form of a Painlev\'{e} V and III equation respectively. When the dimension of the Hermitian matrices from the corresponding deformed Laguerre unitary ensemble is large and $\lambda_k\ge0$, based on Dyson's Coulomb fluid theory, we deduce the equilibrium density for the eigenvalues.