Asymptotics of determinants of Hankel matrices via non-linear difference equations

E. Heine in the 19th century studied a system of orthogonal polynomials associated with the weight $\left[x(x-\alpha)(x-\beta)\right]^{-\frac{1}{2}}$, $x\in[0,\alpha]$, $0<\alpha<\beta$. A related system was studied by C. J. Rees in 1945, associated with the weight $\left[(1-x^2)(1-k^2x^2)\right]^{-\frac{1}{2}}$, $x\in[-1,1]$, $k^2\in(0,1)$. These are also known as elliptic orthogonal polynomials, since the moments of the weights maybe expressed in terms of elliptic integrals. Such orthogonal polynomials are of great interest because the corresponding Hankel determinant, depending on a parameter $k^2$, where $0<k^2<1$ is the $\tau$~function of a particular Painlev\'e VI, the special cases of which are related to enumerative problems arising from string theory. We show that the recurrence coefficients, denoted by $\beta_n(k^2),\;n=1,2,\dots$; and ${\rm p}_1(n,k^2)$, the coefficients of $x^{n-2}$ of the monic polynomials orthogonal with respect to a generalized version of the weight studied by Rees, $$(1-x^2)^{\alpha}(1-k^2x^2)^{\beta},\;\;x\in[-1,1],\;\alpha>-1,\;\beta\in \mathbb{R},$$ satisfy second order non-linear difference equations. The large $n$ expansion based on the difference equations when combined with known asymptotics of the leading terms of the associated Hankel determinant yields a complete asymptotic expansion of the Hankel determinant. The Painlev\'e equation is also discussed as well as the generalization of the linear second order differential equation found by Rees.