Narrow Results

In this paper, we study the asymptotic properties of monic orthogonal polynomials (OPs) with respect to some Freud weights when the degree of the polynomial tends to infinity, including the asymptotics of the recurrence coefficients, the nontrivial leading coefficients of the monic OPs, the associated Hankel determinants and the squares of $L^2$-norm of the monic OPs. These results are derived from the combination of the ladder operator approach, Dyson’s Coulomb fluid approach and some recent results in the literature.

We prove evaluations of Hankel determinants of linear combinations of moments of orthogonal polynomials (or, equivalently, of generating functions for Motzkin paths), thus generalizing known results for Catalan numbers.

Ma–Minda class (of starlike functions) consists of normalized analytic functions $f$ defined on the unit disk for which the image of the function $z{f}^{\prime }(z)/f(z)$ is contained in some starlike region lying in the right-half plane. In this paper, we obtain the best possible bounds on some initial coefficients for the inverse functions of Ma–Minda starlike functions. Further, the bounds on the Fekete–Szegö functional and the second Hankel determinant are computed for such functions. In addition, some sharp radius estimates are also determined.

We consider the generalized Jacobi weight $x^{\alpha }{(1-x)}^{\beta }|x-t{|}^{\gamma },x\in [0,1],t(t-1)>0$, $\alpha >-1,\beta >-1,\gamma \in ℝ$. As is shown in [D. Dai and L. Zhang, Painlevé VI and Henkel determinants for the generalized Jocobi weight, J. Phys. A: Math. Theor.43 (2010), Article ID:055207, 14pp.], the corresponding Hankel determinant is the $\tau$-function of a particular Painlevé VI. We present all the possible asymptotic expansions of the solution of the Painlevé VI equation near $0,\infty$ and $1$ for generic $(\alpha ,\beta ,\gamma )$. For four special cases of $(\alpha ,\beta ,\gamma )$ which are related to the dimension of the Hankel determinant, we can find the exceptional solutions of the Painlevé VI equation according to the results of [A. Eremenko, A. Gabrielov and A. Hinkkanen, Exceptional solutions to the Painlevé VI equation, preprint (2016), arXiv:1602.04694], and thus give another characterization of the Hankel determinant.

We prove evaluations of Hankel determinants of linear combinations of moments of orthogonal polynomials (or, equivalently, of generating functions for Motzkin paths), thus generalizing known results for Catalan numbers.