In the previous work we introduced the higher order $q$-Painlevé system $q$-$P_{(n+1,n+1)}$ as a generalization of the Jimbo-Sakai's $q$-Painlevé VI equation. It is derived from a $q$-analogue of the Drinfeld-Sokolov hierarchy of type $A^{(1)}_{2n+1}$ and admits a particular solution in terms of the Heine's $q$-hypergeometric function ${}_{n+1}ϕ_n$. However the obtained system is insufficient as a generalization of $q$-$P_{\rm{VI}}$ due to some reasons. In this article we rewrite the system $q$-$P_{(n+1,n+1)}$ to a more suitable one.
In this article we formulate a group of birational transformations which is isomorphic to an extended affine Weyl group of type $(A_{2n+1}+A_1+A_1)^{(1)}$ with the aid of mutations and permutations of vertices to a mutation-periodic quiver on a torus. This group provides a class of higher order generalizations of Jimbo-Sakai's $q$-Painlevé VI equation as translations on a root lattice. Then the known three systems are obtained again; the $q$-Garnier system, a similarity reduction of the lattice $q$-UC hierarchy and a similarity reduction of the $q$-Drinfeld-Sokolov hierarchy.
Generalized Charlier Recurrence Coefficients: Hankel Formulas, Laguerre--Freud Dynamics, and Painlevé V Connections
International Chair of Mathematical Physics and Applications (ICMPA–UNESCO Chair)（国际数学物理与应用 chair（ICMPA-UNESCO 讲席））；University of Abomey-Calavi（阿波美卡拉维大学）
Generalized Freud weight, discrete Painlevé I hierarchy and full asymptotics of Hankel determinants
School of Mathematics and Statistics, Hunan University of Science and Technology（湖南科技大学数学与统计学院）；School of Mathematical Sciences, Huaqiao University（华侨大学数学科学学院）
本文研究广义Freud权对应的正交多项式与Hankel行列式，揭示离散Painlevé I hierarchy的统一结构，推导相关微分方程与恒等式，得到这些量的完全渐近展开并以十次Freud权为例说明。
We extend and generalize the construction of Sturm-Liouville problems for a family of Hamiltonians constrained to fulfill a third-order shape-invariance condition and focusing on the "$-2x/3$" hierarchy of solutions to the fourth Painlevé transcendent. Such a construction has been previously addressed in the literature for some particular cases but we realize it here in the most general case. The corresponding potential in the Hamiltonian operator is a rationally extended oscillator defined in terms of the conventional Okamoto polynomials, from which we identify three different zero-modes cons…
This work reports and classifies the most general construction of rational quantum potentials in terms of the generalized Hermite polynomials. This is achieved by exploiting the intrinsic relation between third-order shape-invariant Hamiltonians and the fourth Painlevé equation, such that the generalized Hermite polynomials emerge from the $-1/x$ and $-2x$ hierarchies of rational solutions. Such a relation unequivocally establishes the discrete spectrum structure, which, in general, is composed as the union of a finite- and infinite-dimensional sequence of equidistant eigenvalues separated by …
