Takashi Aoki, Shofu Uchida
Voros coefficients of the generalized hypergeometric differential equations with a large parameter are defined and their explicit forms are given for the origin and for the infinity. It is shown that they are Borel summable in some specified regions in the space of parameters and their Borel sums in the regions are given.
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We show that the each member of the confluent family of the Gauss hypergeometric equations is realized as quantum curves for appropriate spectral curves. As an application, relations between the Voros coefficients of those equations and the free energy of their classical limit computed by the topological recursion are established. We will also find explicit expressions of the free energy and the Voros coefficients in terms of the Bernoulli numbers and Bernoulli polynomials.
In my joint papers with Iwaki and Koike ([IKoT1, IKoT2]) we found an intriguing relation between the Voros coefficients in the exact WKB analysis and the free energy in the topological recursion introduced by Eynard and Orantin in the case of the confluent family of the Gauss hypergeometric differential equations. In this paper we discuss its generalization to the case of the hypergeometric differential equations associated with $2$-dimensional degenerate Garnier systems.
We develop the theory of quantization of spectral curves via the topological recursion. We formulate a quantization scheme of spectral curves which is not necessarily admissible in the sense of Bouchard and Eynard. The main result of this paper and the second part [IKoT] establishes a relation between the Voros coefficients for the quantum curves and the free energy for spectral curves associated with the confluent family of Gauss hypergeometric differential equations. We focus on the Weber equation in this article, and generalize the result for the other members of the confluent family in the…
Elliptic hypergeometric functions: integrals versus series
本文研究椭圆超几何函数，将椭圆贝塔积分表示为无穷级数组合，并讨论其收敛性及终止级数的渐近行为。
A system in a Birkhoff normal form with an irregular singularity of Poincare rank 1 at the origin and a regular singularity at infinity is through the Borel-Laplace transform dual to a system in an Okubo form. Schafke has showed that the Birkhoff system can also be obtained from the Okubo system by a simple limiting procedure. The Okubo system comes naturally with two kinds of mixed solution bases, both of which converge under the limit procedure to the canonical solutions of the limit Birkhoff system on sectors near the irregular singularity at the origin. One can then define Stokes matrices …
Topological recursion and uncoupled BPS structures II: Voros symbols and the $τ$-function
本文从量子曲线视角研究非耦合BPS结构与拓扑递归的对应，证明超几何型谱曲线的Borel重求和Voros符号可求解Bridgeland的BPS Riemann-Hilbert问题，并揭示Bridgeland $τ$-函数与相应势函数及拓扑递归配分函数之间的内在联系。
Zeros of generalized hypergeometric polynomials via finite free convolution. Applications to multiple orthogonality
本文利用有限自由卷积表示法研究了广义超几何多项式零点的弱渐近行为，推导了极限分布的S变换，并分析了多重正交多项式零点的交错性、单调性与渐近性。
We show connection formulae of local solutions of the Ramanujan equation between the origin and the infinity. These solutions are given by the Ramanujan function, the $q$-Airy function and the divergent basic hypergeometric series ${}_2φ_0(0,0;-;q,x)$. We use two different $q$-Borel-Laplace transformations to obtain our connection formulae.
The spherical functions of the noncompact Grassmann manifolds over the real or complex numbers or the quaternions with rank q and dimension parameter p can be seen as Heckman-Opdam hypergeometric functions of type BC, when the double coset space is identified with some Weyl chamber of type B. The associated double coset hypergroups may be embedded into a continuous family of commutative hypergroups with these hypergeometric functions as multiplicative functions with p in some continuous parameter range by a result of Rösler. Several limit theorems for random walks associated with these hypergr…
The aim of this paper is to treat the constant coefficients functional-differential equation $y'(x)=ay(qx)+by(x)$ with the help of the analytic theory of linear $q$-difference equations. When $ab\not=0$, the associated Cauchy problem with $y(0)=1$ admits a unique power series solution, which is the Hadamard product of a usual-hypergeometric series by a basic-hypergeometric series. By means of $θ$-modular relation, it is proved that this entire function can be expressed as linear combination of all the elements of a system of canonical fundamental solutions at infinity. A family of power series…
Strict Total Positivity of a Gauss Hypergeometric Kernel, and the Sharp Threshold for the One-Dimensional $SL(2,\mathbb{R})$ Conformal Block
本文完全解决了Li提出的两个全正性问题，证明高斯超几何核在Δ>0时严格全正，并确定一维SL(2,R)共形块的尖锐阈值为Δ≥1/2，低于该阈值则无法恢复无穷阶全正性。
A Palm hierarchy for determinantal point processes with the confluent hypergeometric kernel, the decomposing measures in the problem of harmonic analysis on the infinite-dimensional unitary group
本文证明无限维酉群调和分析中分解测度参数平移1等价于取无穷远约化Palm测度，通过正交多项式系综的有限维近似及Palm测度与标度极限的交换性完成证明。
We show connection formulae of local solutions of the Ramanujan equation between the origin and the infinity. These solutions are given by the Ramanujan function, the $q$-Airy function and the divergent basic hypergeometric series ${}_2φ_0(0,0;-;q,x)$. We use two different $q$-Borel-Laplace resummation methods to obtain our connection formulae. We also introduce the $q$-Borel-Laplace transformation of level $r-1$, which are higher order extension of these transformations. These methods are useful to obtain an asymptotic formula of a divergent series ${}_rφ_0(0,0,\dots ,0;-;q,x)$.
Asymptotic analysis of the confluent Dotsenko-Fateev system
本文对二维共形场论相关的汇合型Dotsenko-Fateev系统，构造其扇形基本解矩阵并确定斯托克斯矩阵，还研究了它与量子Painlevé方程等的关联及约化性质。
The study of polynomial solutions to the classical Lamé equation in its algebraic form, or equivalently, of double-periodic solutions of its Weierstrass form has a long history. Such solutions appear at integer values of the spectral parameter and their respective eigenvalues serve as the ends of bands in the boundary value problem for the corresponding Schrödinger equation with finite gap potential given by the Weierstrass $\wp$-function on the real line. In this paper we establish several natural (and equivalent) formulas in terms of hypergeometric and elliptic type integrals for the density…
Ground state of the energy-critical Gross-Pitaevskii equation with a harmonic potential can be constructed variationally. It exists in a finite interval of the eigenvalue parameter. The supremum norm of the ground state vanishes at one end of this interval and diverges to infinity at the other end. We explore the shooting method in the limit of large norm to prove that the ground state is pointwise close to the Aubin-Talenti solution of the energy-critical wave equation in near field and to the confluent hypergeometric function in far field. The shooting method gives the precise dependence of …
Let a second order Fuchsian differential equation with only univalued solutions have finite singular points at z_1, ..., z_n with exponents (a_1,b_1), ..., (a_n,b_n). Let the exponents at infinity be (A,B). Then for fixed generic z_1,...,z_n, the number of such Fuchsian equations is equal to the multiplicity of the irreducible sl_2 representation of dimension |A-B| in the tensor product of irreducible sl_2 representations of dimensions |a_1-b_1|, >..., |a_n-b_n|. To show this we count the number of critical points of a suitable function which plays the crucial role in constructions of the hype…
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