Ryoichi Kobayashi, Tatsuya Nishizaka, Shoji Shinzato, Masaaki Yoshida
We propose an affine version of the Schwarz map for the hypergeometric differential equation, and study its image when the monodromy group is finite.
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In the previous paper (math.CA/0609196) we defined a map, called the hyperbolic Schwarz map, from the one-dimensional projective space to the three-dimensional hyperbolic space by use of solutions of the hypergeometric differential equation, and thus obtained closed flat surfaces belonging to the class of flat fronts. We continue the study of such flat fronts in this paper. First, we introduce the notion of derived Schwarz maps of the hypergeometric differential equation and, second, we construct a parallel family of flat fronts connecting the classical Schwarz map and the derived Schwarz map.
The Schwarz map of the hypergeometric differential equation is studied since the beginning of the last century. Its target is the complex projective line, the 2-sphere. This paper introduces the hyperbolic Schwarz map, whose target is the hyperbolic 3-space. This map can be considered to be a lifting to the 3-space of the Schwarz map. This paper studies the singularities of this map, and visualize its image when the monodromy group is a finite group or a typical Fuchsian group. General cases will be treated in a forthcoming paper.
Recently, we formulated the $q$-Garnier system and its variations as translations of an extended affine Weyl group of type $A^{(1)}_{2n+1}\times A^{(1)}_1\times A^{(1)}_1$. On the other hand, those systems admit particular solutions in terms of the basic hypergeometric series ${}_{n+1}ϕ_n$. In this article, we investigate an action of the extended affine Weyl group on ${}_{n+1}ϕ_n$.
Hypergeometric solutions to Schwarzian equations
本文通过求解 Schwarzian 方程并利用等变函数与二维向量值模形式理论，给出了 $m\geq 7$ 时模微分方程 $y''+sE_4y=0$ 的 Gauss 超几何级数解，补全了该方程的研究。
A proof of an unusual summation formula for a basic hypergeometric series associated to the affine root system $\tilde A_n$ that was conjectured by Warnaar is given. It makes use of Milne's $A_n$ extension of Watson's transformation, Ramanujan's $_1ψ_1$-summation, and a determinant evaluation of the author. In addition, a transformation formula between basic hypergeometric series associated to the affine root systems $\tilde A_n$ respectively $\tilde A_m$, which generalizes at the same time the above summation formula and an identity due to Gessel and the author, is proposed as a conjecture.
We extend Schwarz' list of irreducible algebraic Gauss functions to the four classes of Appell-Lauricella functions in several variables and the 14 complete Horn functions in two variables. This gives an example of a family of functions such that for any number of variables there are infinitely many algebraic functions, namely the Lauricella $F_C$ functions.
The hypergeometric functions ${}_nF_{n-1}$ are higher transcendental functions, but for certain parameter values they become algebraic, because the monodromy of the defining hypergeometric differential equation becomes finite. It is shown that many algebraic ${}_nF_{n-1}$'s, for which the finite monodromy is irreducible but imprimitive, can be represented as combinations of certain explicitly algebraic functions of a single variable; namely, the roots of trinomials. This generalizes a result of Birkeland, and is derived as a corollary of a family of binomial coefficient identities that is of i…
A celebrated theorem of Klein implies that any hypergeometric differential equation with algebraic solutions is a pull-back of one of the few standard hypergeometric equations with algebraic solutions. The most interesting cases are hypergeometric equations with tetrahedral, octahedral or icosahedral monodromy groups. We give an algorithm for computing Klein's pull-back coverings in these cases, based on certain explicit expressions (Darboux evaluations) of algebraic hypergeometric functions. The explicit expressions can be computed using contiguous relations and a data base of simplest Darbou…
We consider a $q$-Painlevé III equation and a $q$-Painlevé II equation arising from a birational representation of the affine Weyl group of type $(A_2+A_1)^{(1)}$. We study their hypergeometric solutions on the level of $τ$ functions.
By considering Schwarz's map for the hypergeometric differential equation with parameters $(a,b,c)=(1/6,1/2,1)$ or $(1/12,5/12,1)$, we give some analogies of Jacobi's formula $\vartheta_{00}(τ)^2= F(1/2,1/2,1;λ(τ))$, where $\vartheta_{00}(τ)$ and $λ(τ)$ are the theta constant and the lambda function defined on the upper-half plane, and $F(a,b,c;z)$ is the hypergeometric series defined on the unit disk. As applications of our formulas, we give several functional equations for $F(a,b,c;z)$.
This is basically the text of a survey talk (entitled 'Painleve, Klein and the icosahedron') given at Hitchin's 60th birthday conference. It discusses the search for and construction of algebraic solutions of the sixth Painleve differential equation, which may be viewed as a nonlinear analogue of the Gauss hypergeometric equation. Both algebraic and transcendental methods are used and the story involves affine Weyl groups, braid groups and cubic surfaces. Some emphasis is given to the interpretation of the sixth Painleve equation as the explicit form of the simplest nonabelian Gauss-Manin conn…
Generalized Darboux-Halphen (gDH) systems, which form a versatile class of three-dimensional homogeneous quadratic differential systems (HQDS's), are introduced. They generalize the Darboux-Halphen (DH) systems considered by other authors, in that any non-DH gDH system is affinely but not projectively covariant. It is shown that the gDH class supports a rich collection of rational solution-preserving maps: morphisms that transform one gDH system to another. The proof relies on a bijection between (i) the solutions with noncoincident components of any `proper' gDH system, and (ii) the solutions…
Exactly solvable Schrödinger operators related to the hypergeometric equation
本文研究可通过高斯超几何函数精确求解的一维复势闭薛定谔算子，将其分为三类共九个族，计算了各算子的谱与格林函数，给出相关嬗变恒等式，并说明其源于对称流形（伪）拉普拉斯算子的变量分离。
Dubrovin [Lecture Notes in Math., Vol. 1620, Springer, Berlin, 1996, 120-348] showed that the Chazy XII equation $y'''- 2yy''+3y'^2 = K(6y'-y^2)^2$, $K \in \mathbb{C}$, is equivalent to a projective-invariant equation for an affine connection on a one-dimensional complex manifold with projective structure. By exploiting this geometric connection it is shown that the Chazy XII solution, for certain values of $K$, can be expressed as $y=a_1w_1+a_2w_2+a_3w_3$ where $w_i$ solve the generalized Darboux-Halphen system. This relationship holds only for certain values of the coefficients $(a_1,a_2,a_3…
A note on the inverse problem for finite differential Galois groups
本文提出一种算法，在给定商曲线函数域时，构造常线性微分方程使其 Schwarz 映射参数化不可约有限线性代数群下的不变曲线，推广了已有逆问题解，并证明了不存在微分 Galois 群同构于 $H_{72}^{SL_3}$ 的超几何方程。
An energy-momentum method for ordinary differential equations with an underlying $k$-polysymplectic manifold
本文纠正$k$-多辛Marsden-Weinstein约化理论的既有文献错误，提出新的$k$-多辛能量-动量方法与稳定性分析技术，将其应用于对应$k$-多辛流形的常微分方程哈密顿系统并给出多类算例。
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