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.
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.
Based on Spiridonov's analysis of elliptic generalizations of the Gauss hypergeometric function, we develop a common framework for 7-parameter families of generalized elliptic, hyperbolic and trigonometric univariate hypergeometric functions. In each case we derive the symmetries of the generalized hypergeometric function under the Weyl group of type E_7 (elliptic, hyperbolic) and of type E_6 (trigonometric) using the appropriate versions of the Nassrallah-Rahman beta integral, and we derive contiguous relations using fundamental addition formulas for theta and sine functions. The top level de…
In math.QA/0309252, the author proved a number of multivariate elliptic hypergeometric integrals. The purpose of the present note is to explore more carefully the various limiting cases (hyperbolic, trigonometric, rational, and classical) that exist. In particular, we show (using some new estimates of generalized gamma functions) that the hyperbolic integrals (previously treated as purely formal limits) are indeed limiting cases. We also obtain a number of new trigonometric (q-hypergeometric) integral identities as limits from the elliptic level.
We discuss properties of the index hypergeometric transform (it is named also the Jacobi transform or the Olevsky transform) interpolating analysis of Berezin kernels on rank 1 symmetric spaces. We discuss a unitary intertwining operator from $L^2$ on symmetric space to Berezin deformation of $L^2$. We also find images of some differential operators under the index transform.
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.
A special singular limit $ω_1/ω_2\to 1$ is considered for the Faddeev modular quantum dilogarithm (hyperbolic gamma function) and the corresponding hyperbolic integrals. It brings a new class of hypergeometric identities associated with bilateral sums of Mellin-Barnes type integrals of particular Pochhammer symbol products.
The main aim of the present paper is to establish an integral transform connecting spherical analysis on harmonic NA groups to that of odd dimensional real hyperbolic spaces. Moreover, certain interesting integral identities for the Gauss hypergeometric functions have also been given.
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…
Beijing Institute of Mathematical Sciences and Applications（北京国际数学研究中心）；Laboratory of Theoretical Physics, JINR（约飞联合核子研究所理论物理实验室）；Yerevan Physics Institute（埃里安物理研究所）；National Research University Higher School of Economics（国立研究大学高等经济大学）
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…
The purpose of this article is to demonstrate that i) the framework of elliptic hypergeometric integrals (EHIs) can be extended by input from supersymmetric gauge theory, and ii) analyzing the hyperbolic limit of the EHIs in the extended framework leads to a rich structure containing sharp mathematical problems of interest to supersymmetric quantum field theorists. Both of the above items have already been discussed in the theoretical physics literature. Item i was demonstrated by Dolan and Osborn in 2008. Item ii was discussed in the present author's Ph.D. Thesis in 2016, wherein crucial elem…
We consider a special degeneration limit $ω_1\to - ω_2$ (or $b\to {\rm i}$ in the context of $2d$ Liouville quantum field theory) for the most general univariate hyperbolic beta integral. This limit is also applied to the most general hyperbolic analogue of the Euler-Gauss hypergeometric function and its $W(E_7)$ group of symmetry transformations. Resulting functions are identified as hypergeometric functions over the field of complex numbers related to the ${\rm SL}(2,\mathbb{C})$ group. A new similar nontrivial hypergeometric degeneration of the Faddeev modular quantum dilogarithm (or hyperb…
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…
The main purpose of this paper is to compute all irreducible spherical functions on $G={SL}(2,{\mathbb C})$ of arbitrary type $δ\in \hat K$, where $K={SU}(2)$. This is accomplished by associating to a spherical function $Φ$ on $G$ a matrix valued function $H$ on the three dimensional hyperbolic space ${\mathbb H}=G/K$. The entries of $H$ are solutions of two coupled systems of ordinary differential equations. By an appropriate twisting involving Hahn polynomials we uncouple one of the systems and express the entries of $H$ in terms of Gauss' functions ${}_2F_1$. Just as in the compact instance…
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…
Assume that $p\in(1,\infty]$ and $u=P_{h}[ϕ]$, where $ϕ\in L^{p}(\mathbb{S}^{n-1},\mathbb{R}^{n})$. Then for any $x\in \mathbb{B}^{n}$, we obtain the sharp inequalities $$ |u(x)|\leq \frac{\mathbf{C}_{q}^{\frac{1}{q}}(x)}{(1-|x|^2)^{\frac{n-1 }{p}}} \|ϕ\|_{L^{p}} \quad\text{and}\quad |u(x)|\leq \frac{\mathbf{C}_{q}^{\frac{1}{q}} }{(1-|x|^2)^{\frac{n-1 }{p}}} \|ϕ\|_{L^{p} } $$ for some function $\mathbf{C}_{q}(x)$ and constant $\mathbf{C}_{q}$ in terms of Gauss hypergeometric and Gamma functions, where $q$ is the conjugate of $p$. This result generalize and extend some known result from harmoni…
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…
