Algebraic Gauss hypergeometric functions can be expressed explicitly in several ways. One attractive way is to pull-back their hypergeometric equations (with a finite monodromy) to Fuchsian equations with a finite cyclic monodromy, and express the algebraic solutions as radical functions on the covering curve. This article presents these pull-back transformations of minimal degree for the hypergeometric equations with the tetrahedral, octahedral or icosahedral projective monodromy. The minimal degree is 4, 6 or 12, respectively. The covering curves are called Darboux curves, and they have genus zero or (for some icosahedral Schwarz types) genus one.
Extended double shuffle relations for multiple zeta values are obtained by using the fact that any product of regularized multiple zeta values has two different representations. In this paper, we give two formulas for the generating function of the triple zeta values of any fixed weight by the use of the extended double shuffle relations obtained as two-fold products of double and single zeta values and also as three-fold products of single zeta values. As applications of the formulas, we also obtain parameterized, weighted, and restricted sum formulas for triple zeta values.
Let X be a Hadamard manifold. By applying the stable Jacobi tensor along horospherical foliations of X, a characterization of the real space form is obtained by means of the second fundamental form of the horospheres. The complex space form, the quaternionic space form and the other rank-one symmetric spaces of non-compact type are also similarly characterized. Geometrical characterization of horospheres are also given.
In this paper we study the helicoidal surfaces in the three-dimensional Lorentz-Minkowski space under the condition ΔIIr = Ar, where A is a real 3 × 3 matrix and ΔII is the Laplace operator with respect to the second fundamental form.
In the second author's previous work, we considered an approximation of a catenoid constructed from even truncated cones that maintains minimality in a certain sense. In this paper, we consider such an approximation consisting of odd truncated cones that maintains minimality in the same sense. Through this procedure, we obtain a discrete curve approximating a catenary by exploiting the fact that it is the function that generates a catenoid. In this investigation, the theory of the Gauss hypergeometric functions plays an important role.
We give the monodromy representation and the Pfaffian system of Lauricella's differential equations annihilating the hypergeometric series FD(a,b,c; x) of multivariables. Our representation spaces are twisted homology and cohomology groups associated with integrals representing solutions. Without assigning bases to these groups, we express circuit transformations and components of the connection form in terms of the intersection form of the twisted (co)homology groups. Each of them is characterized by an eigenvector of it.
Motivated by known examples of global integrals which represent automorphic L-functions, this paper initiates the study of a certain two-dimensional array of global integrals attached to any reductive algebraic group, indexed by maximal parabolic subgroups in one direction and by unipotent conjugacy classes in the other. Fourier coefficients attached to unipotent classes, the Gelfand-Kirillov dimension of automorphic representations, and an identity which, empirically, appears to constrain the unfolding process are presented in detail with examples selected from the exceptional groups. Two new Eulerian integrals are included among these examples.
This paper generalizes the notion of local coefficients and fundamental groups of spaces to simplicial coalgebras. We define a Hopf algebra π1(C) from a simplicial coalgebra C as a generalization of a fundamental group, and show that a module over π1(C) corresponds to a local coefficient of C. As a consequence, the Hoschild cohomology of a Hopf algebra H with a coefficient M coincides with the cohomology of the nerve simplicial coalgebra of H with the local coefficient M∗ associated with M.
We introduce a new type of multiple zeta function, which we call a bilateral zeta function. We prove that the bilateral zeta function has a nice Fourier series expansion and the Barnes zeta function can be expressed as a finite sum of bilateral zeta functions. By these properties of the bilateral zeta functions, we obtain simple proofs of some formulas, for example, the reflection formula for the multiple gamma function, the inversion formula for the Dedekind η-function, Ramanujan's formula, Fourier expansion of the Barnes zeta function and multiple Iseki's formula.