5,164 research outputs found
Nonlinear Differential Equations Satisfied by Certain Classical Modular Forms
A unified treatment is given of low-weight modular forms on \Gamma_0(N),
N=2,3,4, that have Eisenstein series representations. For each N, certain
weight-1 forms are shown to satisfy a coupled system of nonlinear differential
equations, which yields a single nonlinear third-order equation, called a
generalized Chazy equation. As byproducts, a table of divisor function and
theta identities is generated by means of q-expansions, and a transformation
law under \Gamma_0(4) for the second complete elliptic integral is derived.
More generally, it is shown how Picard-Fuchs equations of triangle subgroups of
PSL(2,R) which are hypergeometric equations, yield systems of nonlinear
equations for weight-1 forms, and generalized Chazy equations. Each triangle
group commensurable with \Gamma(1) is treated.Comment: 40 pages, final version, accepted by Manuscripta Mathematic
Properties of generalized univariate hypergeometric functions
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
degenerations of the hyperbolic and trigonometric hypergeometric functions are
identified with Ruijsenaars' relativistic hypergeometric function and the
Askey-Wilson function, respectively. We show that the degeneration process
yields various new and known identities for hyperbolic and trigonometric
special functions. We also describe an intimate connection between the
hyperbolic and trigonometric theory, which yields an expression of the
hyperbolic hypergeometric function as an explicit bilinear sum in trigonometric
hypergeometric functions.Comment: 46 page
Positive convolution structure for a class of Heckman-Opdam hypergeometric functions of type BC
In this paper, we derive explicit product formulas and positive convolution
structures for three continuous classes of Heckman-Opdam hypergeometric
functions of type . For specific discrete series of multiplicities these
hypergeometric functions occur as the spherical functions of non-compact
Grassmann manifolds over one of the (skew) fields We write the product formula of these spherical
functions in an explicit form which allows analytic continuation with respect
to the parameters. In each of the three cases, we obtain a series of hypergroup
algebras which include the commutative convolution algebras of -biinvariant
functions on
Moduli spaces of flat tori and elliptic hypergeometric functions
In the genus one case, we make explicit some constructions of Veech on flat
surfaces and generalize some geometric results of Thurston about moduli spaces
of flat spheres as well as some equivalent ones but of an
analytico-cohomological nature of Deligne-Mostow, which concern the monodromy
of Appell-Lauricella hypergeometric functions.
In the twin paper arXiv:1604.01812, we follow Thurston's approach and study
moduli spaces of flat tori with conical singularities and prescribed holonomy
by means of geometrical methods relying on surgeries for flat surfaces.
In the present paper, we study the same objects making use of analytical and
cohomological methods, more in the spirit of Deligne-Mostow's paper.Comment: 156 pages, 20 figures. Preliminary version. Comments are welcom
The Chazy XII Equation and Schwarz Triangle Functions
Dubrovin [Lecture Notes in Math., Vol. 1620, Springer, Berlin, 1996, 120-348]
showed that the Chazy XII equation , , 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 , can be expressed as where
solve the generalized Darboux-Halphen system. This relationship holds
only for certain values of the coefficients and the
Darboux-Halphen parameters , which are enumerated in
Table 2. Consequently, the Chazy XII solution is parametrized by a
particular class of Schwarz triangle functions
which are used to represent the solutions of the Darboux-Halphen system.
The paper only considers the case where . The associated
triangle functions are related among themselves via rational maps that are
derived from the classical algebraic transformations of hypergeometric
functions. The Chazy XII equation is also shown to be equivalent to a
Ramanujan-type differential system for a triple
Modular Solutions to Equations of Generalized Halphen Type
Solutions to a class of differential systems that generalize the Halphen
system are determined in terms of automorphic functions whose groups are
commensurable with the modular group. These functions all uniformize Riemann
surfaces of genus zero and have --series with integral coefficients.
Rational maps relating these functions are derived, implying subgroup relations
between their automorphism groups, as well as symmetrization maps relating the
associated differential systems.Comment: PlainTeX 36gs. (Formula for Hecke operator corrected.
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