811 research outputs found
Elliptic Hypergeometric Laurent Biorthogonal Polynomials with a Dense Point Spectrum on the Unit Circle
Using the technique of the elliptic Frobenius determinant, we construct new
elliptic solutions of the -algorithm. These solutions can be interpreted as
elliptic solutions of the discrete-time Toda chain as well. As a by-product, we
obtain new explicit orthogonal and biorthogonal polynomials in terms of the
elliptic hypergeometric function . Their recurrence coefficients
are expressed in terms of the elliptic functions. In the degenerate case we
obtain the Krall-Jacobi polynomials and their biorthogonal analogs
Ising n-fold integrals as diagonals of rational functions and integrality of series expansions
We show that the n-fold integrals of the magnetic susceptibility
of the Ising model, as well as various other n-fold integrals of the "Ising
class", or n-fold integrals from enumerative combinatorics, like lattice Green
functions, correspond to a distinguished class of function generalising
algebraic functions: they are actually diagonals of rational functions. As a
consequence, the power series expansions of the, analytic at x=0, solutions of
these linear differential equations "Derived From Geometry" are globally
bounded, which means that, after just one rescaling of the expansion variable,
they can be cast into series expansions with integer coefficients. We also give
several results showing that the unique analytical solution of Calabi-Yau ODEs,
and, more generally, Picard-Fuchs linear ODEs, with solutions of maximal
weights, are always diagonal of rational functions. Besides, in a more
enumerative combinatorics context, generating functions whose coefficients are
expressed in terms of nested sums of products of binomial terms can also be
shown to be diagonals of rational functions. We finally address the question of
the relations between the notion of integrality (series with integer
coefficients, or, more generally, globally bounded series) and the modularity
of ODEs.Comment: This paper is the short version of the larger (100 pages) version,
available as arXiv:1211.6031 , where all the detailed proofs are given and
where a much larger set of examples is displaye
Geometry of q-Hypergeometric Functions as a Bridge between Yangians and Quantum Affine Algebras
The rational quantized Knizhnik-Zamolodchikov equation (qKZ equation)
associated with the Lie algebra is a system of linear difference
equations with values in a tensor product of Verma modules. We solve the
equation in terms of multidimensional -hypergeometric functions and define a
natural isomorphism between the space of solutions and the tensor product of
the corresponding quantum group Verma modules, where the parameter
is related to the step of the qKZ equation via .
We construct asymptotic solutions associated with suitable asymptotic zones
and compute the transition functions between the asymptotic solutions in terms
of the trigonometric -matrices. This description of the transition functions
gives a new connection between representation theories of Yangians and quantum
loop algebras and is analogous to the Kohno-Drinfeld theorem on the monodromy
group of the differential Knizhnik-Zamolodchikov equation.
In order to establish these results we construct a discrete Gauss-Manin
connection, in particular, a suitable discrete local system, discrete homology
and cohomology groups with coefficients in this local system, and identify an
associated difference equation with the qKZ equation.Comment: 66 pages, amstex.tex (ver. 2.1) and amssym.tex are required;
misprints are correcte
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