2,462 research outputs found
Square lattice Ising model susceptibility: Series expansion method and differential equation for
In a previous paper (J. Phys. A {\bf 37} (2004) 9651-9668) we have given the
Fuchsian linear differential equation satisfied by , the
``three-particle'' contribution to the susceptibility of the isotropic square
lattice Ising model. This paper gives the details of the calculations (with
some useful tricks and tools) allowing one to obtain long series in polynomial
time. The method is based on series expansion in the variables that appear in
the -dimensional integrals representing the -particle contribution to
the isotropic square lattice Ising model susceptibility . The
integration rules are straightforward due to remarkable formulas we derived for
these variables. We obtain without any numerical approximation as
a fully integrated series in the variable , where , with the conventional Ising model coupling constant. We also
give some perspectives and comments on these results.Comment: 28 pages, no figur
Bases in the solution space of the Mellin system
Local holomorphic solutions z=z(a) to a univariate sparse polynomial equation
p(z) =0, in terms of its vector of complex coefficients a, are classically
known to satisfy holonomic systems of linear partial differential equations
with polynomial coefficients. In this paper we investigate one of such systems
of differential equations which was introduced by Mellin. We compute the
holonomic rank of the Mellin system as well as the dimension of the space of
its algebraic solutions. Moreover, we construct explicit bases of solutions in
terms of the roots of p and their logarithms. We show that the monodromy of the
Mellin system is always reducible and give some factorization results in the
univariate case
An algorithm to obtain global solutions of the double confluent Heun equation
A procedure is proposed to construct solutions of the double confluent Heun
equation with a determinate behaviour at the singular points. The connection
factors are expressed as quotients of Wronskians of the involved solutions.
Asymptotic expansions are used in the computation of those Wronskians. The
feasibility of the method is shown in an example, namely, the Schroedinger
equation with a quasi-exactly-solvable potential
Transformations of Heun's equation and its integral relations
We find transformations of variables which preserve the form of the equation
for the kernels of integral relations among solutions of the Heun equation.
These transformations lead to new kernels for the Heun equation, given by
single hypergeometric functions (Lambe-Ward-type kernels) and by products of
two hypergeometric functions (Erd\'elyi-type). Such kernels, by a limiting
process, also afford new kernels for the confluent Heun equation.Comment: This version was published in J. Phys. A: Math. Theor. 44 (2011)
07520
Multiple zeta values and the WKB method
The multiple zeta values ζ(d1, . . . , dr ) are natural generalizations
of the values ζ(d) of the Riemann zeta functions at integers d. They have many applications, e.g. in knot theory and in quantum physics. It turns out that some generating functions for the multiple zeta values, like fd(x) = 1 − ζ(d)xd + ζ(d, d)x2d − . . . , are related with hypergeometric equations. More precisely, fd(x) is the value at t = 1 of some hypergeometric series dFd−1(t) = 1 − x t + . . ., a solution to a hypergeometric equation of degree d with parameter x. Our idea is to represent fd(x) as some connection coeffi- cient between certain standard bases of solutions near t = 0 and near t = 1. Moreover, we assume that |x| is large. For large complex x the above basic solutions are represented in terms of so-called WKB solutions. The series which define the WKB solutions are divergent and are subject to so-called Stokes phenomenon. Anyway it is possible to treat them rigorously. In the paper we review our results about application of the WKB method to the generating functions
f
x), focusing on the cases d = 2 and d = 3
Binomial D-modules
We study quotients of the Weyl algebra by left ideals whose generators
consist of an arbitrary Z^d-graded binomial ideal I along with Euler operators
defined by the grading and a parameter in C^d. We determine the parameters for
which these D-modules (i) are holonomic (equivalently, regular holonomic, when
I is standard-graded); (ii) decompose as direct sums indexed by the primary
components of I; and (iii) have holonomic rank greater than the generic rank.
In each of these three cases, the parameters in question are precisely those
outside of a certain explicitly described affine subspace arrangement in C^d.
In the special case of Horn hypergeometric D-modules, when I is a lattice basis
ideal, we furthermore compute the generic holonomic rank combinatorially and
write down a basis of solutions in terms of associated A-hypergeometric
functions. This study relies fundamentally on the explicit lattice point
description of the primary components of an arbitrary binomial ideal in
characteristic zero, which we derive in our companion article arxiv:0803.3846.Comment: This version is shorter than v2. The material on binomial primary
decomposition has been split off and now appears in its own paper
arxiv:0803.384
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