45 research outputs found
Lattice Green's Functions of the Higher-Dimensional Face-Centered Cubic Lattices
We study the face-centered cubic lattice (fcc) in up to six dimensions. In
particular, we are concerned with lattice Green's functions (LGF) and return
probabilities. Computer algebra techniques, such as the method of creative
telescoping, are used for deriving an ODE for a given LGF. For the four- and
five-dimensional fcc lattices, we give rigorous proofs of the ODEs that were
conjectured by Guttmann and Broadhurst. Additionally, we find the ODE of the
LGF of the six-dimensional fcc lattice, a result that was not believed to be
achievable with current computer hardware.Comment: 16 pages, final versio
Diagonals of rational functions, pullbacked 2F1 hypergeometric functions and modular forms (unabrigded version)
We recall that diagonals of rational functions naturally occur in lattice
statistical mechanics and enumerative combinatorics. We find that a
seven-parameter rational function of three variables with a numerator equal to
one (reciprocal of a polynomial of degree two at most) can be expressed as a
pullbacked 2F1 hypergeometric function. This result can be seen as the simplest
non-trivial family of diagonals of rational functions. We focus on some
subcases such that the diagonals of the corresponding rational functions can be
written as a pullbacked 2F1 hypergeometric function with two possible rational
functions pullbacks algebraically related by modular equations, thus showing
explicitely that the diagonal is a modular form. We then generalise this result
to eight, nine and ten parameters families adding some selected cubic terms at
the denominator of the rational function defining the diagonal. We finally show
that each of these previous rational functions yields an infinite number of
rational functions whose diagonals are also pullbacked 2F1 hypergeometric
functions and modular forms.Comment: 39 page
On Christol's conjecture
We show that the unresolved examples of Christol's conjecture
\, _3F_{2}\left([2/9,5/9,8/9],[2/3,1],x\right) and
, are indeed diagonals of
rational functions. We also show that other \, _3F_2 and \, _4F_3
unresolved examples of Christol's conjecture are diagonals of rational
functions. Finally we give two arguments that show that it is likely that the
\, _3F_2([1/9, 4/9, 5/9], \, [1/3,1], \, 27 \cdot x) function is a diagonal
of a rational function.Comment: 13 page
Heun functions and diagonals of rational functions (unabridged version)
International audienceWe provide a set of diagonals of simple rational functions of four variables that are seen to be squares of Heun functions. Each time, these Heun functions, obtained by creative telescoping, turn out to be pullbacked 2 F 1 hypergeometric functions and in fact classical modular forms. We even obtained Heun functions that are automorphic forms associated with Shimura curves as solutions of telescopers of rational functions
Diagonals of rational functions, pullbacked 2 F 1 hypergeometric functions and modular forms (unabrigded version)
International audienceWe recall that diagonals of rational functions naturally occur in lattice statistical mechanics and enumerative combinatorics. We find that a seven-parameter rational function of three variables with a numerator equal to one (reciprocal of a polynomial of degree two at most) can be expressed as a pullbacked 2 F 1 hypergeometric function. This result can be seen as the simplest non-trivial family of diagonals of rational functions. We focus on some subcases such that the diagonals of the corresponding rational functions can be written as a pullbacked 2 F 1 hypergeometric function with two possible rational functions pullbacks algebraically related by modular equations, thus showing explicitely that the diagonal is a modular form. We then generalise this result to eight, nine and ten parameters families adding some selected cubic terms at the denominator of the rational function defining the diagonal. We finally show that each of these previous rational functions yields an infinite number of rational functions whose diagonals are also pullbacked 2 F 1 hypergeometric functions and modular forms
Echelons of power series and Gabrielov's counterexample to nested linear Artin Approximation
Gabrielov's famous example for the failure of analytic Artin approximation in
the presence of nested subring conditions is shown to be due to a growth
phenomenon in standard basis computations for echelons, a generalization of the
concept of ideals in power series rings.Comment: To appear in Bulletin of the London Mathematical Societ
Computer-Assisted Proofs of Some Identities for Bessel Functions of Fractional Order
We employ computer algebra algorithms to prove a collection of identities
involving Bessel functions with half-integer orders and other special
functions. These identities appear in the famous Handbook of Mathematical
Functions, as well as in its successor, the DLMF, but their proofs were lost.
We use generating functions and symbolic summation techniques to produce new
proofs for them.Comment: Final version, some typos were corrected. 21 pages, uses svmult.cl