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Publication Details
AFRICAN RESEARCH NEXUS
SHINING A SPOTLIGHT ON AFRICAN RESEARCH
mathematics
Ising n-fold integrals as diagonals of rational functions and integrality of series expansions
Journal of Physics A: Mathematical and Theoretical, Volume 46, No. 18, Article 185202, Year 2013
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Description
We show that the n-fold integrals χ(n) 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 functions generalizing 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 diagonals 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. © 2013 IOP Publishing Ltd.
Authors & Co-Authors
Bostan, Alin
France, Palaiseau
Inria Saclay
Boukraa, Salah
Algeria, Blida
Université Blida 1
Christol, Gilles
France, Paris
Institut de Mathématiques de Jussieu-paris Rive Gauche
Hassani, Saoud
Algeria, Algiers
Commissariat à L'energie Atomique
Maillard, Jean Marie
France, Paris
Laboratoire de Physique Théorique de la Matière Condensée
Statistics
Citations: 42
Authors: 5
Affiliations: 5
Identifiers
Doi:
10.1088/1751-8113/46/18/185202
ISSN:
17518113
e-ISSN:
17518121