Solutions for the General, Confluent and Biconfluent Heun equations and their connection with Abel equations

In a recent paper, the canonical forms of a new multi-parameter class of Abel differential equations, so-called AIR, all of whose members can be mapped into Riccati equations, were shown to be related to the differential equations for the hypergeometric 2F1, 1F1 and 0F1 functions. In this paper, a connection between the AIR canonical forms and the Heun General (GHE), Confluent (CHE) and Biconfluent (BHE) equations is presented. This connection fixes the value of one of the Heun parameters, expresses another one in terms of those remaining, and provides closed form solutions in terms of pFq functions for the resulting GHE, CHE and BHE, respectively depending on four, three and two irreducible parameters. This connection also turns evident what is the relation between the Heun parameters such that the solutions admit Liouvillian form, and suggests a mechanism for relating linear equations with N and N-1 singularities through the canonical forms of a non-linear equation of one order less.Comment: Original version submitted to Journal of Physics A: 16 pages, related to math.GM/0002059 and math-ph/0402040. Revised version according to referee's comments: 23 pages. Sign corrected (June/17) in formula (79). Second revised version (July/25): 25 pages. See also http://lie.uwaterloo.ca/odetools.ht

Hypergeometric representation of the two-loop equal mass sunrise diagram

A recurrence relation between equal mass two-loop sunrise diagrams differing in dimensionality by 2 is derived and it's solution in terms of Gauss' 2F1 and Appell's F_2 hypergeometric functions is presented. For arbitrary space-time dimension d the imaginary part of the diagram on the cut is found to be the 2F1 hypergeometric function with argument proportional to the maximum of the Kibble cubic form. The analytic expression for the threshold value of the diagram in terms of the hypergeometric function 3F2 of argument -1/3 is given.Comment: 10 page

Darboux evaluations of algebraic Gauss hypergeometric functions

This paper presents explicit expressions for algebraic Gauss hypergeometric functions. We consider solutions of hypergeometric equations with the tetrahedral, octahedral and icosahedral monodromy groups. Conceptually, we pull-back such a hypergeometric equation onto its Darboux curve so that the pull-backed equation has a cyclic monodromy group. Minimal degree of the pull-back coverings is 4, 6 or 12 (for the three monodromy groups, respectively). In explicit terms, we replace the independent variable by a rational function of degree 4, 6 or 12, and transform hypergeometric functions to radical functions.Comment: The list of seed hypergeometric evaluations (in Section 2) reduced by half; uniqueness claims explained; 34 pages; Kyushu Journal of Mathematics, 201

Beyond series expansions: mathematical structures for the susceptibility of the square lattice Ising model

We first study the properties of the Fuchsian ordinary differential equations for the three and four-particle contributions $\chi^{(3)}$ and $\chi^{(4)}$ of the square lattice Ising model susceptibility. An analysis of some mathematical properties of these Fuchsian differential equations is sketched. For instance, we study the factorization properties of the corresponding linear differential operators, and consider the singularities of the three and four-particle contributions $\chi^{(3)}$ and $\chi^{(4)}$, versus the singularities of the associated Fuchsian ordinary differential equations, which actually exhibit new ``Landau-like'' singularities. We sketch the analysis of the corresponding differential Galois groups. In particular we provide a simple, but efficient, method to calculate the so-called ``connection matrices'' (between two neighboring singularities) and deduce the singular behaviors of $\chi^{(3)}$ and $\chi^{(4)}$. We provide a set of comments and speculations on the Fuchsian ordinary differential equations associated with the $n$-particle contributions $\chi^{(n)}$ and address the problem of the apparent discrepancy between such a holonomic approach and some scaling results deduced from a Painlev\'e oriented approach.Comment: 21 pages Proceedings of the Counting Complexity conferenc

Algebraic transformations of Gauss hypergeometric functions

This article gives a classification scheme of algebraic transformations of Gauss hypergeometric functions, or pull-back transformations between hypergeometric differential equations. The classification recovers the classical transformations of degree 2, 3, 4, 6, and finds other transformations of some special classes of the Gauss hypergeometric function. The other transformations are considered more thoroughly in a series of supplementing articles.Comment: 29 pages; 3 tables; Uniqueness claims and Remark 7.1 clarified by footnotes; formulas (28), (29) correcte

Square lattice Ising model susceptibility: Series expansion method and differential equation for χ(3)\chi^{(3)}

In a previous paper (J. Phys. A {\bf 37} (2004) 9651-9668) we have given the Fuchsian linear differential equation satisfied by $\chi^{(3)}$, 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 $(n-1)$-dimensional integrals representing the $n$-particle contribution to the isotropic square lattice Ising model susceptibility $\chi$. The integration rules are straightforward due to remarkable formulas we derived for these variables. We obtain without any numerical approximation $\chi^{(3)}$ as a fully integrated series in the variable $w=s/2/(1+s^{2})$, where $s =sh (2K)$, with $K=J/kT$ the conventional Ising model coupling constant. We also give some perspectives and comments on these results.Comment: 28 pages, no figur

Ising n-fold integrals as diagonals of rational functions and integrality of series expansions

We show that the n-fold integrals $\chi^{(n)}$ 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 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