9,758 research outputs found
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
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
Bivariate second--order linear partial differential equations and orthogonal polynomial solutions
In this paper we construct the main algebraic and differential properties and
the weight functions of orthogonal polynomial solutions of bivariate
second--order linear partial differential equations, which are admissible
potentially self--adjoint and of hypergeometric type. General formulae for all
these properties are obtained explicitly in terms of the polynomial
coefficients of the partial differential equation, using vector matrix
notation. Moreover, Rodrigues representations for the polynomial eigensolutions
and for their partial derivatives of any order are given. Finally, as
illustration, these results are applied to specific Appell and Koornwinder
orthogonal polynomials, solutions of the same partial differential equation.Comment: 27 page
Diagonal Ising susceptibility: elliptic integrals, modular forms and Calabi-Yau equations
We give the exact expressions of the partial susceptibilities
and for the diagonal susceptibility of the Ising model in terms
of modular forms and Calabi-Yau ODEs, and more specifically,
and hypergeometric functions. By solving the connection problems we
analytically compute the behavior at all finite singular points for
and . We also give new results for .
We see in particular, the emergence of a remarkable order-six operator, which
is such that its symmetric square has a rational solution. These new exact
results indicate that the linear differential operators occurring in the
-fold integrals of the Ising model are not only "Derived from Geometry"
(globally nilpotent), but actually correspond to "Special Geometry"
(homomorphic to their formal adjoint). This raises the question of seeing if
these "special geometry" Ising-operators, are "special" ones, reducing, in fact
systematically, to (selected, k-balanced, ...) hypergeometric
functions, or correspond to the more general solutions of Calabi-Yau equations.Comment: 35 page
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