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19
Transformations of some Gauss hypergeometric functions
 J. Comput. Appl. Math
"... This paper presents explicit algebraic transformations of some Gauss hypergeometric functions. Specifically, the considered transformations apply to hypergeometric solutions of hypergeometric differential equations with local exponent differences 1/k, 1/ℓ, 1/m such that k, ℓ,m are positive integers ..."
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Cited by 8 (4 self)
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This paper presents explicit algebraic transformations of some Gauss hypergeometric functions. Specifically, the considered transformations apply to hypergeometric solutions of hypergeometric differential equations with local exponent differences 1/k, 1/ℓ, 1/m such that k, ℓ,m are positive integers and 1/k + 1/ℓ + 1/m < 1. All algebraic transformations of these Gauss hypergeometric functions are considered. We show that apart from classical transformations of degree 2, 3, 4, 6 there are several
EXPLICIT FORMULA FOR THE GENERATING SERIES OF DIAGONAL 3D ROOK PATHS
 SÉMINAIRE LOTHARINGIEN DE COMBINATOIRE 66 (2011), ARTICLE B66A
, 2011
"... Let an denote the number of ways in which a chess rook can move from a corner cell to the opposite corner cell of an n × n × n threedimensional chessboard, assuming that the piece moves closer to the goal cell at each step. We describe the computerdriven discovery and proof of the fact that the ge ..."
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Cited by 6 (2 self)
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Let an denote the number of ways in which a chess rook can move from a corner cell to the opposite corner cell of an n × n × n threedimensional chessboard, assuming that the piece moves closer to the goal cell at each step. We describe the computerdriven discovery and proof of the fact that the generating series G(x) = ∑ n≥0 anxn admits the following explicit expression in terms of a Gaussian hypergeometric function:
Darboux evaluations of algebraic Gauss hypergometric functions
, 2004
"... 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 pullback such a hypergeometric equation onto its Darboux curve so that the pul ..."
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Cited by 5 (2 self)
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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 pullback such a hypergeometric equation onto its Darboux curve so that the pullbacked equation has a cyclic monodromy group. Minimal degree of the pullback 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. 1
Remarks Towards Classification of RS2 4 (3)Transformations and Algebraic Solutions of the Sixth Painlevé Equation
 Proceedings of the Angers Conference “Asymptotic Theories and Painlevé Equations” (June 01–05, 2004). Sèminaires et Congrès 14 (2006
"... We introduce a notion of the divisor type for rational functions and show that it can be effectively used for the classification of the deformations of dessins d’enfants related with the construction of the algebraic solutions of the sixth Painlevé equation via the method of RStransformations. ..."
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Cited by 5 (0 self)
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We introduce a notion of the divisor type for rational functions and show that it can be effectively used for the classification of the deformations of dessins d’enfants related with the construction of the algebraic solutions of the sixth Painlevé equation via the method of RStransformations.
Solving second order linear differential equations with Klein’s theorem
 In ISSAC’05
, 2005
"... Given a second order linear differential equations with coefficients in a field k = C(x), the Kovacic algorithm finds all Liouvillian solutions, that is, solutions that one can write in terms of exponentials, logarithms, integration symbols, algebraic extensions, and combinations thereof. A theorem ..."
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Cited by 4 (2 self)
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Given a second order linear differential equations with coefficients in a field k = C(x), the Kovacic algorithm finds all Liouvillian solutions, that is, solutions that one can write in terms of exponentials, logarithms, integration symbols, algebraic extensions, and combinations thereof. A theorem of Klein states that, in the most interesting cases of the Kovacic algorithm (i.e when the projective differential Galois group is finite), the differential equation must be a pullback (a change of variable) of a standard hypergeometric equation. This provides a way to represent solutions of the differential equation in a more compact way than the format provided by the Kovacic algorithm. Formulas to make Klein’s theorem effective were given in [4, 2, 3]. In this paper we will give a simple algorithm based on such formulas. To make the algorithm more easy to implement for various differential fields k, we will give a variation on the earlier formulas, namely we will base the formulas on invariants of the differential Galois group instead of semiinvariants. 1.
Schlesinger transformations for algebraic Painlevé VI solutions
"... Various Schlesinger transformations can be combined with a direct pullback of a hypergeometric 2×2 system to obtain RS2 4pullback transformations to isomonodromic 2 × 2 Fuchsian systems with 4 singularities. The corresponding Painlevé VI solutions are algebraic functions, possibly in different orb ..."
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Cited by 4 (3 self)
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Various Schlesinger transformations can be combined with a direct pullback of a hypergeometric 2×2 system to obtain RS2 4pullback transformations to isomonodromic 2 × 2 Fuchsian systems with 4 singularities. The corresponding Painlevé VI solutions are algebraic functions, possibly in different orbits under Okamoto transformations. This paper demonstrates a direct computation of Schlesinger transformations acting on several apparent singular points, and presents an algebraic procedure (via syzygies) of computing algebraic Painlevé VI solutions without deriving full RSpullback transformations.
On rationally parametrized modular equations
"... Abstract. The classical theory of elliptic modular equations is reformulated and extended, and many new rationally parametrized modular equations are discovered. Each arises in the context of a family of elliptic curves attached to a genuszero congruence subgroup Γ0(N), as an algebraic transformati ..."
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Cited by 3 (0 self)
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Abstract. The classical theory of elliptic modular equations is reformulated and extended, and many new rationally parametrized modular equations are discovered. Each arises in the context of a family of elliptic curves attached to a genuszero congruence subgroup Γ0(N), as an algebraic transformation of elliptic curve periods, which are parametrized by a Hauptmodul (function field generator). Since the periods satisfy a Picard–Fuchs equation, which is of hypergeometric, Heun, or more general type, the new equations can be viewed as algebraic transformation formulas for special functions. The ones for N = 4,3, 2 yield parametrized modular transformations of Ramanujan’s elliptic integrals of signatures 2, 3,4. The case of signature 6 will require an extension of the present theory, to one of modular equations for general elliptic surfaces.
Transformations of algebraic Gauss hypergeometric functions
, 2003
"... A celebrated theorem of Klein implies that any hypergeometric differential equation with algebraic solutions is a pullback of one of the few standard hypergeometric equations with algebraic solutions. The most interesting cases are hypergeometric equations with tetrahedral, octahedral or icosahedra ..."
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Cited by 2 (2 self)
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A celebrated theorem of Klein implies that any hypergeometric differential equation with algebraic solutions is a pullback of one of the few standard hypergeometric equations with algebraic solutions. The most interesting cases are hypergeometric equations with tetrahedral, octahedral or icosahedral monodromy groups. We give an algorithm for computing Klein’s pullback coverings in these cases, based on certain explicit expressions (Darboux evaluations) of algebraic hypergeometric functions. The explicit expressions can be computed from a data base (covering the Schwarz table) and using contiguous relations. Klein’s pullback transformations also induce algebraic transformations between hypergeometric solutions and a standard hypergeometric function with the same finite monodromy group.
Renormalization, isogenies and rational symmetries of differential equations
, 911
"... Abstract. We give an example of infinite order rational transformation that leaves a linear differential equation covariant. This example can be seen as a nontrivial but still simple illustration of an exact representation of the renormalization group. ..."
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Cited by 1 (1 self)
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Abstract. We give an example of infinite order rational transformation that leaves a linear differential equation covariant. This example can be seen as a nontrivial but still simple illustration of an exact representation of the renormalization group.
Closed Form Solutions for Linear Differential and Difference Equations, Project Description
, 2007
"... Many scientists use computer algebra systems to solve linear differential or difference equations. These programs check if an equation can be matched ..."
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Cited by 1 (0 self)
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Many scientists use computer algebra systems to solve linear differential or difference equations. These programs check if an equation can be matched