## Algebraic transformations of Gauss hypergeometric functions, preprint (2004)

Citations: | 22 - 9 self |

### BibTeX

@MISC{Vidūnas04algebraictransformations,

author = {Raimundas Vidūnas},

title = {Algebraic transformations of Gauss hypergeometric functions, preprint},

year = {2004}

}

### OpenURL

### Abstract

The paper classifies algebraic transformations of Gauss hypergeometric functions and pull-back transformations between hypergeometric differential equations. This classification recovers the classical transformations of degree 2, 3, 4, 6, and finds other transformations of some special classes of the Gauss hypergeometric function.

### Citations

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Citation Context ...2 shows that any pull-back transformation (1/2, 1/4, 1/4) d ←− (1/2, 1/4, 1/4) is induced by an endomorphism of E1. The ring of endomorphisms of E1 is isomorphic to the ring Z[i] of Gaussian integers =-=[Sil86]-=-. We identify i ∈ Z[i] with the endomorphism (x, y) ↦→ (−x, iy). Addition of endomorphisms is equivalent 19to the chord-and-tangent addition law on E1, so all endomorphisms are computable. For exampl... |

84 |
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Citation Context ... and local exponents there. The linear space of solutions is determined by the same data; it can be defined without reference to hypergeometric equations as a local system on the projective line; see =-=[Kat96]-=-, [Gra86, Section 1.4]. The notation for it is ⎧ ⎪⎨ P ⎪⎩ α β γ a1 b1 c1 a2 b2 c2 z ⎫ ⎪⎬ , (9) ⎪⎭ where α, β, γ ∈ P 1 z are the singular points, and a1, a2; b1, b2; c1, c2 are the local exponents at th... |

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Citation Context ...ve the same meaning as in formula (1). Geometrically, this is a pull-back transformation of equation (4) with respect to the finite covering ϕ : P 1 → P 1 determined by the rational function ϕ(x). In =-=[Kit03]-=- these transformations are called RS-transformations. Recall that a rational function on a Riemann surface is a Belyi function [Sha00, Kre03] if it has at most 3 critical values, or equivalently, if t... |

19 |
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Citation Context ...h branching pattern. Possible compositions of small degree coverings are easy to list and identify. Ultimately, Table 1 yields precisely the classical transformations of degree 3, 4, 6 due to Goursat =-=[Gou81]-=-. Formulas (3)–(5) are examples of classical transformations for the three indecomposable coverings. The two degenerate cases are: • k1 = 2, k2 = 2, d any. The monodromy group of H1 is a dihedral grou... |

12 |
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Citation Context ...substituted t ↦→ x −2 . We recognize an integral of a holomorphic differential form on the genus 1 curve y 2 = x 3 − x. Let E1 denote the corresponding elliptic curve in the standard Weierstrass form =-=[Sil86]-=-. If (ψx, ψy) is an endomorphism of E1, then the substitution x ↦→ ψx(x, √ x 3 − x) in (12) gives an integral of a holomorphic differential form again. Since the linear space of holomorphic differenti... |

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Citation Context ...+1/k3 > 1. The monodromy groups of H1 and H2 are finite, the hypergeometric functions are algebraic. The degree d is unbounded. The most important transformations are those implied by Klein’s theorem =-=[Kle78]-=-. In particular, any hypergeometric equation with the tetrahedral, octahedral or icosahedral monodromy group is a pull-back transformation of a standard hypergeometric equation with that monodromy gro... |

8 | Some examples of RS2 3 (3)-transformations of ranks 5 and 6 as the higher order transformations for the hypergeometric function
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Citation Context ...ther similar example is the following formula, proved in [BBG95, Theorem 2.3]: ( 1 c, c + 2F1 ∣ x3 ) = ( 1 + 2x ) ( 1 −3c c, c+ (1−x)3 2F1 1 − ∣ (1+2x) 3 ) . (8) The case c = 1/3 was found earlier in =-=[BB91]-=-. 3 3c+1 2 A pull-back transformation between hypergeometric equations usually gives several identities like (1) between some of the 24 Kummer’s solutions of both equations. Therefore it is appropriat... |

5 |
hypergeometric function
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Citation Context ...n scheme is presented in Section 3. We follow the approach of Riemann and Papperitz [AAR99, Sections 2.3 and 3.9]. For basic theory of hypergeometric functions and Fucshian equations we also refer to =-=[Beu02]-=- or [vdW02, Chapters 1 and 2]. In Section 4 we outline more interesting types of algebraic transformations. All non-classical special cases are extensively considered in separate papers [Vid04a], [Vid... |

5 | evaluations of algebraic Gauss hypergeometric functions, arXiv:math 0504264
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(Show Context)
Citation Context ...fer to [Beu02] or [vdW02, Chapters 1 and 2]. In Section 4 we outline more interesting types of algebraic transformations. All non-classical special cases are extensively considered in separate papers =-=[Vid04a]-=-, [Vid04b], [Vid03], [Vid04c]. 2 Preliminaries The hypergeometric differential equation is [AAR99, Formula (2.3.5)]: z (1 − z) d2y(z) dz2 + � C − (A+B+1) z � dy(z) − A B y(z) = 0. (6) dz This is a Fuc... |

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5 | Degenerate Gauss hypergeometric functions
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(Show Context)
Citation Context ...rities; their monodromy group is abelian. The explicit classification scheme of Section 3 refers to this case three times. These equations form a special sample of degenerate hypergeometric equations =-=[Vid04a]-=-. Transformations of this Section will be described more thoroughly in a separate paper, along with rendition of algebraic transformations for all classes of degenerate Gauss hypergeometric functions.... |

4 |
Contiguous relations of hypergeometric series
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- 2003
(Show Context)
Citation Context ...1, 2, . . ., k − 1}. (47) 15Note that this kind of function can be evaluated by (43) if n = m = ℓ = 0 or n = 0, m = ℓ = −1. Using those two evaluations and contiguous relations [AAR99, Section 2.5], =-=[Vid03]-=- express function (47) in the form ( √ ) p/k 1 + x R(x) , with R(x) ∈ C(x). 2 2. Write R(x) k (1 + √ x) p in the form θ1(x) + θ2(x) √ x with θ1(x), θ2(x) ∈ C[x]. Just like for transformations with gen... |

3 | Lamé Equations with Finite Monodromy - Waall - 2002 |

3 |
A detail in Conformal Representation
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Citation Context ...urth column of Table 3. The last column indicates existence of Coxeter decompositions described at the end of Section 3. The three cases which admit a Coxeter decomposition are implicitly obtained in =-=[Hod20]-=- and [Beu02]. Here we give rational functions defining the indecomposable pull-back transformations, and examples of corresponding algebraic transformations of Gauss hypergeometric functions. 21Local... |

3 | On families of geometric parasitic solutions for Belyi systems of genus zero - Kreines |

2 |
An application of conformal representation to certain hypergeometric series
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(Show Context)
Citation Context ...sformations of hyperbolic hypergeometric functions. The list of these transformations is finite [Vid04c], the maximal degree of their coverings is 24. Some of these transformations are anticipated in =-=[Hod18]-=-, [Beu02]. 4 Explicit transformations Solutions of hypergeometric equations with an abelian or dihedral monodromy group are very explicit. Their transformations are extensively considered in [Vid04b],... |

2 |
Transformations os some Gauss hypergeometric functions. 2004. Submited for the proceedings of the 7th international symposium “Orthogonal Polynomials, Special Functions and Applications
- Vidūnas
- 2005
(Show Context)
Citation Context ...nd to the endomorphisms of the corresponding elliptic curve. • 1/k1 + 1/k2 + 1/k3 < 1. Here we have transformations of hyperbolic hypergeometric functions. The list of these transformations is finite =-=[Vid04c]-=-, the maximal degree of their coverings is 24. Some of these transformations are anticipated in [Hod18], [Beu02]. 4 Explicit transformations Solutions of hypergeometric equations with an abelian or di... |

1 | Transformations of hypergeometric elliptic integrals
- Vidūnas
- 2003
(Show Context)
Citation Context ...dW02, Chapters 1 and 2]. In Section 4 we outline more interesting types of algebraic transformations. All non-classical special cases are extensively considered in separate papers [Vid04a], [Vid04b], =-=[Vid03]-=-, [Vid04c]. 2 Preliminaries The hypergeometric differential equation is [AAR99, Formula (2.3.5)]: z (1 − z) d2y(z) dz2 + � C − (A+B+1) z � dy(z) − A B y(z) = 0. (6) dz This is a Fuchsian equation with... |

1 |
Degenerate and dihedral Gauss hypergeometric functions
- Vidūnas
- 2004
(Show Context)
Citation Context ...u02] or [vdW02, Chapters 1 and 2]. In Section 4 we outline more interesting types of algebraic transformations. All non-classical special cases are extensively considered in separate papers [Vid04a], =-=[Vid04b]-=-, [Vid03], [Vid04c]. 2 Preliminaries The hypergeometric differential equation is [AAR99, Formula (2.3.5)]: z (1 − z) d2y(z) dz2 + � C − (A+B+1) z � dy(z) − A B y(z) = 0. (6) dz This is a Fuchsian equa... |

1 | Sur l'equation dierentielle lineaire qui adment pour integrale la serie hypergeometrique - Goursat |

1 | Divisible tilings in the hyperbolic plane
- Broughton, Haney, et al.
(Show Context)
Citation Context ... hyperbolic triangles and quadrangles into hyperbolic triangles are classified in [Fel98, BHMM], where they are called Coxeter decompositions and divisible tilings respectively. The classification in =-=[BHMM]-=- is incomplete; for instance, it misses triangulation (b) in Figure 1. We adopt the terminology of [Fel98]. The last columns of Tables 1 and 2 tell us which transformations of hypergeometric equations... |