Hostname: page-component-cd9895bd7-fscjk Total loading time: 0 Render date: 2024-12-25T18:15:48.093Z Has data issue: false hasContentIssue false
  • English
  • Français

Generalizations of Clausen's Formula and algebraic transformations of Calabi–Yau differential equations

Part of: Families, fibrations Singularities Complex manifolds Surfaces and higher-dimensional varieties Combinatorics

Published online by Cambridge University Press:  30 March 2011

Gert Almkvist ,
Duco van Straten  and
Wadim Zudilin
Gert Almkvist
Affiliation:
Matematikcentrum, Lunds Universitet, Matematik MNF, Box 118, 22100 Lund, Sweden (gert@maths.lth.se)
Duco van Straten
Affiliation:
Fachbereich Mathematik 08, Institut für Mathematik, AG Algebraische Geometrie, Johannes Gutenberg-Universität, 55099 Mainz, Germany (straten@mathematik.uni-mainz.de)
Wadim Zudilin
Affiliation:
School of Mathematical and Physical Sciences, The University of Newcastle, Callaghan, NSW 2308, Australia (wadim.zudilin@newcastle.edu.au)
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We provide certain unusual generalizations of Clausen's and Orr's theorems for solutions of fourth-order and fifth-order generalized hypergeometric equations. As an application, we present several examples of algebraic transformations of Calabi–Yau differential equations.

Keywords

Calabi–Yau differential equationgeneralized hypergeometric seriesmonodromyalgebraic transformation

MSC classification

Secondary: 33C20: Generalized hypergeometric series, $_pF_q$ 05A10: Factorials, binomial coefficients, combinatorial functions 05A19: Combinatorial identities, bijective combinatorics 14D05: Structure of families (Picard-Lefschetz, monodromy, etc.) 14J32: Calabi-Yau manifolds 32Q25: Calabi-Yau theory 32S40: Monodromy; relations with differential equations and $D$-modules
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 54 , Issue 2 , June 2011 , pp. 273 - 295
Copyright
Copyright © Edinburgh Mathematical Society 2011

References

1.Almkvist, G., Calabi–Yau differential equations of degree 2 and 3 and Yifan Yang's pullback, Preprint (http://arxiv.org/abs/math.AG/0612215; 2006).Google Scholar
2.Almkvist, G. and Zudilin, W., Differential equations, mirror maps and zeta values, in Mirror symmetry V (ed. Yui, N., Yau, S.-T. and Lewis, J. D.), AMS/IP Studies in Advanced Mathematics, Volume 38, pp. 481–515 (American Mathematical Society and International Press, Providence, RI, 2006).Google Scholar
3.Almkvist, G., van Enckevort, C., van Straten, D. and Zudilin, W., Tables of Calabi–Yau equations, Preprint (http://arxiv.org/abs/math.AG/0507430; 2005).Google Scholar
4.André, Y., G-functions and geometry, Aspects of Mathematics, Volume 13 (Vieweg & Sohn, Braunschweig, 1989).Google Scholar
5.Apéry, R., Irrationalité de ζ(2) et ζ(3), in Journées arithmétiques de Luminy, Astérisque, Volume 61, pp. 11–13 (Société Mathématique de France, Paris, 1979).Google Scholar
6.Beauville, A., Les familles stables de courbes elliptiques sur ℙ1 admettant quatre fibres singuliéres, C. R. Acad. Sci. Paris Sér. I 294 (1982), 657–660.Google Scholar
7.Beukers, F., Irrationality of π2, periods of an elliptic curve and Λ1(5), in Diophantine approximations and transcendental numbers, Progress in Mathematics, Volume 31, pp. 47–66 (Birkhäuser, 1983).Google Scholar
8.Beukers, F., On Dwork's accessory parameter problem, Math. Z. 241 (2002), 425–444.CrossRefGoogle Scholar
9.Bogner, M., Differentielle Galoisgruppen und Transformationstheorie für Calabi-Yau-Operatoren vierter Ordnung, Diploma-Thesis, Institut für Mathematik, Johannes Gutenberg-Universität, Mainz (2008).Google Scholar
10.Golyshev, V. V., Classification problems and mirror duality, in Surveys in geometry and number theory: reports on contemporary Russian mathematics, London Mathematical Society Lecture Note Series, Volume 338, pp. 88–121 (Cambridge University Press, 2007).Google Scholar
11.Krattenthaler, C. and Rivoal, T., Multivariate p-adic formal congruences and integrality of Taylor coefficients of mirror maps, in Théories galoisiennes et arithmétiques des équations différentielles (ed. Di Vizio, L. and Rivoal, T.), Séminaires et Congrès (SociétéMathématique de France, Paris (in press).Google Scholar
12.Lian, B. H. and Yau, S.-T., Differential equations from mirror symmetry, in Surveys in differential geometry: differential geometry inspired by string theory, Surveys in Differential Geometry, Volume 5, pp. 510–526 (International Press, Boston, MA, 1999).Google Scholar
13.Miranda, R. and Persson, U., On extremal rational elliptic surfaces, Math. Z. 193 (1986), 537–558.CrossRefGoogle Scholar
14.Peters, C., Monodromy and Picard–Fuchs equations for families of K3-surfaces and elliptic curves, Annales Sci. École Norm. Sup. (4) 19 (1986), 583–607.CrossRefGoogle Scholar
15.Peters, C. and Stienstra, J., A pencil of K3-surfaces related to Apéry's recurrence for ζ(3) and Fermi surfaces for potential zero, in Arithmetic of complex manifolds, Lecture Notes in Mathematics, Volume 1399, pp. 110–127 (Springer, 1989).Google Scholar
16.Schmickler-Hirzebruch, U., Elliptische Flächen über ℙ1ℂ mit drei Ausnahmefasernund die hypergeometrische Differentialgleichung, Schriftenreihe des Mathematischen Instituts der Universität Münster, 2. Serie 33 (Universität Münster, Mathematisches Institut, Münster, 1985).Google Scholar
17.Slater, L. J., Generalized hypergeometric functions (Cambridge University Press, 1966).Google Scholar
18.van Enckevort, C. and van Straten, D., Monodromy calculations of fourth-order equations of Calabi–Yau type, in Mirror symmetry V (ed. Yui, N., Yau, S.-T. and Lewis, J. D.), AMS/IP Studies in Advanced Mathematics, Volume 38, pp. 539–559 (American Mathematical Society and International Press, Providence, RI, 2006).Google Scholar
19.Whittaker, E. T. and Watson, G. N., A course of modern analysis, 4th edn (Cambridge University Press, 1927).Google Scholar
20.Zagier, D., Integral solutions of Apéry-like recurrence equations, in Groups and symmetries: from Neolithic Scots to John McKay, CRM Proceedings and Lecture Notes, Volume 47, pp. 349–366 (American Mathematical Society, Providence, RI, 2009).Google Scholar
21.Zudilin, W., Quadratic transformations and Guillera's formulas for 1/π2, Math. Notes 81 (2007), 297–301.CrossRefGoogle Scholar