Tsinghua lectures on hypergeometric functions unfinished and. We study the symplectic geometry of moduli spaces m r of polygons with fixed side lengths in euclidean space. Hyperbolic monodromy groups for the hypergeometric equation. This is a reasonably popular topic, with goursats original 9page contribution 18 as the starting point. The idea is that one can extend a complexanalytic function from here on called simply analytic function along curves starting in the original domain of the function and ending in the larger set. The recursions they satisfy gives rise to a system of partial di. Suppose that a second order linear di erential equation has only one regular singular point at z 0 and in nity is an ordinary point. These functions generalize the euler gauss hypergeometric function for the rank one root system and the elementary spherical functions on a real semisimple lie group for particular parameter values. Algebraic transformations of gauss hypergeometric functions.
There is also a classical method to compute monodromy with respect to an explicit basis of functions using socalled mellinbarnes integrals, see f. Suppose that a machine shop orders 500 bolts from a supplier. Hypergeometric distribution proposition the mean and variance of the hypergeometric rv x having pmf hx. Monodromy of hypergeometric systems and analytic complexity of algebraic functions abstract. Monodromy of hypergeometric systems and analytic complexity of algebraic functions. We shall introduce the basics of classical gauss hypergeometric equation and to study its monodromy problem. To determine whether to accept the shipment of bolts,the manager of the facility randomly selects 12 bolts. The monodromy representation and twisted period relations for. Generalized euler integrals and a hypergeometric functions. Monodromy of ahypergeometric functions internet archive. We should point out that algebraic transformations of hypergeometric functions, in particular, of modular origin, are related to the monodromy of the underlying linear di erential equations. However, it does cover aspects such as monodromy calculations for this system and a moduli interpretation of the underlying geometry. The hypergeometric solutions are algebraic functions. Citeseerx scientific documents that cite the following paper.
Algebraic ahypergeometric functions and their monodromy algebraische ahypergeometrische functies en hun monodromie meteensamenvattinginhetnederlands. Dec, 2018 department of mathematics, graduate school of science, kobe university, 11, rokkodai, nadaku, kobe, japan. The parameters of the hypergeometric function are zeros and poles of this rational function. We describe results of levelt and beukersheckman on the explicit computation of monodromy for generalised. Archived from the original pdf on 20170624 cs1 maint. Beukers, frits 2002, gauss hypergeometric function. Monodromy of hypergeometric functions and nonlattice. If we replace m n by p, then we get ex np and vx n n n 1 np1 p.
Monodromy of hypergeometric functions and nonlattice integral monodromy deligne, pierre. To prove it we use the classification of irreducible perverse sheaves by means of delignegoresky macpherson extensions of local systems bbd. Maximally reducible monodromy of bivariate hypergeometric systems. Monodromy of hypergeometric functions and nonlattice integral. In this article, we settle the arithmeticity problem for the fourteen monodromy groups, by showing that, the remaining four are arithmetic. Monodromy representations associated with the generalized. Bravthomas in show that seven of the remaining eleven are thin. Monodromy, riemannhilbert problem eulergaus 2f1, first properties monodromy and schwarzklein theory including schwarzs list and criterium for arithmeticity of the monodromy group connection with modular forms and functions then we continue with more recent work. Monodromy of hypergeometric systems and analytic complexity. Example 3 using the hypergeometric probability distribution problem. Monodromy of hypergeometric functions and nonlattice integral monodromy. The combinatorial tractability of a hypergeometric systems, combined with existing deep results on monodromy of classical hypergeometric functions see, e. Heckman 2 1 university of utrecht, department of mathematics, budapestlaan 6, nl3508 ta utrecht, the netherlands 2 university of leiden, department of mathematics, niels bohrweg 1, nl2333 al leiden, the netherlands contents 1. Timur sadykov, independent university of moscow title.
Monodromy groups of hypergeometric functions satisfying algebraic equations kato, mitsuo and noumi, masatoshi, tohoku mathematical journal, 2003. Singularities, monodromy, local behavior of solutions, normal forms. Eulers hypergeometric series are now often called gauss hypergeometric functions. Thus, we can obtain almost without calculation explicit determination of many polynomials and hypergeometric power series related to. Hyperbolic hypergeometric monodromy groups and geometric finiteness. However, the literature in this direction is sparse but, see 6, 26. A hypergeometric function, namely the aomotosystems x2,4 and x3,6. Hypergeometric function these keywords were added by machine and not by the authors. Using mellinbarnes integrals we give a method to compute a relevant subgroup of the monodromy group of an a hypergeometric system of differential equations. Using mellinbarnes integrals we give a method to compute a relevant. Algebraic ahypergeometric functions and their monodromy. Our result are general and will be described only by the con. Monodromy of the hypergeometric differential equation of.
Hypergeometric functions reading problems introduction the hypergeometric function fa, b. Hyperbolic monodromy groups for the hypergeometric equation and cartan involutions elena fuchs chen meiri peter sarnak y september 10, 20 to nicholas katz with admiration abstract we give a criterion which ensures that a group generated by cartan involutions in the automorph group of a rational quadratic form of signature n 1. Keywords hypergeometric function generalized binomial function monodromy group. In mathematics, the gaussian or ordinary hypergeometric function 2f1a,b. Monodromy, hypergeometric polynomials and diophantine. Hypergeometric equation and monodromy groups 5 remark 2. A natural generalization of the eulergauss hypergeometric function is. Monodromy for the hypergeometric function department of. Beukers and others published monodromy for the hypergeometric function nfn1 find, read and cite all the research you need on researchgate. The monodromy of a hypergeometric equation describes how fundamental solutions change when.
This process is experimental and the keywords may be updated as the learning algorithm improves. We also assume that m nsince otherwise the corresponding hypergeometric series 2. This function was generalized to functions with more parame ters cla28, tho70 and more variables app82, app80, lau93, hor89, hor31. In the cas e of o nev ar iable hyper geometric functions there is 8 which g ive a n characterisation of mono dro my gr oups as complex re.
Hyperbolic schwarz map of the confluent hypergeometric differential equation saji, kentaro, sasaki, takeshi, and yoshida, masaaki, journal of the mathematical society of japan, 2009. Department of mathematics, graduate school of science, kobe university, 11, rokkodai, nadaku, kobe, japan. The hypergeometric probability distribution is used in acceptance sampling. Springer nature is making sarscov2 and covid19 research free. The monodromy s study of fuchsian hypergeometric di erential equation provides a natural framework for the explicit determination of rational approximations of polylogarithmic and hypergeometric functions.477 1504 1062 126 705 1047 93 239 456 110 118 1271 1489 1071 416 660 1503 1179 1128 464 1568 361 1015 427 1295 632 100 104 669 846 1286 252 216 29