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**Numerical Methods for the Computation of the Confluent and Gauss Hypergeometric Functions(****arXiv****)**

**Author : **John W. Pearson, Sheehan Olver, Mason A. Porter

**Abstract : **The two most commonly used hypergeometric functions are the confluent hypergeometric function and the Gauss hypergeometric function. We review the available techniques for accurate, fast, and reliable computation of these two hypergeometric functions in different parameter and variable regimes. The methods that we investigate include Taylor and asymptotic series computations, Gauss-Jacobi quadrature, numerical solution of differential equations, recurrence relations, and others. We discuss the results of numerical experiments used to determine the best methods, in practice, for each parameter and variable regime considered. We provide ‘roadmaps’ with our recommendation for which methods should be used in each situation.

**2. New Series Expansions of the Gauss Hypergeometric Function(****arXiv****)**

**Author :** José Luis López, Nico M. Temme

**Abstract : **The Gauss hypergeometric function 2F1(a,b,c;z) can be computed by using the power series in powers of z,z/(z−1),1−z,1/z,1/(1−z),(z−1)/z. With these expansions 2F1(a,b,c;z) is not completely computable for all complex values of z. As pointed out in Gil, {it et al.} [2007, S2.3], the points z=e±iπ/3 are always excluded from the domains of convergence of these expansions. Bühring [1987] has given a power series expansion that allows computation at and near these points. But, when b−a is an integer, the coefficients of that expansion become indeterminate and its computation requires a nontrivial limiting process. Moreover, the convergence becomes slower and slower in that case. In this paper we obtain new expansions of the Gauss hypergeometric function in terms of rational functions of z for which the points z=e±iπ/3 are well inside their domains of convergence . In addition, these expansion are well defined when b−a is an integer and no limits are needed in that case. Numerical computations show that these expansions converge faster than Bühring’s expansion for z in the neighborhood of the points e±iπ/3, especially when b−a is close to an integer number

**3.Triality in SU(2) Seiberg-Witten theory and Gauss hypergeometric function (****arXiv****)**

**Author : **Ta-Sheng Tai

**Abstract :** Through AGT conjecture, we show how triality observed in N=2 SU(2) N_f=4 QCD can be interpreted geometrically as the interplay among six of Kummer’s twenty-four solutions belonging to one fixed Riemann scheme in the context of hypergeometric differential equations. We also stress that our presentation is different from the usual crossing symmetry of Liouville conformal blocks, which is described by the connection coefficient in the case of hypergeometric functions. Besides, upon solving hypergeometric differential equations at the zeroth order by means of the WKB method, a curve (thrice-punctured Riemann sphere) emerges. The permutation between these six Kummer’s solutions then boils down to the outer automorphism of the associated curve

**4.Dihedral Gauss hypergeometric functions (****arXiv****)**

**Author : **Raimundas Vidunas

**Abstract :** Gauss hypergeometric functions with a dihedral monodromy group can be expressed as elementary functions, since their hypergeometric equations can be transformed to Fuchsian equations with cyclic monodromy groups by a quadratic change of the argument variable. The paper presents general elementary expressions of these dihedral hypergeometric functions, involving finite bivariate sums expressible as terminating Appell’s F2 or F3 series. Additionally, trigonometric expressions for the dihedral functions are presented, and degenerate cases (logarithmic, or with the monodromy group Z/2Z) are considered.

**5. Transformations of algebraic Gauss hypergeometric functions(****arXiv****)**

**Author : **Raimundas Vidunas

**Abstract : **A celebrated theorem of Klein implies that any hypergeometric differential equation with algebraic solutions is a pull-back 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 pull-back coverings in these cases, based on certain explicit expressions (Darboux evaluations) of algebraic hypergeometric functions. The explicit expressions can be computed using contiguous relations and a data base of simplest Darboux evaluations (covering the Schwarz table). Klein’s pull-back transformations also induce algebraic transformations between hypergeometric solutions and a standard hypergeometric function with the same finite monodromy group

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