2013
New series expansions of the Gauss hypergeometric function
Publication
Publication
Advances in Computational Mathematics , Volume 39  Issue 2 p. 349 365
The Gauss hypergeometric function ${}_2F_1(a,b,c;z)$ can be computed by using the power series in powers of $z, z/(z1), 1z, 1/z, 1/(1z), (z1)/z$. With these expansions ${}_2F_1(a,b,c;z)$ is not completely computable for all complex values of $z$. As pointed out in Gil, {\it et al.} [2007, \S2.3], the points $z=e^{\pm i\pi/3}$ are always excluded from the domains of convergence of these expansions. B\"uhring [1987] has given a power series expansion that allows computation at and near these points. But, when $ba$ 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 $z=e^{\pm i\pi/3}$ are well inside their domains of convergence. In addition, these expansion are well defined when $ba$ is an integer and no limits are needed in that case. Numerical computations show that these expansions converge faster than B\"uhring's expansion for $z$ in the neighborhood of the points $e^{\pm i\pi/3}$, especially when $ba$ is close to an integer number.
Additional Metadata  

Gauss hypergeometric function. Approximation by rational functions. Two and threepoint Taylor expansions  
Classical hypergeometric functions, ${}_2F_1$ (msc 33C05)  
Springer  
Advances in Computational Mathematics  
Organisation  Computational Dynamics 
López, J.L, & Temme, N.M. (2013). New series expansions of the Gauss hypergeometric function. Advances in Computational Mathematics, 39(2), 349–365.
