New series expansions of the Gauss hypergeometric function
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New series expansions of the Gauss hypergeometric function José L. López · Nico M. Temme
Received: 8 November 2011 / Accepted: 13 September 2012 © Springer Science+Business Media New York 2012
Abstract The Gauss hypergeometric function 2 F1 (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), and (z − 1)/z. With these expansions, 2 F1 (a, b , c; z) is not completely computable for all complex values of z. As pointed out in Gil et al. (2007, §2.3), the points z = e±iπ/3 are always excluded from the domains of convergence of these expansions. Bühring (SIAM J Math Anal 18:884–889, 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 expansions 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.
Communicated by Juan Manuel Peña. J. L. López (B) Departamento de Ingeniería Matemática e Informática, Universidad Pública de Navarra, 31006 Pamplona, Spain e-mail: [email protected] N. M. Temme CWI, Science Park 123, 1098 XG Amsterdam, The Netherlands e-mail: [email protected]
J.L. López, N.M. Temme
Keywords Gauss hypergeometric function · Approximation by rational functions · Two- and three-point Taylor expansions Mathematics Subject Classifications (2010) 33C05 · 41A58 · 41A20 · 65D20
1 Introduction The power series of the Gauss hypergeometric function 2 F1 (a, b , c; z), 2 F1 (a, b , c; z)
=
∞ (a)n (b )n n=0
(c)n n!
zn ,
(1)
converges inside the unit disk. For numerical computations, we can use the right-hand side of (1) to compute 2 F1 (a, b , c; z) only in the disk |z| ≤ ρ < 1, with ρ depending on numerical requirements, such as precision and efficiency. From [3, §§2.3.1 and 2.3.2] or [7, Eq. 15.2.1 and §§15.8(i) and 15.8(ii)], we see that the Gauss hypergeometric function 2 F1 (a, b , c; z) may be written in terms of one or two other 2 F1 functions with any of the following arguments: 1 , z
1 − z,
1 , 1−z
z , 1−z
z−1 . z
(2)
As explained in [3, §2.3.2], when these formulas are combined with the series expansion (1), we obtain a set of series expansions of 2 F1 (a, b , c; z) in powers of some of the rational functions given in (2). The domains of convergence of the whole set of the expansions obtained in this way are the regions |z| ≤ ρ < 1, 1 1 − z ≤ ρ < 1,
1 ≤ ρ < 1, |1 − z| ≤ ρ < 1, z z z − 1 ≤ ρ < 1, 1 − z z ≤ ρ < 1.
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