Computing hypergeometric functions rigorously

06/22/2016

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by   Fredrik Johansson, et al.

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We present an efficient implementation of hypergeometric functions in arbitrary-precision interval arithmetic. The functions _0F_1, _1F_1, _2F_1 and _2F_0 (or the Kummer U-function) are supported for unrestricted complex parameters and argument, and by extension, we cover exponential and trigonometric integrals, error functions, Fresnel integrals, incomplete gamma and beta functions, Bessel functions, Airy functions, Legendre functions, Jacobi polynomials, complete elliptic integrals, and other special functions. The output can be used directly for interval computations or to generate provably correct floating-point approximations in any format. Performance is competitive with earlier arbitrary-precision software, and sometimes orders of magnitude faster. We also partially cover the generalized hypergeometric function _pF_q and computation of high-order parameter derivatives.