Convergent Asymptotic Expansions of Charlier, Laguerre and Jacobi Polynomials

Convergent expansions are derived for three types of orthogonal polynomials: Charlier, Laguerre and Jacobi. The expansions have asymptotic properties for large values of the degree. The expansions are given in terms of functions that are special cases of the given polynomials. The method is based on expanding integrals in one or two points of the complex plane, these points being saddle points of the phase functions of the integrands.Comment: 20 pages, 5 figures. Keywords: Charlier polynomials, Laguerre polynomials, Jacobi polynomials, asymptotic expansions, saddle point methods, two-points Taylor expansion

Bessel function expansions of Coulomb wave functions

From the convergence properties of the expansion of the function Φ_l∝r^(−l−1)F_l in powers of the energy, we successively obtain the expansions of F_l and G_l as single series of modified Bessel functions I_(2l+1+n) and K_(2l+1+n), respectively, as well as corresponding asymptotic approximations of G_l for ‖η‖→∞. Both repulsive and attractive fields are considered for real and complex energies as well. The expansion of F_l is not new, but its convergence is given a simpler and corrected proof. The simplest form of the asymptotic approximations obtained for G_l, in the case of a repulsive field and for low positive energies, is compared to an expansion obtained by Abramowitz

Computing hypergeometric functions rigorously

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.Comment: v2: corrected example in section 3.1; corrected timing data for case E-G in section 8.5 (table 6, figure 2); adjusted paper siz