Inequalities satisfied by the zeros of the solutions of second order hypergeometric equations are derived through a systematic use of Liouville transformations together with the application of classical Sturm theorems. This systematic study allows us to improve previously known inequalities and to extend their range of validity as well as to discover… (More)
Algorithms for the numerical evaluation of the incomplete gamma function ratios P (a, x) = γ(a, x)/Γ(a) and Q(a, x) = Γ(a, x)/Γ(a) are described for positive values of a and x. Also, inversion methods are given for solving the equations P (a, x) = p, Q(a, x) = q, with 0 < p, q < 1. Both the direct computation and the inversion of the incomplete gamma… (More)
GammaCHI: a package for the inversion and computation of the gamma and chi-square cumulative distribution functions (central and noncentral) 1 Abstract A Fortran 90 module GammaCHI for computing and inverting the gamma and chi-square cumulative distribution functions (central and noncentral) is presented. The main novelty of this package are the reliable… (More)
Two Fortran 77 routines for the evaluation of Airy functions of complex arguments <i>Ai</i>(<i>z</i>), <i>Bi</i>(<i>z</i>) and their first derivatives are presented. The routines are based on the use of Gaussian quadrature, Maclaurin series and asymptotic expansions. Comparison with a previous code by D. E. Amos  is provided.
Integral representations are considered of solutions of the inhomo-geneous Airy differential equation w − z w = ±1/π. The solutions of these equations are also known as Scorer functions. Certain functional relations for these functions are used to confine the discussion to one function and to a certain sector in the complex plane. By using steepest descent… (More)
Each family of Gauss hypergeometric functions fn = 2F1(a + ε1n, b + ε2n; c + ε3n; z), n ∈ Z , for fixed εj = 0, ±1 (not all εj equal to zero) satisfies a second order linear difference equation of the form Anfn−1 + Bnfn + Cnfn+1 = 0. Because of symmetry relations and functional relations for the Gauss functions , many of the 26 cases (for different εj… (More)