Inequalities for generalized hypergeometric functions

AbstractIt is shown that some well-known Padé approximations for a particular form of the Gaussian hypergeometric function and two of its confluent forms give upper and lower bounds for these functions under suitable restrictions on the parameters and variable. With the aid of the beta and Laplace transforms, two-sided inequalities are derived for the generalized hypergeometric function pFq, p = q or p = q + 1, and for a particular form of Meijer's G-function. Several examples are developed. These include upper and lower bounds for certain elementary functions, complete elliptic integrals, the incomplete gamma function, modified Bessel functions, and parabolic cylinder functions

Predictor-corrector formulas based on rational interpolants

AbstractTitle formulas are developed for particular use in the solution of nonlinear differential equations which in general offer no clue as to the presence of singularities on or near the path of integration. Procedure is advantageous since the approximations ascertain the existence of zeros and poles and locate these data with great accuracy. The function y = J1(x)/Jo(x) where Jn(x) is the Bessel function of the first kind satisfies a first order nonlinear differential equation of the Riccati type, and has an infinite number of zeros and poles on the positive real axis. A numerical example is provided to illustrate computation of these singular points in 0 < x < 100. Some other examples are also given