Canonical vector polynomials for the computation of complex order Bessel functions with the tau method

A modification of the Lanczos Tau Method for the approximate solution of second-order differential equations with polynomial coefficients was proposed by the author earlier. This modification chooses a minimum perturbation, among all possible, on the right-hand side of the corresponding Volterra integral equation. A Tau Method computational scheme is applied to the approximate solution of a system of differential equations related to the differential equation of hypergeometric type. Various vector perturbations are discussed. Our choice of the perturbation term is a shifted Chebyshev polynomial with a special form of selected transition and normalization. The minimality conditions for the perturbation term are found for one equation. They are sufficiently simple for verification in a number of important cases. Ortizʹs recursive approach to the Tau Method is extended to canonical vector-polynomial coefficients, and to the coefficients of the linear combination that generate an approximate solution. The advantages of the proposed approach are shown for modified Bessel functions computation. Seven digit tables of modified Bessel function K1/2+iβ(x) have been computed and published. The computer programs for the calculations may be found in the State Fund of Algorithms and Programs and in the United Library of Numerical Analysis of the Moscow State University.