Chebyshev polynomial approximations for some hypergeometric systems

The hypergeometric type differential equations of the second order with polynomial coefficients and their systems are considered. The realization of the Lanczos Tau Method with minimal residue is proposed for the approximate solution of the second order differential equations with polynomial coefficients. The scheme of Tau method is extended for the systems of hypergeometric type differential equations. 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. The case of the discrete systems may be considered also. 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 the verification in a number of important cases. Several approaches for the approximation of kernels of Kontorovich–Lebedev integral transforms modified Bessel functions of the second kind with pure imaginary order and with complex order are elaborated. The codes of the evaluation are constructed and tables of the modified Bessel functions K1/2+iβ(x) are published. The advantages of discussed algorithms in accuracy and timing are shown. The effective applications for the solution of some mixed boundary value problems in wedge domains are given.