A note on the tau-method approximations for the Bessel functions Y0(z) and Y1(z)

This paper is to complete and improve the work reported in [1,2], using the Lanczos τ-method (in Colemanʹs version) to approximate the Bessel functions Y0(z) and Y1(z). We introduce symbolic representations of the scaled Faber polynomials on any fan-shaped section of the complex plane. These Faber polynomials are used as the perturbation terms in the τ-method. Numerical comparison among the power series, the Chebyshev series and the τ-method are conducted to show the accuracy improvement achieved by this new version of the τ-method. Some concluding remarks and suggestions on future research are given.