Rational approximations of the Arrhenius integral using Jacobi fractions and gaussian quadrature

The aim of this work is to find approaches for the Arrhenius integral by using the n-th convergent of the Jacobi fractions. The n-th convergent is a rational function whose numerator and denominator are polynomials which can be easily computed from three-term recurrence relations. It is noticed that such approaches are equivalent to the one established by the Gauss quadrature formula and it can be seen that the coefficients in the quadrature formula can be given as a function of the coefficients in the recurrence relations. An analysis of the relative error percentages in the approximations is also presented.