A high accurate algorithm of calculating the associated Legendre functions for integral order and complex degree is presented. On a microcomputer and using double precision, the calculated results indicate the precision of at least 9 significant digits is attained.

The basic formulas of the emission and transmission for non-local thermodynamic equlibrium(NLTE)planetary atmospheres are derived.By assuming that rotational levels within a vibrational level remain in thermodynamic equilibrium,an accurate and efficient line-by-line algorithm of NLTE emission and transmission is developed to evaluate the radiance due to infrared active species in the upper planetary atmospheres.The limb viewing spectral radiance,line integrated radiance,band integrated radiance and synthetic spectral radiance of the Earth's atmosphere for the 9.6μm bands of ¹⁶O₃ under daytime and nighttime and under NLTE and LTE conditions are calculated by using this algorithm.The results of calculations reveal that NLTE radiance obviously departs from LTE.

The molecular velocity distribution function and Boltzmann equation in the kinetic theory of gases are expanded in basis of the orthogonal Burnett functions. Hence it is shown that the calculation of Boltzmann collision matrix elements can be realized by computing the reduced integrals, Clebsch-Gordan coefficients and Brody-Moshinsky coefficients. Finally the computatimal mathod of high efficiency and numerical examples are given.