Abstract:There are some inaccuracies in the derivation of the sectorial harmonic term (order is equal to degree) and the quasi-sectorial harmonic term (order is equal to degree minus 1) in the related literature, and we give a strict derivation method. Based on the two derivative formulas of fnALFs, we give the standard forward column recurrence formula of its definite integration; the results are completely consistent with those given by predecessors, but the derivation process is more concise, which can be recursed from the second degree. In addition, we derive two other useful definite integration formulas of fnALFs.