A Classification of the Mathematical Expressions of the Asymptotic Series Expansion of the Associated Legendre Function

The mathematical expressions of the asymptotic series expansion of the associated Legendre function, which have been widely described in the mathematical literature, are here shown to be classifiable into two types according to their manner of expression. The Type-A mathematical expressions are represented by the first three higher terms of the asymptotic series, while the Type-B expressions are represented by the infinite sequence. As a result, the Type-B mathematical expressions differ slightly from one another, despite having identical computation results. The distinct representation of the Type-B expressions arises from (1) the treatment of the gamma functions including the colatitude angle in the spherical coordinate system and the azimuthal order number of the associated Legendre function and (2) the method of expressing the phase angle term. Moreover, one of the mathematical expressions of the Type-B is derived using a hypergeometric formula different from other usual one.