Trigonometric Representations of Legendre Functions

Abstract

Although the trigonometric representation of associated Legendre functions has been considered in literature, here we give a new insight into the trigonometric reduction of Legendre polynomials. We show that Legnedre polynomials can be calculated up to an ultra-high degree, e.g., n = 10⁶ and beyond without recursive relations and this can be used as a basis for the calculation of associated Legendre functions. The approach presented here was reported for the first time in Švehla (2008) and in Svehla (2010). In addition, we derive orthogonal geometrical forms of associated Legendre functions. However, in terms of performance, our geometrical approach based on the addition theorem of Legendre functions and geometrical rotations along the equator (previous section) is significantly more elegant.