Summing Pn(cos θ)/ p( n) for certain polynomials p( n)

Abstract

Procedures are given to sum P n (cos θ)/ p( n), where P n (cos θ) is a Legendre function, p( n) is a polynomial in (2 n+1) 2, with roots that are perfect squares, and p( n) ≠ 0 for all n ≥ N, the lower limit of the infinite series. These series are needed in problems of computational chemistry, and may be of use in other applications of potential theory. They converge too slowly for direct numerical summation, when the degree of p( n) is small. The results are extended to coefficients (2 n+1) p( n) and p( n)/(2 n+1).