THE OVERLAP INTEGRAL OF ASSOCIATED LEGENDRE FUNCTIONS - COMMENT

Closed formulas have been derived for the definite integral ∫π0 ℒknℒ k+1m sin ρdρ where 0 ≤ k ≤ n and 0 ≤ k+1 ≤ m involving two normalised associated Legendre functions. It has been shown that the selection rules are independent of the parity of k and depend only on the parities of l, n, and m. These results complete those obtained by Salem and Wio (1989) who have given closed expressions for the special k=0 case. As side results, closed formulae of the definite integral ∫π 0 (sin ρ)P(cos ρ)M dρ where P and M are integers and 0 ≤ P, M as well as sum rules involving combinations of binomial coefficients have been obtained.