Inequalities for the associated legendre functions

Abstract

In this paper bounds for the associated Legendre functions of the first kind Pnm(x) for real x ∈ [ -1, 1] and integers m, n are proved. A relation is derived that allows us to generalize known bounds of the Legendre polynomials P n(x) ≡ Pn0(x) for the Legendre functions Pnm(x) of non-zero order m. Furthermore, upper and lower bounds of the type A(α, n, m) ≤ maxx ∈[ -1, 1 ] |(1 - x2)α/2 Pnm(x)| ≤ B(α, n, m) are proved for all 0 ≤ α ≤ 1/2 and 1 ≤ |m| ≤ n. For α = 0 and α = 1/2 these upper bounds are improvements and simplifications of known results. © 1998 Academic Press.