Legendre functions with both parameters large

Abstract
By application of the theory for second-order linear differential equations with two turning points developed in the preceding paper, some new asymptotic approximations are obtained for the associated Legendre functions when both the degree n and order m are large. The approximations are expressed in terms of parabolic cylinder functions, and are uniformly valid with respect to x ∈ ( − 1 , 1 ) and m / ( n + 1 2 ) ∈ [ δ , 1 + Δ ] where δ and ∆ are arbitrary fixed numbers such that 0 < δ < 1 and ∆ > 0. The values of m and n + ½ are either both real, or both purely imaginary. In all cases explicit bounds are supplied for the error terms associated with the approximations.

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