Large-sphere limits of the Mie-scattering functions

For x/n ≫ 1, the following relations between the Mie-scattering functions an (x,m) and bn(x,m) are satisfied: a1 (x,m) = b2 (x,m) = a3 (x,m) = ⋯ = an−1(x,m) = bn(x,m) and b1(x,m) = a2 (x,m) = b3 (x,m) = ⋯ = bn−1 (x,m) = an(x,m) for arbitrary refractive index m. By use of these relations, the Van de Hulst and Deirmendjian conjectures about the x → ∞ behavior of the scattering functions or their linear and bilinear combinations, as well as several new relations, are rigorously proved.