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.
Petr Chýlek, "Large-sphere limits of the Mie-scattering functions," J. Opt. Soc. Am. 63, 699-706 (1973)