Everywhere differentiable quasi-classical approximations of spherical functions. (Q1852517)

The authors obtain everywhere differentiable approximations of normalized spherical functions \(\Theta_{lm}(\theta)\), in terms of elementary functions, based on the generalized quasi-classical approximation. The spherical functions under consideration are related to the associated Legendre functions of the first kind \(P_l^m(x)\) of degree \(l\) and order \(m\) as \[ \Theta_{lm}(\theta)= (-1)^{(m+| m|)/2} \Biggl[ \frac {(2l+1)}{2} \frac {(l-| m|)!} {(l+| m|)!} \Biggr]^{1/2} P_l^{| m|} (\cos\theta). \]