Lame polynomials functions

The generating function of fhe Lame polynomials and of fhe spherical harmonics are familiar since the study of electrostatics... 

Table II in Ref. [6] illustrates the eigenvalues h for the cases of = 4 and 5 for the respective species and types of the Lam6 polynomials for molecules with the different asymmetry distributions. Figure 1 in Ref. [6] shows the variations of the Lame polynomials A% A% A (three of each), and A " (two) as functions of their argument and of the asymmetry distribution.

The numerical results for the eigenenergies and eigenfunctions evaluated in Refs. [5] and [6] for molecules with different asymmetry distributions and states are accurate and consistent. The zeros of the individual Lame functions can be determined with high accuracy, and are illustrated in Figure 1 in Ref. [6]. They allow writing the Lame polynomial in product forms presented in Sections 2.1 and 2.2. They are also the key to implement the boundary condition for the rotations of molecules confined by elliptical cones as discussed in Section 3. 

Both works [2] and [3] show the separations of the eigenvalue equations for H and H, and H and H, in their respective spheroconal coordinates, into Lame differential equations in the individual elliptical cone angular coordinates. The corresponding solutions are Lam6 spheroconal polynomials included in the classic book of Whittaker and Watson [12]. In practice, the numerical evaluation of such Lame functions was not developed in an efficient manner so that the exact formulation of Ref. [2] did not prosper. Consequently, the analysis of rotations of asymmetric molecules took the route of perturbation theory using the familiar basis of spherical harmonics. 

I even and Vl - w, /w, Vl + w, w - xo ) for I odd define the eight types of Lame functions in their different species, as in Section 2.1, upon multiplication by their respective polynomials P(w). 

Notice that the coefficient + 1) in the original equation becomes — 1) + 2), ( — 2)( + 3), and ( — 3) ( + 4) for the successive species being the same as Eqs. (52-56), while the eigenvalues hf are also shifted depending on the coefficients involved in the second derivatives of fA(Xi)- These changes are reminiscent of the familiar ones for the ordinary and associated Legendre polynomials, and their connections with the actions of ladder operafors. We are exploring the possibilities for the Lame functions themselves and their connections with Section 4.2.2. 

For the mathematically inclined Volkmer s Chapter 29 on Lame functions in the NIST Handbook of Mathematical Functions [50], and the article "A new basis for the representations of the rotational group. Lame and Heun polynomials" [51] are parts of the toolbox. 

Section 2.6 recognizes that for the hydrogen atom, its Hamiltonian also commutes with and H correspondingly, it also admits solutions with Lame spheroconal harmonics polynomial eigenfunctions. It also shares the same radial eigenfunction with the familiar solution with spherical harmonics, and additionally both can be obtained from a common generating function and both satisfy the addition theorem. 

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