Spherical functions recurrence relations

In general case Eqs. (4.60) and (4.61) present infinite sets of the five-term (pentadiagonal) recurrence relations with respect to the index l. In certain special cases (t - 0 or a - 0), they reduce to three-term (tridiagonal) recurrence relations. In this section the sweep procedure for solving such relations is described. This method, also known as the Thomas algorithm, is widely used for recurrence relations entailed by the finite-difference approximation in the solution of differential equations (e.g., see Ref. 61). In our case, however, the recurrence relation follows from the exact expansion (4.60) of the distribution function in the basis of orthogonal spherical functions and free of any seal of proximity, inherent to finite-difference method. Moreover, in our case, as explained below, the sweep method provides the numerical representation of the exact solution of the recurrence relations. 

The following recurrent relations obey the modified spherical Bessel functions ... 

The computation of the spherical Bessel functions may be enhanced by using the recurrence relations ... 

The quantities r Yij 9,) and Yi,m(9,)/r + appearing in equations (67) and (68) are known as solid spherical harmonics. Because of their central role in multipole approximations, it is important to have optimized procedures for their generation and manipulation. An equivalent and efficient reformulation of the solid spherical harmonics has been published recently, with the interesting property of having very simple derivatives with respect to Cartesian coordinates, which are required for the computation of forces, or for obtaining useful recurrence relations for the integrals of solid spherical harmonic with Gaussian basis functions. 

In practice, the generalized spherical functions can be found from the recurrence relation [162]... 

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