Evaluation of associated Legendre functions off the cut and parabolic cylinder functions. (English) Zbl 0961.65017

Segura, Javier; Gil, Amparo

ETNA, Electron. Trans. Numer. Anal. 9, 137-146 (1999).

The authors review a set of algorithms for evaluating the associated Legendre functions off the cut. In particular, they consider prolate spheroidal, oblate spheroidal and toroidal harmonics. Another family of functions that be computed in a similar wy are the parabolic cylinder functions. Aspects of forward and backward recursion schemes for computing solutions of three-term recursion relations are discussed. Numerical test of existing algorithms are also given.

Reviewer: N.M.Temme (Amsterdam)

MSC:

65D20	Computation of special functions and constants, construction of tables
33-04	Software, source code, etc. for problems pertaining to special functions
33C05	Classical hypergeometric functions, \({}_2F_1\)
33C45	Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
33C10	Bessel and Airy functions, cylinder functions, \({}_0F_1\)
33C55	Spherical harmonics

Keywords:

computation of special functions; Legendre functions; parabolic cylinder functions; prolate spheroidal harmonics; oblate spheroidal harmonics; algorithms; toroidal harmonics; forward and backward recursion schemes; three-term recursion relations; numerical examples

Software:

Algorithm 644