Efficient computation of Laguerre polynomials

An efficient algorithm and a Fortran 90 module (LaguerrePol) for computing Laguerre polynomials Ln(α)(z) are presented. The standard three-term recurrence relation satisfied by the polynomials and different types of asymptotic expansions valid for n large and α small, are used depending on the parameter region.Based on tests of contiguous relations in the parameter α and the degree n satisfied by the polynomials, we claim that a relative accuracy close to or better than 10−12 can be obtained using the module LaguerrePol for computing the functions Ln(α)(z) in the parameter range z≥0, −1<α≤5, n≥0.Program summaryProgram Title: Module LaguerrePolProgram Files doi:http://dx.doi.org/10.17632/3jkk659cn8.1Licensing provisions: CC by 4.0Programming language: Fortran 90Nature of problem: Laguerre polynomials Ln(α)(z) appear in a vast number of applications in physics, such as quantum mechanics, plasma physics, etc.Solution method: The algorithm uses asymptotic expansions or recurrence relations for computing the function values depending on the range of parameters.Restrictions: The admissible input parameter ranges for computing the Laguerre Ln(α)(z) are z≥0, −1<α≤5, n≥0.