The use of generalized Laguerre polynomials in spectral methods for nonlinear differential equations

The expansion of products of generalized Laguerre polynomials Lνn(x) in terms of a series of generalized Laguerre polynomials is considered. The expansion coefficients, which are equal to triple-product integrals of generalized Laguerre polynomials, are expressed in terms of a three-index recurrence relation. This is reduced to a one-index relation which facilitates computation of the expansion coefficients. The results are useful in the solution of nonlinear differential equations when it is desired to express products of generalized Laguerre polynomials as a linear series of these functions. As an application, we use the results to compute a spectral solution of a nonlinear boundary-value problem, namely the Blasius equation on a semi-infinite interval. By using a truncated series containing the first eight polynomials L1/2n(x), a solution is obtained within 4% accuracy.