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Explicit formulae for the coefficients of integrated expansions of Laguerre and Hermite polynomials and their integrals. (English) Zbl 1169.42314
Summary: Two new formulae expressing explicitly the integrals of Laguerre (Hermite) polynomials of any degree and for any order in terms of the Laguerre (Hermite) polynomials themselves are proved. Another two new explicit formulae relating the Laguerre (Hermite) coefficients of an expansion for an infinitely differentiable function that has been integrated an arbitrary number of times in terms of the coefficients of the original expansion of the function are also established. An application of these formulae for solving ordinary differential equations with varying coefficients is discussed.
##### MSC:
 42C10 Fourier series in special orthogonal functions 65L50 Mesh generation and refinement (ODE) 65L10 Boundary value problems for ODE (numerical methods)