Asymptotic summation of Hermite series

Abstract

A new method for the numerical evaluation of slowly convergent or even divergent series involving the Hermite functions is presented. The author considers series with either of the forms where cm decays algebraically as m to infinity . The first series is a Fourier-Hermite series, while the second arises in the representation of Green functions for problems whose eigenfunctions involve the Hermite functions. By use of the Poisson summation formula, the author derives rapidly convergent asymptotic expansions for the remainders of these series after a sufficiently large number of terms. The series can then be evaluated as a partial sum plus an asymptotic approximation to its remainder. The asymptotic expansion for the remainder of G(x,y) also reveals the nature of the possible singular behaviour of this series near x=y.