Saturday, 10 December 2011

special functions - modular arithmetic of Hermite polynomials

I wonder if there is anything known (formula, asymptotics, etc) of computing the remainder



Rk,mequivHk modHm



for k>m, where Hm denotes the mth Hermite polynomial (orthogonal under the weight w(x)=ex2) and degRk,mleqm1. I haven't been able to find anything online, neither could compute it through the recurrence relation of Hermite polynomials...



Update:



The motivation for my question is as follows. The m-point Gauss-quadrature is obtained by placing the nodes at the roots of Hm and choosing the weights accordingly such that integrating any polynomial (with respect to weight w) of order leq2m1 is exact. Now I want to know the error formula for polynomials of degree kgeqm, especially Hk. By computing Hk modulo Hm, the integration error is given by the integration of the remainder Rk,m.

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