It is a well-known identity that $$ 2^{\frac n 2}H_n(x+y)=\sum_{k=0}^n {n\choose k} H_{n-k}(x\sqrt{2})H_k(y\sqrt{2}) $$ My question is whether something similar is known for $$H_n(x+y)=\sum_{k=0}^n \sqrt{{n\choose k}} H_{n-k}(x\sqrt{2})H_k(y\sqrt{2})$$ I am interested in this because this would allow to evaluate sums of hermite functions which are solutions to the harmonic oscillator in quantum mechanics. Such a formula would then allow an easy transformation into relative coordinates, for example.
2026-03-25 14:39:45.1774449585
Summation formula for Hermite polynomials
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