Word Count : 258

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To establish the relation between \( H_n(x) \) and \( H_n(-x) \) using generating functions, let’s go step by step. First, recall that the Hermite polynomials \( H_n(x) \) are defined via their generating function: \[ e^{2xt - t^2} = \sum_{n=0}^{\infty} H_n(x) \frac{t^n}{n!} \] ### Step 1: Substitute \( -x \) into the generating function We want to find a relationship between \( H_n(x) _____ _____ ___ ___ _____ ____ _________ ____ _________.
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