Prove Part 2 of Theorem 1.14. That is, prove that for any non-negative integer \(k \geq 2\),

\[H_{k}(x)=x H_{k-1}(x)-(k-1) H_{k-2}(x) .\]

The simplest approach is to use Definition 1.6.

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Theorem 1.14. Let H(x) be the kth Hermite polynomial, then 1. For any positive integer k, [k/2] Hk (x) = (-1) (2) (k) i=0 2ii! 2i k-2i X where [k/2] is the greatest integer less than or equal to k/2. 2. For any integer k> 2, Hk(x) = xHk-1(x) - (k-1)Hk-2(x). 3. For any positive integer k, d Hk(x) = kHk-1(2). dx