Prove Part 1 of Theorem 1.14 using induction. That is, prove that for any non-negative integer \(k\),

\[H_{k}(x)=\sum_{i=0}^{\lfloor k / 2floor}(-1)^{i} \frac{(2 i) !}{2^{i} i !}\left(\begin{array}{c}k \\2 i\end{array}ight) x^{k-2 i}\]

where \(\lfloor k / 2floor\) is the greatest integer less than or equal to \(k / 2\). It may prove useful to use the result of Exercise 32.

Exercise 32.

Prove Part 3 of Theorem 1.14 using only Definition 1.6. That is, prove that for any non-negative integer \(k\),

\[\frac{d}{d x} H_{k}(x)=k H_{k-1}(x)\]

Do not use the result of Part 1 of Theorem 1.14.

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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