5.4 EXPLICIT EXPRESSIONS FOR, AND SPECIAL VALUES OF, THE HERMITE POLYNOMIALS We may use either the definition (5.3), or

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5 4 Explicit Expressions For And Special Values Of The Hermite Polynomials We May Use Either The Definition 5 3 Or 1 (81.72 KiB) Viewed 61 times

5.4 EXPLICIT EXPRESSIONS FOR, AND SPECIAL VALUES OF, THE HERMITE POLYNOMIALS We may use either the definition (5.3), or theorem 5.2 or theorem 5.3 to write down an explicit expression for the Hermite polynomial of any order we choose. For the first few orders we obtain * H.(x) = 1 H/(x) = 2x H(x) = 4x2 2 Hz(x) = 8x3 12x H4(x) = 16x4 48x2 + 12 H(x) = 32x5 – 160x3 + 120x. = = Theorem 5.4 H.(0) = (–1)n (2n); H3 +(0) = 0. ! H2n= H2n1= 0 !