Now given the exponential generating function of the Hermite polynomial e <mrow class="

Jaycee Mathis

Jaycee Mathis

Answered question

2022-05-28

Now given the exponential generating function of the Hermite polynomial
e u x ( 1 / 2 ) u 2 = 1 + n = 1 u n n ! h n ( x ) ,

Answer & Explanation

Norah Baxter

Norah Baxter

Beginner2022-05-29Added 10 answers

It is a simple convolution:
e u x = a 0 x a a ! u a , e u 2 / 2 = b 0 ( 1 ) b 2 b b ! u 2 b
hence the coefficient of u n in the product between e u x and e u 2 / 2 (i.e. n ! h n ( x )) is given by
a + 2 b = n x a ( 1 ) b a ! b ! 2 b = b = 0 n / 2 x n 2 b ( 1 ) b b ! ( n 2 b ) ! 2 b .

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