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**StefanM** $\displaystyle H_{n}(x)=-xH_{n-1}(x)-(n-1)H_{n-2}(x) ,for\ n\geq2 H_{0}(x)=1\ and H_{1}(x)=-x $.

a)Show that $\displaystyle H_{n}(x)$ is an even function when n is even and an odd function when n is odd.

Also show by induction that:

b)$\displaystyle H_{2k}(x)=(-1)^k(2k-1)(2k-3)...1$.

What is the value of $\displaystyle H_{n}(0) $when n is odd?

a)Now I proved that $\displaystyle H_{n}(x)$ is an even function when n is even and an odd function when n is odd. for the base case but I'm stuck whit the general case as when :

1.n is even=>n+1 odd and by the recurrence relation I'm stuck with the difference between an even function and an odd one.

2.n is odd=>n+1 even and by the recurrence relation again I get the difference between an odd function and an even one.

The induction here works as follows: for n even, then n+2 even, and for

n odd, n+2 odd. You have to do these two cases separatedly.

b)I think it has something to do with a)

Show the formula in (b) by induction, again. And then remember that for any odd function, if it is defined in zero

then its value there must be zero.

Tonio