How Do Polynomial Recurrence Relations Determine Function Parity?

[TiberiusK]

Homework Statement

[H_{n}(x)=-xH_{n-1}(x)-(n-1)H_{n-2}(x) ,for,n>=2 H_{0}(x)=1\ and H_{1}(x)=-x
a)Show that 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)H_{2k}(x)=(-1)^k(2k-1)(2k-3)...1
hat is the value o H_{n}(0) when n is odd

Homework Equations

The Attempt at a Solution

a)Show that is an even function when n is even and an odd function when n is odd.
Also show by induction that:
b).
What is the value of when n is odd?
a)Now I proved that 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.
b)I think it has something to do with a)

 

[Mark44]

Homework Statement

[H_{n}(x)=-xH_{n-1}(x)-(n-1)H_{n-2}(x) ,for,n>=2 H_{0}(x)=1\ and H_{1}(x)=-x
a)Show that 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)H_{2k}(x)=(-1)^k(2k-1)(2k-3)...1
hat is the value o H_{n}(0) when n is odd

Homework Equations

The Attempt at a Solution

a)Show that is an even function when n is even and an odd function when n is odd.
Also show by induction that:
b).
What is the value of when n is odd?
a)Now I proved that 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.
b)I think it has something to do with a)

I think that you are mixing up the two parts. Part a doesn't have anything to do with induction, but what you're saying about base cases implies that you think it does.

Part b definitely should be done using induction. For the base case you can use k = 1, and evaluate H₂(x).

BTW, your partial LaTeX is difficult to read. Since all you're using are exponents and subscripts, you can write your stuff using the X² and X₂ buttons. Click Go Advanced to see the expanded menu across the top of the entry area.