# Thread: Help with exponential generating function proof

1. ## Help with exponential generating function proof

Use exponential generating functions to prove that:
P(n) = P(n-1) + n, with n >= 1 and P(0) = 1
will result in this equation: P(n) = (1/2)n2 + (1/2)n + 1

I know how to do this with generating functions, you just plug in n=1, n=2, n=3,....
P(1) will result in (1/2)(1)2 + (1/2)(1) +1 which is 2
P(2) will result in P(2-1)+2 which is P(1) + 2.. and so on
but I'm kind of stuck on "exponential" generating functions.
Can someone help me out?

2. ## Re: Help with exponential generating function proof

Originally Posted by chinknig
Use exponential generating functions to prove that:
P(n) = P(n-1) + n, with n >= 1 and P(0) = 1
will result in this equation: P(n) = (1/2)n2 + (1/2)n + 1

I know how to do this with generating functions, you just plug in n=1, n=2, n=3,....
P(1) will result in (1/2)(1)2 + (1/2)(1) +1 which is 2
P(2) will result in P(2-1)+2 which is P(1) + 2.. and so on
but I'm kind of stuck on "exponential" generating functions.
Can someone help me out?
First here is a good FREE source textbook.

Now, just from reading your post, I for one do not understand what is required.
It seems as if you placed us into the middle of some working.
May I suggest that you give the context or setting along with the complete and exact wording of the exercise.

3. ## Re: Help with exponential generating function proof

the exponential generating function is

$\displaystyle f(x)=\sum _{n=0}^{\infty } \frac{p_n}{n!}x^n$

compute

$\displaystyle f'(x)-f(x)=e^x+x e^x$

more computation

$\displaystyle f(x)=\frac{1}{2}e^x\left(x^2+2x+2\right)$

more computation

4. ## Re: Help with exponential generating function proof

I've proven how to derive the second equation from the eqution before using generating functions. That's why I know that the answer should be P(n) = (1/2)n^2 + (1/2)n + 1.
But know the instructor wants the class to use 'exponential' generating functions.

5. ## Re: Help with exponential generating function proof

Originally Posted by Idea
the exponential generating function is

$\displaystyle f(x)=\sum _{n=0}^{\infty } \frac{p_n}{n!}x^n$

compute

$\displaystyle f'(x)-f(x)=e^x+x e^x$

more computation

$\displaystyle f(x)=\frac{1}{2}e^x\left(x^2+2x+2\right)$

more computation

6. ## Re: Help with exponential generating function proof

First, solve the differential equation to get $\displaystyle f(x)$

Next,

$\displaystyle f(x)=\left(\frac{1}{2}x^2+x+1\right)e^x$

$\displaystyle =\left(1+x+\frac{1}{2}x^2\right)\left(1+\frac{x}{1 !}+\frac{x^2}{2!}+\text{...}\right)$

$\displaystyle =1 + 2x +\text{...}+ \left(\frac{1}{n!}+\frac{1}{(n-1)!}+\frac{1/2}{(n-2)!}\right)x^n+\text{...}$

so that

$\displaystyle \frac{p_n}{n!}=\frac{1}{n!}+\frac{1}{(n-1)!}+\frac{1/2}{(n-2)!}$

and

$\displaystyle p_n=1+\frac{n}{2}+\frac{n^2}{2}$