Find a simple form for the EGF (in terms of $f(x)$)

Question

For a sequence $(a_n) = (a_0 ,a_1, ...)$ supposed that $f(x)$ is its EGF (Exponential Geneterating Function). For each of the following sequences, find a simple form for the EGF

For a sequence $(a_0,a_1,a_2,...),$ its EGF is $\sum_{n=0}^{\infty}a_n\frac{x^n}{n!}$

1. $(0,a_1, a_2,...)$

2. $(a_1, a_2,...)$

3. $(0,0,1,a_3,a_4,...)$

4. $(a_0,0,a_2,0,a_4,0,...)$

I am completely lost because I keep getting 0=0 or something along those lines. For something like #1, I have

$A(x) = 0 + \sum_{n=1}^{\infty}a_n\frac{x^n}{n!} = (-a_0 + a_0) + \sum_{n=1}^{\infty}a_n\frac{x^n}{n!} = -a_0 + \sum_{n=0}^{\infty}a_n\frac{x^n}{n!}$

$A(x) = -a_0 + A(x)$

This is where I'm lost

Answer 1

You used the same function symbol $A$ for both generating functions. On the right you should have replaced the complete series by $f(x)$, not $A(x)$, yielding the correct result $A(x)=f(x)-a_0$.

For #$2$, you can shift the sequence by taking the derivative.

I think #$3$ will be clear from #$1$.

For #$4$, consider $f(-x)$ and how you might combine it with $f(x)$ to get rid of the odd-numbered terms.