1. Finding a generating function

Here's the problem:

Let $\displaystyle g(x)$ be a generating function for the sequence $\displaystyle (a_n)$. Find a sequence $\displaystyle (b_n)$ whose generating function is $\displaystyle (1-x)g(x)$.

***

We know that, given a sequence $\displaystyle (a_n)_{n \in \mathbb{N}_0}$, the generating function of that sequence is $\displaystyle g(x)= \sum_{n=0}^{\infty} a_n x^n$.
Let's define $\displaystyle h(x):=(1-x)g(x)$ and it is known that $\displaystyle h(x)$ is a generating function for the sequence $\displaystyle (b_n)$, i.e. $\displaystyle h(x)=\sum_{n=0}^{\infty} b_n x^n=(1-x)g(x)=(1-x)\sum_{n=0}^{\infty} a_n x^n$.
Therefore we have the relationship $\displaystyle \sum_{n=0}^{\infty} b_n x^n=(1-x)\sum_{n=0}^{\infty} a_n x^n$ and I'm a little lost finding what exactly is $\displaystyle (b_n)$.

Any help will be greatly appreciated!

2. Originally Posted by gusztav
Here's the problem:

Let $\displaystyle g(x)$ be a generating function for the sequence $\displaystyle (a_n)$. Find a sequence $\displaystyle (b_n)$ whose generating function is $\displaystyle (1-x)g(x)$.

***

We know that, given a sequence $\displaystyle (a_n)_{n \in \mathbb{N}_0}$, the generating function of that sequence is $\displaystyle g(x)= \sum_{n=0}^{\infty} a_n x^n$.
Let's define $\displaystyle h(x):=(1-x)g(x)$ and it is known that $\displaystyle h(x)$ is a generating function for the sequence $\displaystyle (b_n)$, i.e. $\displaystyle h(x)=\sum_{n=0}^{\infty} b_n x^n=(1-x)g(x)=(1-x)\sum_{n=0}^{\infty} a_n x^n$.
Therefore we have the relationship $\displaystyle \sum_{n=0}^{\infty} b_n x^n=(1-x)\sum_{n=0}^{\infty} a_n x^n$ and I'm a little lost finding what exactly is $\displaystyle (b_n)$.

Any help will be greatly appreciated!

Observe $\displaystyle \sum_{n=0}^{\infty} b_n x^n=(1-x)\sum_{n=0}^{\infty} a_n x^n = \sum_{n=0}^{\infty} a_n x^n - \sum_{n=0}^{\infty} a_n x^{n+1} = a_0 + \sum_{n=1}^{\infty} (a_n - a_{n-1}) x^{n}$

Thus $\displaystyle \sum_{n=0}^{\infty} b_n x^n = a_0 + \sum_{n=1}^{\infty} (a_n - a_{n-1}) x^{n}$

3. Hello,

$\displaystyle (1-x) \sum_{n=0}^\infty a_n x^n=\sum_{n=0}^\infty a_n x^n-\sum_{n=0}^\infty a_n x^{n+1}$
(by expanding)

Change the indice of the second sum :

$\displaystyle =\sum_{n=0}^\infty a_n x^n-\sum_{n=1}^\infty a_{n-1} x^n$

Make the indices be the same :

$\displaystyle =a_0+\sum_{n=1}^\infty a_n x^n-\sum_{n=1}^\infty a_{n-1} x^n$

$\displaystyle =a_0+\sum_{n=1}^\infty (a_n-a_{n-1}) x^n$

Hence $\displaystyle b_0=a_0$ and $\displaystyle b_n=a_n-a_{n-1}$ for $\displaystyle n \geq 1$

If you cannot visualize some steps, just write the first few terms of the sums

Edit : too slow