Formula for $\sum_{n=0}^\infty x_{n+1}z^n$

Question

I have a recurrence:

$x_{n+1} = ax_n + b ,x_0 = 1,n=0,1....$

I have $g(z) = \sum_{n=0}^{\infty} x_nz^n$

So, multiplying the recurrence by $z^n$ and summing up, I get

$\sum_{n=0}^{\infty} x_{n+1}z^n = a \sum_{n=0}^{\infty} x_n z^n + b\sum_{n=0}^{\infty} z^n$

I get

$? - x_0= a g(z) + \frac{b}{1-z}$

$? - 1= a g(z) + \frac{b}{1-z}$

Can anyone tell me what is the value for (?) and if the simplification is correct?

Thanks

Answer 1

We have,

$$\sum_{n=0}^{\infty} x_{n+1}z^n$$

$$=\frac{1}{z} \sum_{n=0}^{\infty} x_{n+1}z^{n+1}$$

$$=\frac{1}{z} \sum_{n=1}^{\infty} x_{n} z^n$$

$$=\frac{1}{z}(g(z)-x_0)$$

Everything you write is correct preceding the ?.