Generation function for recurrence

Question

Could you tell me how I can find the generation function for recurrence $\sum_{n = 0}^{ \infty} n a_n t^n$ if I know $A(t)$ - generation function for $a_0, a_1, a_2 \dots$ .

Thanks

Answer 1

You know that:

$$\sum_{n=0}^{\infty}a_nt^n=A(t)$$

Derive both sides (left side term by term).You get:

$$\sum_{n=0}^{\infty}na_nt^{n-1}=A'(t)$$

Now multiply both sides by $t$:

$$\sum_{n=0}^{\infty}na_nt^{n}=tA'(t)$$