Adding summation with different start-values

Question

I'm trying to find the recurrence relation, through adding the summations below;

$$\sum_{n=0}^\infty n(n-1)a_nx^{n-2} + \sum_{n=0}^\infty a_nx^n(n-1)(n-2)$$

Now I'm a bit confused about how to add these to summations, since the term x has different exponents. If I would rewrite the first term, so that it's exponent is n, instead of n-2, I would get;

$$\sum_{n=2}^\infty (n+2)(n+1)a_{n+2}x^n$$

But the question is, how do I add together two summations, one that start with n=0, and one that start with n=2?

Answer 1

Hint: Write out the first two terms in the second sum:

$$\sum_{n=0}^{\infty}a_nx^n(n-1)(n-2)=a_0x^0(0-1)(0-2)+a_1x^1(1-1)(1-2)+\sum_{n=2}^{\infty}a_nx^n(n-1)(n-2)$$ $$\sum_{n=0}^{\infty}a_nx^n(n-1)(n-2)=2a_0+\sum_{n=2}^{\infty}a_nx^n(n-1)(n-2)$$