# Thread: Sequence question

1. ## Sequence question

Hi, I am currently stuck on this sequence question, I tried playing around with it for a while now, can't seem to solve it. Any help would be appreciated!

What are a necessary and sufficient set of conditions on these integer sequences $\displaystyle \{a_n\}_{n \ge 0}$ and $\displaystyle \{b_n\}_{n \ge 0}$ in order to satisfy the equation $\displaystyle A(x) \times B(x) = 1$ for all $\displaystyle x \in \mathbb{R}$ such that $\displaystyle A(x), B(x)$ are defined. Where $\displaystyle A(x) = a_0+a_1x+a_2x^2+ \cdots$ and $\displaystyle B(x) = b_0+b_1x+b_2x^2+ \cdots$

Thanks again!

2. Thanks Opalg, excellent explanations!

Also I'm stuck on another extension of this problem. It says "Assuming $\displaystyle A(x) \times B(x) = 1$, find a recurrence formula for the integer sequence {b_n} in terms of the integer sequence {a_n}

Now I've gotten this far:

$\displaystyle a_0b_0 = 1 so b_0 = \frac{1}{a_0}$

By the same principles $\displaystyle b_2 = \frac{a_1^2a_0-a_2}{a_0^2}$

But as you keep going b_3, b_4 get extremely messy and I can not find any relationship or pattern between the b_n's and a_n's.

Could anyone shed some light?

Thanks!

3. Originally Posted by usagi_killer
Thanks Opalg, excellent explanations!

Also I'm stuck on another extension of this problem. It says "Assuming $\displaystyle A(x) \times B(x) = 1$, find a recurrence formula for the integer sequence {b_n} in terms of the integer sequence {a_n}
I read this and gave it a go but couldn't get far. I see what you mean by it getting very messy. I am sorry I couldn't help but now I really want to find out how this is done.

4. I think that the only thing resembling a recurrence relation that is satisfied by these coefficients is the relation $\displaystyle \sum_{k=0}^na_kb_{n-k} = 0$, which implies that $\displaystyle b_n = -a_0^{-1}\sum_{k=1}^na_kb_{n-k}$. That hardly counts as recurrence relation in the usual sense, because the number of terms in the relation varies with n.

Even for a function like $\displaystyle \cosh x$, which has a very simple power series, the coefficients of the reciprocal function $\displaystyle 1/\cosh x$ are not easy to determine. In fact, they involve the Euler numbers, which do not satisfy any straightforward recurrence relation.

5. An additional question which follows from the previous questions.
If for all n≥0 and , then what are the ?

Thanks for the help.