# Sequence question

• Aug 30th 2010, 09:14 AM
usagi_killer
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 $\{a_n\}_{n \ge 0}$ and $\{b_n\}_{n \ge 0}$ in order to satisfy the equation $A(x) \times B(x) = 1$ for all $x \in \mathbb{R}$ such that $A(x), B(x)$ are defined. Where $A(x) = a_0+a_1x+a_2x^2+ \cdots$ and $B(x) = b_0+b_1x+b_2x^2+ \cdots$

Thanks again!
• Aug 30th 2010, 10:12 AM
Opalg
• Aug 30th 2010, 12:33 PM
usagi_killer
Thanks Opalg, excellent explanations!

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

Now I've gotten this far:

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

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

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!
• Sep 1st 2010, 12:11 AM
Nguyen
Quote:

Originally Posted by usagi_killer
Thanks Opalg, excellent explanations!

Also I'm stuck on another extension of this problem. It says "Assuming $A(x) \times B(x) = 1$, 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.
• Sep 1st 2010, 05:47 AM
Opalg
I think that the only thing resembling a recurrence relation that is satisfied by these coefficients is the relation $\sum_{k=0}^na_kb_{n-k} = 0$, which implies that $b_n = -a_0^{-1}\sum_{k=1}^na_kb_{n-k}$. 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 $\cosh x$, which has a very simple power series, the coefficients of the reciprocal function $1/\cosh x$ are not easy to determine. In fact, they involve the Euler numbers, which do not satisfy any straightforward recurrence relation.
• Sep 1st 2010, 07:26 AM
koyyew
An additional question which follows from the previous questions.
If http://1.bp.blogspot.com/_Gaxb8WAXd-...DE/s1600/a.png for all n≥0 and http://www.mathhelpforum.com/math-he...36a92e4f88.png, then what are the http://4.bp.blogspot.com/_Gaxb8WAXd-...WMQ/s320/b.png?

Thanks for the help.