By user8603

2011-03-28 05:05:29 8 Comments

Say if I define a power series over some arbitrary field $F$ as

$$a = \sum^{ \infty }_{i = 0} a_{i} X^{i} $$

Then can I say:

$$ab = \sum^{ \infty }_{i = 0} \sum^{ \infty }_{j = 0} a_{i} b_{j} X^{i + j} $$


@Bill Dubuque 2011-03-28 17:24:20

Yes, using the natural notion of convergence for formal power series, the stated sum does indeed converge to the Cauchy product. One should beware - as exemplified in this thread - that there is widespread confusion about formal vs. functional power series - even by some experts (in other fields). Rota frequently told jokes in his lectures about certain distinguished mathematicians who published complete nonsense based on such confusion (Indiscrete Thoughts!)

In any case the basic ideas are quite simple if you merely take off your analyst hat and, instead, put on your algebraist or combinatorist hat. In particular, you should be able to find a correct discussion of convergence of formal power series in almost any good book on combinatorics or generating functions, e.g. here is an excerpt from Stanley's classic $\: $ Enumerative Combinatorics I.

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@Robert Israel 2011-03-28 05:10:13

Yes, you can: this is the definition of the product of the two formal power series $a$ and $b =\sum_{i=0}^\infty b_i X^i$.

@joriki 2011-03-28 05:14:27

No, this is not the definition of the product of the formal power series; in fact it's not a power series. The product is $\sum_{k=0}^\infty(\sum_{i=0}^k a_ib_{k-i})X^k$; see and (Of course the two coincide if all the limits exist, but the point about formal power series is that you can manipulate them like this without worrying about whether the limits exist.) By the way, note that if you talk about "power series over some arbitrary field", these are actually just polynomials if the field is finite.

@Bill Dubuque 2011-03-28 17:41:04

@Joriki: No, this is in fact correct (but far too terse). It appears that you are confusing formal power series with functional power series. In particular, formal power series over finite fields are not "just polynomials".

@Alex Becker 2011-03-28 05:13:59

Not if you are talking about multiplying $a$ by a similarly defined formal power series $b$. The multiplication of formal power series can be written as: $ab = \sum_{i=0}^{\infty}(\sum_{j=0}^ia_jb_{i-j})X^i$

@joriki 2011-03-28 05:41:06

It should be $X^i$; but also see my answer.

@Alex Becker 2011-03-28 05:43:29

@joriki: Thanks, but I think that's obviously enough a type not to call it an "incorrect answer".

@joriki 2011-03-28 05:45:21

I'm sorry, that didn't refer to the typo but to the fact that your answer basically says "no" when it should have been "yes and no" -- but in fact I weakened that to "incorrect or partial" even before I'd seen your comment -- sorry that the initial criticism was a bit too strong.

@Alex Becker 2011-03-28 05:52:23

@joriki: That's why I said "not if... formal power series". But I know you didn't mean to blast me, and I agree that your answer is much fuller and more informative.

@Marc van Leeuwen 2016-05-03 04:09:21

This answer is wrong, in the sense that while it gives a decent definition of multiplication of formal power series, but wrongly claims the one in the OP is not right. The two are in fact equivalent definitions. In the OP formula, the coefficient of some $X^k$ is $\sum_{i+j=k}a_ib_j = \sum_{i=0}^ka_ib_{k-i}$ and after some renaming of indices the formula above follows.

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