Infinite products

Let us now extend our FPS playground somewhat. We have made sense of infinite sums. What about infinite products?

3.11.1. An example

We start with a motivating example (due to Euler, in [Euler48, §328–329]), which we shall first discuss informally.

Assume (for the time being) that the infinite product

i

∏

∈_N

1+x²ⁱ

=1+x¹ 1+x² 1+x⁴ 1+x⁸

· · · (105) in the ring K[[x]] is meaningful, and that such products behave as nicely as finite products. Can we simplify this product?

We can observe that eachi ∈_Nsatisfies 1+x²ⁱ = ¹−x²ⁱ⁺¹

1−x²ⁱ (since 1−x²ⁱ⁺¹ = 1−x²ⁱ2

= ₁−x²ⁱ 1+x²ⁱ

). Multiplying these equalities over all i ∈ _N, we obtain The product on the right hand side here is a telescoping product– meaning that each numerator is cancelled by the denominator of the following fraction. As-suming (somewhat plausibly, but far from rigorously) that we are allowed to cancel infinitely many factors from an infinite product, we thus end up with a single 1−x¹ factor in the denominator. That is, our product simplifies to

1

This was not very rigorous, so let us try to compute the product ∏

i∈_N

1+x²ⁱ in a different way. Namely, we recall a simple fact about finite products: If a0,a1, . . . ,amare finitely many elements of a commutative ring, then the product

∏

m i=0

(1+ai) = (1+a0) (1+a1)· · ·(1+am) (107) equals the sum³³

i₁<i2<···<

∑

i_k≤m

a_i1a_i2· · ·a_ik

of all the 2^m+1 “sub-products” of the product a₀a₁· · ·am (because this sum is what we obtain if we expand ∏^m

i=0

(1+a_i)by repeatedly applying distributivity).

For example, form=2, this is saying that (1+a0) (1+a1) (1+a2)

=1+a0+a1+a2+a0a1+a0a2+a1a2+a0a1a2. Now, it is plausible to expect the same formula ∏^m

i=₀

(1+a_i) = _∑

I⊆{0,1,...,m} ∏

i∈I

a_i to hold if “mis ∞” (that is, if the product ranges over all i ∈N), provided that the product is meaningful. In other words, it is plausible to expect that

i

∏

∈_N

(1+a_i) =

∑

i₁<i₂<···<i_k

a_i1a_i2· · ·a_ik (108) for any infinite sequence a0,a1,a2, . . . as long as ∏

i∈_N

(1+ai) makes sense. (It’s a little bit more complicated than that, but we aren’t trying to be fully rigorous yet. The correct condition is that the sequence (a₀,a₁,a₂, . . .) is summable.) If we now apply (108) toai =x²ⁱ, then we obtain

i

∏

∈N

1+x²ⁱ

=

∑

i₁<i₂<···<i_k

x²ⁱ¹x²ⁱ² · · ·x^2ik =

∑

i₁<i₂<···<i_k

x^{2i1+2i2+···+2ik}

=

∑

n∈_N

qnxⁿ, (109)

where qn is the # of ways to write the integer n as a sum 2ⁱ¹ +2ⁱ² +· · ·+2^ik with nonnegative integersi₁,i₂, . . . ,i_k satisfyingi₁ <i₂ < · · · < i_k. Comparing this with (106), we obtain

n

∑

∈_N

qnxⁿ =1+x+x²+x³+· · · =

∑

n∈_N

xⁿ,

33The indicesi₁,i2, . . . ,i_kof the sum are supposed to be nonnegative integers.

at least if our assumptions were valid. Comparing coefficients, this would mean that qn = 1 for each n ∈ N. In other words, each n ∈ _N can be written in exactly one way as a sum 2ⁱ¹ +2ⁱ² +· · · +2^ik with nonnegative integers i₁,i₂, . . . ,i_k satisfying i₁ < i₂ < · · · < i_k. In other words, each n ∈ _N can be written uniquely as a finite sum of distinct powers of 2.

Is this true? Yes, because this is just saying that each n ∈ _N has a unique binary representation. For example, 21=2⁴+2²+2⁰corresponds to the binary representation 21= (10101)₂.

Thus, the two results we have obtained in (106) and (109) are actually equal, which is reassuring. Yet, this does not replace a formal definition of infinite products that rigorously justifies the above arguments.

3.11.2. A rigorous definition

One way of rigorously defining infinite products of FPSs can be found in [Loehr11, §7.5]. However, this definition only defines infinite products of the form ∏

i∈_Nor ∏^∞

i=1

, but not (for example) of the form ∏

I⊆_Nor ∏

(i,j)∈_N×_N

. Another def-inition of infinite products uses the Log and Exp bijections from Defdef-inition 3.7.6 to turn products into sums; but this requiresKto be aQ-algebra (since Log and Exp aren’t defined otherwise). Thus, we shall give a different definition here.

We recall our definition of infinite sums of FPSs (Definition 3.2.9):

Definition 3.2.9 (repeated). A (possibly infinite) family (a_i)_i∈I of FPSs is said to besummableif

for each n∈ N, all but finitely manyi ∈ I satisfy [xⁿ]a_i =_0.

In this case, the sum ∑

i∈I

a_i is defined to be the FPS with

[xⁿ]

∑

i∈I

a_i

!

=

∑

i∈I

[xⁿ]a_i

| {z }

an essentially finite sum

for all n ∈_N.

This is how we defined infinite sums of FPSs. We cannot use the same defi-nition for infinite products, because usually

wedon’texpect to have [xⁿ]

∏

i∈I

a_i

!

=

∏

i∈I

[xⁿ]a_i

(after all, multiplication of FPSs is not defined coefficientwise). The condition

“all but finitely many i ∈ I satisfy [xⁿ]a_i = 0” is therefore not what we are looking for.

Let us instead go back to the idea behind Definition 3.2.9. Let us fix some n ∈ N. What was the actual purpose of the “all but finitely many i ∈ I sat-isfy [xⁿ]a_i = 0” condition? The purpose was to ensure that the coefficient [xⁿ]

is determined by finitely many of the a_i’s. In other words, the purpose was to ensure that there is a finite partial sum of ∑

i∈I

a_i such that if we add any further a_i’s to this partial sum, then the coefficient of xⁿ does not change any more. Here is a way to restate this condition more rigorously: There is afinitesubset MofI such that every finite subset J ofI satisfying M⊆ J ⊆ I

(The subset M here is the indexing set of our finite partial sum, and the set J is what it becomes if we add some furthera_i’s to this partial sum.)

This condition is a mouthful; this is why we found it easier to boil it down to the simple “all but finitely many i ∈ I satisfy [xⁿ]a_i = 0” condition in the case of infinite sums. However, in the case of infinite products, we cannot boil it down to something this simple; thus, we have to live with it.

Fortunately, we can simplify our life by giving this condition a name:

Definition 3.11.1. Let (a_i)_i∈I ∈ K[[x]]^I be a (possibly infinite) family of FPSs.

Let n∈ _{N. Let} M be a finite subset of I.

(a) We say that M determines the xⁿ-coefficient in the sum of (a_i)_i∈I if every finite subset J of I satisfying M ⊆ J ⊆ I satisfies

[xⁿ]

∑

(You can think of this condition as saying “If you add any further a_is to the sum ∑

i∈M

a_i, then thexⁿ-coefficient stays unchanged”, or, more informally: “If you want to know thexⁿ-coefficient of ∑

i∈I

a_i, it suffices to take the partial sum over alli ∈ M”.)

(b)We say that M determines the xⁿ-coefficient in the product of (a_i)_i∈I if every finite subset J of I satisfying M ⊆ J ⊆ I satisfies

[xⁿ]

∏

(You can think of this condition as saying “If you multiply any furthera_is to the product ∏

i∈M

a_i, then the xⁿ-coefficient stays unchanged”, or, more infor-mally: “If you want to know the xⁿ-coefficient of ∏

i∈I

a_i, it suffices to take the partial product over alli∈ M”.)

Example 3.11.2. (a)Consider the family

i∈N, because every finite subset J of N satisfying {1, 2, 3} ⊆ J ⊆_Nsatisfies

(this is simply a consequence of the fact that the only three entries of our family that have a nonzero x³-coefficient are the entries x+x²i

for i ∈ {1, 2, 3}). Thus, any finite subset of N that contains {1, 2, 3} as a subset determines the x³-coefficient in the sum of this family

x+x²i

i∈_N, because every finite subset J of N satisfying {0, 1, 2, 3} ⊆ J ⊆_Nsatisfies

(This is because multiplying an FPS by any of the polynomials 1+_x⁴_{, 1}+ x⁵, 1+x⁶, . . . leaves its x³-coefficient unchanged.) Thus, any finite subset

(The philosophical reason is that, even though the monomial x³ itself does not appear in any of the entries 1+x¹ and 1+x², it does emerge in the product of these two entries with the constant term of ∏

i∈{0,3}

1+xⁱ

= (1+1) 1+x³

.)

Definition 3.11.3. Let (a_i)_i∈I ∈ K[[x]]^I be a (possibly infinite) family of FPSs.

Let n∈ _N.

(a) We say that the xⁿ-coefficient in the sum of (a_i)_i∈I is finitely determined if there is a finite subset Mof I that determines thexⁿ-coefficient in the sum of (a_i)_i∈I.

(b)We say that the xⁿ-coefficient in the product of (a_i)_i∈I is finitely determined if there is a finite subset M of I that determines the xⁿ-coefficient in the product of (a_i)_i∈I.

Using these concepts, we can now reword our definition of infinite sums as follows:

Proposition 3.11.4. Let (a_i)_i∈I ∈ K[[x]]^I be a (possibly infinite) family of FPSs. Then:

(a)The family(a_i)_i∈I is summable if and only if each coefficient in its sum is finitely determined (i.e., for each n ∈ _{N, the} xⁿ-coefficient in the sum of (a_i)_i∈I is finitely determined).

(b) If the family (a_i)_i∈I is summable, then its sum ∑

i∈I

a_i is the FPS whose xⁿ-coefficient (for any n ∈ N) can be computed as follows: If n ∈ _{N, and} if M is a finite subset of I that determines the xⁿ-coefficient in the sum of (a_i)_i∈I, then

[xⁿ]

∑

i∈I

a_i

!

= [xⁿ]

∑

i∈M

a_i

! .

Proof. Easy and LTTR.

The rest of Chapter 3 needs more details.

TODO: Push this border stone further down this chapter.

Inspired by Proposition 3.11.4, we can now define infinite products of FPSs at last:

Definition 3.11.5. Let (a_i)_i∈I be a (possibly infinite) family of FPSs. Then:

(a)The family(a_i)_i∈I is said to bemultipliableif and only if each coefficient in its product is finitely determined.

(b) If the family (a_i)_i∈I is multipliable, then its product ∏

i∈I

a_i is defined to be the FPS whosexⁿ-coefficient (for anyn ∈N) can be computed as follows:

If n∈ _{N, and if} M is a finite subset of I that determines thexⁿ-coefficient in the product of (a_i)_i∈I, then

Proposition 3.11.6. This definition of ∏

i∈I

a_iis well-defined – i.e., the coefficient [xⁿ] M1and M2 are two finite subsets of I that each determine the xⁿ-coefficient in the product of(a_i)_i∈I, then finite subset J of I satisfying M₁ ⊆ J ⊆I satisfies

[xⁿ]

∏

argument (with the roles of M1 and M2 swapped) yields

[xⁿ]

∏

The left hand sides of the equalities (111) and (112) are equal (since M₁∪M₂ = M2∪ M1). Thus, the right hand sides are equal as well. In other words, [xⁿ] _∏

i∈M₁

a_i

!

= [xⁿ] _∏

i∈M₂

a_i

!

. Thus, we have proved (110), and with it Propo-sition 3.11.6.

The attentive (and pedantic) reader might notice that there is one more thing that needs to be checked in order to make sure that Definition 3.11.5(b) is le-gitimate. In fact, this definition does not merely define (some) infinite products

i∏∈I

a_i of FPSs, but also “accidentally” gives a new meaning to finite products

i∏∈I

a_i (since a finite family (a_i)_i∈I of FPSs is always multipliable). We therefore need to check that this new meaning does not conflict with the original defini-tion of a finite product of elements of a commutative ring. In other words, we need to prove the following:

Proposition 3.11.7. Let (a_i)_i∈I be a finite family of FPSs. Then, the product

i∏∈I

a_i defined according to Definition 3.11.5(b)equals the finite product ∏

i∈I

a_i defined in the usual way (i.e., defined as in any commutative ring).

Proof. Argue that I itself is a subset of I that determines all coefficients in the product of(a_i)_i∈I. See Section B.2 for a detailed proof.

Let us now see that this legitimizes our product ∏

i∈_N

1+x²ⁱ

from Subsec-tion 3.11.1. Indeed,

i

∏

∈N

1+x²ⁱ

=1+x¹ 1+x² 1+x⁴ 1+x⁸

· · · .

If you want to compute the x⁶-coefficient in this product, you only need to multiply the first 3 factors 1+x¹

1+x²

1+x⁴

; none of the other factors will change this coefficient in any way, because multiplying an FPS by 1+x^m (for some m > 0) does not change its first m coefficients³⁴. Likewise, if you want to compute the x¹³-coefficient of the above product, then you only need to multiply the first 4 factors; none of the others will have any effect on this coefficient. The same logic applies to the xⁿ-coefficient for any n ∈ _{N; it is} determined by the firstblog₂nc+1 factors of the product. Thus, each coefficient

34For example, let us check this form=3: If we multiply an FPSa0x⁰+a1x¹+a2x²+· · · by 1+x³, then we obtain

a0x⁰+a₁x¹+a2x²+· · · 1+x³

=a0x⁰+a1x¹+a2x²+ (a3+a0)x³+ (a4+a1)x⁴+ (a5+a2)x⁵+· · ·, and so the first 3 coefficients are left unchanged.

in the product is finitely determined. This means that the family is multipliable;

thus, its product makes sense.

In contrast, the product

(1+0x) (1+1x) (1+2x) (1+3x) (1+4x)· · · =

∏

i∈_N

(1+ix)

does not make sense. Indeed, its x¹-coefficient is not finitely determined (any of the factors other than 1+0x affects it), so the family (1+ix)_i∈N is not mul-tipliable.

Recall our reasoning that we used above to prove that the family

1+x²ⁱ

i∈_N

is multipliable. The core of this reasoning was the observation that multiplying an FPS by 1+x^m (for some m > 0) does not change its first m coefficients.

This can be generalized: If f ∈ K[[x]] is an FPS whose first m coefficients are 0 (for example, f can be x^m, in which case we recover the statement in our preceding sentence), then multiplying an FPS a by 1+ f does not change its first m coefficients (that is, the first m coefficients of a(1+ f) are the first m coefficients ofa). This is a useful fact, so let us state it as a lemma (renamingm asn+1):

Lemma 3.11.8. Let a, f ∈ K[[x]]be two FPSs. Letn ∈N. Assume that

[x^m] f =0 for each m∈ {0, 1, . . . ,n}. (113) Then,

[x^m] (a(1+ f)) = [x^m]a for each m∈ {0, 1, . . . ,n}.

Proof of Lemma 3.11.8. The FPS a f is a multiple of f (since a f = f a). Hence, Lemma 3.3.21 (applied tou= f and v =a f) yields that

[x^m] (a f) =0 for each m∈ {0, 1, . . . ,n} (114) (since we have assumed that[x^m] f =0 for each m∈ {0, 1, . . . ,n}).

Now, for eachm ∈ {0, 1, . . . ,n}, we have

[x^m]





a(1+ f)

| {z }

=a+a f





= [x^m] (a+a f) = [x^m]a+ [x^m] (a f)

| {z }

=0 (by (114))

(by (18))

= [x^m]a.

This proves Lemma 3.11.8.

For convenience, let us extend Lemma 3.11.8 to products of several factors:

Lemma 3.11.9. Let a ∈ K[[x]] be an FPS. Let (f_i)_i∈J ∈ K[[x]]^J be a finite family of FPSs. Letn∈ N. Assume that eachi∈ J satisfies

[x^m] (f_i) =0 for eachm∈ {0, 1, . . . ,n}. (115) Then,

[x^m] a

∏

i∈J

(1+ fi)

!

= [x^m]a for each m∈ {0, 1, . . . ,n}.

Proof of Lemma 3.11.9. This is just Lemma 3.11.8, applied several times (specifi-cally,|J|many times). See Section B.2 for a detailed proof.

Now, using Lemma 3.11.9, we can obtain the following convenient criterion for multipliability:

Theorem 3.11.10. Let(f_i)_i∈I ∈ K[[x]]^I be a (possibly infinite) summable fam-ily of FPSs. Then, the famfam-ily(1+ f_i)_i∈I is multipliable.

Proof of Theorem 3.11.10. This is an easy consequence of Lemma 3.11.9. See Sec-tion B.2 for a detailed proof.

We notice two simple sufficient (if rarely satisfied) criteria for multipliability:

Proposition 3.11.11. If all but finitely many entries of a family (a_i)_i∈I ∈ K[[x]]^I equal 1 (that is, if all but finitely many i ∈ I satisfy a_i = 1), then this family is multipliable.

Proof. LTTR. (See Section B.2 for a detailed proof.)

Remark 3.11.12. If a family (a_i)_i∈I ∈ _K[[x]]^I contains 0 as an entry (i.e., if there exists an i ∈ I such that a_i = 0), then this family is automatically multipliable, and its product is 0.

Proof. Assume that the family (a_i)_i∈I contains 0 as an entry. That is, there exists some j ∈ I such that a_j = 0. Consider this j. Now, it is easy to see that the subset {j} of I determines all coefficients in the product of (a_i)_i∈I. The details are LTTR.

Working with multipliable families gets slightly easier using the following notion:

Definition 3.11.13. Let (a_i)_i∈I ∈ K[[x]]^I be a family of FPSs. Letn ∈ _{N. An} xⁿ-approximator for (a_i)_i∈I means a finite subset M of I that determines the first n+1 coefficients in the product of (a_i)_i∈I. (In other words, Mhas to de-termine thex^m-coefficient in the product of(a_i)_i∈I for eachm ∈ {0, 1, . . . ,n}.) The name “xⁿ-approximator” is supposed to hint at the fact that if M is an xⁿ-approximator for a multipliable family (a_i)_i∈I, then the (finite) subproduct

i∈∏M

a_i “approximates” the full product ∏

i∈I

a_i up until thexⁿ-coefficient (i.e., the first n+1 coefficients of ∏

i∈M

a_i equal the respective coefficients of ∏

i∈I

a_i). See Proposition 3.11.16(b)below for the precise statement of this fact.

Clearly, an xⁿ-approximator for a family (a_i)_i∈I always determines the xⁿ -coefficient in the product of(a_i)_i∈I. But the converse is not true, as the following example shows:

Example 3.11.14. Consider the family

1+x²ⁱ

i∈N =1+x¹, 1+x², 1+x⁴, 1+x⁸, . . .

of FPSs. The finite subset {1, 2} of N determines the x⁶-coefficient in the product of this family (indeed, the x⁶-coefficient of the product

1+x²

1+x⁴

is 1, and this does not change if we multiply any further fac-tors onto this product), but is not an x⁶-approximator for this family (since, e.g., it does not determine thex⁵-coefficient in its product).

Lemma 3.11.15. Let (a_i)_i∈I ∈ K[[x]]^I be a multipliable family of FPSs. Let n ∈N. Then, there exists anxⁿ-approximator for (a_i)_i∈I.

Proof of Lemma 3.11.15 (sketched). This is an easy consequence of the fact that a union of finitely many finite sets is finite. A detailed proof can be found in Section B.2.

As promised above, we can use xⁿ-approximators to “approximate” infinite products of FPSs (in the sense of: compute the first n+1 coefficients of these products). Here is why this works:³⁵

Proposition 3.11.16. Let (a_i)_i∈I ∈ K[[x]]^I be a family of FPSs. Let n∈ _{N. Let} M be anxⁿ-approximator for (a_i)_i∈I. Then:

(a)Every finite subset J of I satisfying M⊆ J ⊆ I satisfies

∏

i∈J

a_i ^x

n

≡

∏

i∈M

a_i.

35See Definition 3.10.1 for the meaning of the symbol “^x

n

≡” appearing in this proposition.

(b)If the family(a_i)_i∈I is multipliable, then

∏

i∈I

a_i ^x

n

≡

∏

i∈M

a_i.

Proof. This follows easily from Definition 3.11.13 and Definition 3.11.5 (b). See Section B.2 for a detailed proof.

Here are some further properties of multipliable families:

Proposition 3.11.17. Let (a_i)_i∈I ∈ K[[x]]^I be a family of FPSs. Let J be a subset of I. Assume that the subfamilies(a_i)_i∈J and(a_i)_i∈I\J are multipliable.

Then:

(a)The entire family (a_i)_i∈I is multipliable.

(b)We have

∏

i∈I

a_i =

∏

i∈J

a_i

!

·





∏

i∈I\J

a_i



.

Proof of Proposition 3.11.17 (sketched). Here is the idea: Fix n ∈ _N. Lemma 3.11.15 (applied to J instead of I) shows that there exists anxⁿ-approximatorU for (a_i)_i∈J. Consider this U. Lemma 3.11.15 (applied to J instead of I) shows that there exists anxⁿ-approximator V for(a_i)_i∈I\J. Consider thisV. Note that U∪V is finite (sinceU and V are finite). Now, it is not hard to see thatU∪V determines the xⁿ-coefficient in the product of (a_i)_i∈I (indeed, it is not much harder to see that U ∪V is an xⁿ-approximator for (a_i)_i∈I). Hence, the xⁿ -coefficient in the product of(a_i)_i∈I is finitely determined (sinceU∪V is finite).

Now, forget that we fixedn, and conclude that the family(a_i)_i∈I is multipliable.

This proves part(a). Part(b)easily follows using Proposition 3.11.16(b).

The details of this proof can be found in Section B.2.

Proposition 3.11.18. Let (a_i)_i∈I ∈ K[[x]]^I and (b_i)_i∈I ∈ K[[x]]^I be two multi-pliable families of FPSs. Then:

(a)The family (a_ib_i)_i∈I is multipliable.

(b)We have

∏

i∈I

(a_ib_i) =

∏

i∈I

a_i

!

·

∏

i∈I

b_i

! .

Proof of Proposition 3.11.18 (sketched). Here is the idea: Fix n ∈ _N. Lemma 3.11.15 shows that there exists anxⁿ-approximator U for (a_i)_i∈I. Consider this U. Lemma 3.11.15 (applied to b_i instead of a_i) shows that there exists an xⁿ -approximatorV for(b_i)_i∈I. Consider this V. Note thatU∪V is finite (since U

and V are finite). Now, it is not hard to see that U∪V is an xⁿ-approximator for(a_ib_i)_i∈I. From here, proceed as in the proof of Proposition 3.11.17.

The details of this proof can be found in Section B.2.

Proposition 3.11.19. Let (a_i)_i∈I ∈ K[[x]]^I be a multipliable family of invert-ible FPSs. Then, any subfamily of (a_i)_i∈I is multipliable.

Proof of Proposition 3.11.19 (sketched). This is another proof in the tradition of the proofs of Proposition 3.11.17 and Proposition 3.11.18. We must show that the family(a_i)_i∈J is multipliable wheneverJis a subset ofI. The idea is to show that ifU is an xⁿ-approximator for(a_i)_i∈I, then U∩ _J determines the xⁿ-coefficient in the product of (a_i)_i∈J (and, in fact, is an xⁿ-approximator for (a_i)_i∈J). This relies on the invertibility of ∏

i∈M\J

a_i; this is why the FPS a_i are required to be invertible in the proposition.

The details of this proof can be found in Section B.2.

Remark 3.11.20. Proposition 3.11.19 would not hold if we did not assume that thea_iare invertible. For example, the family(0, 1, 2, 3, . . .)is multipliable, but its subfamily (1, 2, 3, . . .)is not.

To justify our work from Subsection 3.11.1, we still need to argue that the products we have defined are well-behaved – i.e., satisfy the usual rules that finite products satisfy, with some minor caveats. To stay on the safe side, we state them only for products of invertible FPSs, to avoid the nasty surprises we know from Remark 3.11.20:

Proposition 3.11.21. Products of multipliable families of FPSs satisfy the usual rules for products, as long as we assume that our families consist of invertible FPSs. For example, the following facts hold:

• Let(a_s)_s∈S ∈ K[[x]]^S and(b_s)_s∈S ∈ K[[x]]^S be two multipliable families of invertible FPSs. Then, the family (asbs)_s∈S is multipliable as well, and satisfies

∏

s∈S

(asbs) =

∏

s∈S

as

!

·

∏

s∈S

bs

! .

• Let (as)_s∈S ∈ K[[x]]^S be a multipliable family of invertible FPSs. LetX andY be two subsets ofS such that X∩Y =_∅and X∪Y =S. Then,

∏

s∈S

a_s =

∏

s∈X

a_s

!

·

∏

s∈Y

a_s

! .

• Let (as)_s∈S ∈ K[[x]]^S be a multipliable family of invertible FPSs. Let W be a set. Let f : S→W be a map. Then,

∏

s∈S

a_s =

∏

w∈W

∏

s∈S;

f(s)=w

a_s.

(In particular, the right hand side is well-defined – i.e., the fam-ily (as)_{s∈S; f(s)=w} is multipliable for each w ∈ W, and the family





 ∏

s∈S;

f(s)=w

as







w∈W

is also multipliable.)

• Let I and J be two sets. Let a_(i,j)

(i,j)∈I×J ∈K[[x]]^I×J be a multipliable family of invertible FPSs. Then,

∏

i∈I

∏

j∈J

a_(i,j) =

∏

(i,j)∈I×J

a_(i,j) =

∏

j∈J

∏

i∈I

a_(i,j).

(In particular, all the products appearing in this equality are well-defined.)

Proof. In essence, all of these facts can be proved by reducing them to the cor-responding properties of finite products. This relies on the fact that all co-efficients in multipliable products are finitely determined (conveniently using xⁿ-approximators, which determine several coefficients at the same time).

TODO: Add details!

Proposition 3.11.21 justifies most of our manipulations in Subsection 3.11.1 (except possibly for the telescope principle, which is somewhat subtle and needs some qualifications³⁶; it is better to argue using more fundamental rules³⁷).

36Here is an example of how not to use the telescope principle:

1 2 ·²

2·² 2 ·²

2· · · · 6=1.

The infinitely many 2’s don’t factor each other out completely.

37To wit, we can argue as follows: We have

i∈

∏

N

1+x²ⁱ

=

∏

i∈N

1−x²ⁱ⁺¹ 1−x²ⁱ =

i∈∏N

1−x²ⁱ⁺¹

i∈∏N

1−x²ⁱ , where the last step used the fact that both families

1−x²ⁱ⁺¹

i∈N and

1−x²ⁱ

i∈N are multipliable (this is important, but very easy to check in this case) and that the product

The only thing we still need to justify is the equality (108). This is what we will do next.

3.11.3. Product rules (generalized distributive laws)

The equality (108) is an instance of a product rule – a statement of the form

“a product of sums can be expanded into one big sum”. The simplest prod-uct rules are the distributive laws a(b+c) = ab+ac and (a+b)c = ac+bc (here, one of the sums being multiplied is a one-addend sum); one of the next-simplest is (a+b) (c+d) = ac+ad+bc+bd. As far as finite sums and finite products are concerned, the following product rule is one of the most general:³⁸ Proposition 3.11.22. Let L be a commutative ring. For every n ∈ _{N, let} [n] denote the set{1, 2, . . . ,n}.

We can rewrite (116) in a less abstract way as follows:

p1,1+p1,2+· · ·+p1,m₁

(p2,1+p2,2+· · ·+p2,m₂)· · ·(pn,1+pn,2+· · ·+pn,m_n)

= p1,1p2,1· · ·pn,1+p1,1p2,1· · ·pn−1,1pn,2+· · ·+p1,m₁p2,m₂· · ·pn,mn,

i∈∏N

1−x²ⁱ

is invertible (indeed, its constant term is 1). However, splitting off the factor fori=0 from the product ∏ foriin the product)

= (1−x)·

∏

1−x. So we don’t need the telescope principle to justify this equality.

38Keep in mind that an empty Cartesian product (i.e., a Cartesian product of 0 sets) is always a 1-element set; its only element is the 0-tuple (). Thus, a sum ranging over an empty Cartesian product has exactly 1 addend.

where the right hand side is the sum of allm₁m₂· · ·mn many ways to multiply one addend from each of the factors on the left hand side.

See [Grinbe15, solution to Exercise 6.9] for a formal proof of Proposition 3.11.22. (The idea is to reduce it to the casen =2 by induction, then to use the discrete Fubini rule.)

Let us now move on to product rules for infinite sums and products. First, let us extend Proposition 3.11.22 to a finite product of infinite sums (which are now required to be inK[[x]] in order to have a notion of summability):

Proposition 3.11.23. For everyn∈ _{N, let} [n] denote the set{1, 2, . . . ,n}.

Proposition 3.11.23. For everyn∈ _{N, let} [n] denote the set{1, 2, . . . ,n}.

Im Dokument Algebraic Combinatorics (Seite 126-149)