Posets,
Integrals &
Sums
Michael E.
Hoffman
Outline
Labeled Posets,
Iterated Integrals, and Nested Sums
Yamamoto’s
formalism
An algebra of
2-labeled
posets
Integerlabeled posets
and nested
sums
From iterated
integrals to
nested sums
Extension to
(r + 1)-labeled
posets
Michael E. Hoffman
U. S. Naval Academy
AMS Eastern Section
Special Session on Iterated Integrals & Applications
Georgetown University, Washington DC
8 March 2015
Michael E. Hoffman
Posets, Integrals & Sums
Outline
Posets,
Integrals &
Sums
Michael E.
Hoffman
1 Yamamoto’s formalism
Outline
Yamamoto’s
formalism
2 An algebra of 2-labeled posets
An algebra of
2-labeled
posets
3 Integer-labeled posets and nested sums
Integerlabeled posets
and nested
sums
4 From iterated integrals to nested sums
From iterated
integrals to
nested sums
5 Extension to (r + 1)-labeled posets
Extension to
(r + 1)-labeled
posets
Michael E. Hoffman
Posets, Integrals & Sums
Yamamoto’s Formalism
Posets,
Integrals &
Sums
Michael E.
Hoffman
Outline
Yamamoto’s
formalism
An algebra of
2-labeled
posets
Integerlabeled posets
and nested
sums
From iterated
integrals to
nested sums
Extension to
(r + 1)-labeled
posets
Recently S. Yamamoto developed an elegant way of
representing iterated intgrals by 2-labeled posets. The iterated
integrals in question are all over some subset of [0, 1]n , and
involve only the two forms
dt
dt
.
ω0 (t) = , ω1 (t) =
t
1−t
Let (X , δ) be a 2-labeled poset, i.e., a finite partially ordered
set X with a function δ : X → {0, 1}. Call (X , δ) admissible if
δ(x) = 1 for all minimal x ∈ X and δ(x) = 0 for all maximal
x ∈ X . For an admissible 2-labeled poset (X , δ), define the
associated integral by
ωδ(x) (tx ),
(1)
I (X ) =
Δ(X ) x∈X
where Δ(X ) = {(tx )x∈X ∈ [0, 1]X | tx < ty if x < y in X }.
Michael E. Hoffman
Posets, Integrals & Sums
Graphical Representation
Posets,
Integrals &
Sums
Michael E.
Hoffman
Outline
Yamamoto’s
formalism
An algebra of
2-labeled
posets
Integerlabeled posets
and nested
sums
From iterated
integrals to
nested sums
Extension to
(r + 1)-labeled
posets
We can represent a 2-labeled poset (X , δ) graphically by its
Hasse diagram, with an open dot ◦ for those elements x ∈ X
with δ(x) = 0, and a closed dot • for those x ∈ X with
δ(x) = 1. For example, the 2-labeled poset X = {x1 , x2 , x3 , x4 }
with x1 > x2 > x4 , x1 > x3 > x4 , δ(x1 ) = δ(x2 ) = 0, and
δ(x3 ) = δ(x4 ) = 1 has graphical representation
and associated integral
I (X ) =
t1 >t2 >t4
t1 >t3 >t4
Michael E. Hoffman
dt1 dt2 dt3 dt4
.
t1 t2 1 − t3 1 − t4
Posets, Integrals & Sums
2-labeled Posets and MZVs
Posets,
Integrals &
Sums
Michael E.
Hoffman
Outline
Yamamoto’s
formalism
An algebra of
2-labeled
posets
Integerlabeled posets
and nested
sums
Admissibility of (X , δ) guarantees convergence of the integral
I (X ). If X is a chain, then I (X ) is just the well-known iterated
integral representation of a multiple zeta value (MZV). For
example,
dt1 dt2 dt3
=
=
I
t1 >t2 >t3 t1 1 − t2 1 − t3
0
1
dt1
t1
t1
0
From iterated
integrals to
nested sums
Extension to
(r + 1)-labeled
posets
dt2
1 − t2
t2
0
∞ ∞
1
=
0
j=1 i =1
dt3
=
1 − t3
1
0
dt1
t1
0
∞ i +j−1
∞ t1 t2
j=1 i =1
i
dt2
t1i +j−1
1
dt1 =
= ζ(2, 1).
i (i + j)
i (i + j)2
Michael E. Hoffman
i ,j≥1
Posets, Integrals & Sums
2-labeled Posets and MZVs cont’d
Posets,
Integrals &
Sums
Michael E.
Hoffman
Our notation for MZVs is
ζ(k1 , . . . , kr ) =
Outline
Yamamoto’s
formalism
An algebra of
2-labeled
posets
Integerlabeled posets
and nested
sums
From iterated
integrals to
nested sums
Extension to
(r + 1)-labeled
posets
1
k1
n1 >n2 >···>nk ≥1 n1
· · · nrkr
,
with k1 , . . . , kr positive integers and k1 > 1 required for
convergence. The representation of MZVs by iterated integrals
is also encoded by the polynomial algebra Qx, y , where x and
dt
y are noncommuting variables encoding dtt and 1−t
respectively; convergent integrals correspond to monomials that
start with x and end with y . The MZV ζ(k1 , . . . , kr ) is
represented by the monomial x k1 −1 y · · · x kr −1 y . Shuffle
on H = Qx, y ,
product gives a commutative operation
which corresponds to multiplication of the integrals; indeed we
can think of ζ as a homomorphism (H0 , ) → R.
Michael E. Hoffman
Posets, Integrals & Sums
Properties of the Integral
Posets,
Integrals &
Sums
Michael E.
Hoffman
Outline
Yamamoto’s
formalism
An algebra of
2-labeled
posets
Integerlabeled posets
and nested
sums
From iterated
integrals to
nested sums
Extension to
(r + 1)-labeled
posets
Yamamoto proved the following theorem, which follows easily
from general properties of iterated integrals.
Theorem
Let X be an admissible 2-labeled poset.
1
2
3
If Y is another admissible 2-labeled poset,
I (X Y ) = I (X )I (Y ).
If a, b ∈ X are incomparable, let Xa<b be X with the
additional relation a < b. Then
I (X ) = I (Xa<b ) + I (Xb<a ).
If X ∨ is X with reversed order and new labeling function
δ∨ (x) = 1 − δ(x), then I (X ∨ ) = I (X ).
Michael E. Hoffman
Posets, Integrals & Sums
Using the Properties
Posets,
Integrals &
Sums
Michael E.
Hoffman
Outline
Yamamoto’s
formalism
An algebra of
2-labeled
posets
Integerlabeled posets
and nested
sums
From iterated
integrals to
nested sums
Extension to
(r + 1)-labeled
posets
It is evident that combining two disjoint chains via Part 2 is
equivalent to shuffle product in Qx, y . For example,
=I
+ 3I
+ 6I
I
is exactly parallel to
xy
x 2y = xyx 2y + 3x 2yxy + 6x 3y 2,
both showing that
ζ(2)ζ(3) = ζ(2, 3) + 3ζ(3, 2) + 6ζ(4, 1).
Michael E. Hoffman
Posets, Integrals & Sums
Using the Properties cont’d
Posets,
Integrals &
Sums
Michael E.
Hoffman
Outline
Yamamoto’s
formalism
An algebra of
2-labeled
posets
Integerlabeled posets
and nested
sums
From iterated
integrals to
nested sums
Extension to
(r + 1)-labeled
posets
In fact, use of Part 2 allows us to write I (X ) as a sum of
multiple zeta values for any admissible 2-labeled poset X . For
example,
=I
+I
= ζ(3, 1) + ζ(2, 2).
I
By the way, the “sum theorem” for MZVs implies that the
latter sum is ζ(4), and more generally that
= ζ(n),
I
where the diagram has n − k open dots and k closed ones, for
2 ≤ k ≤ n − 2.
Michael E. Hoffman
Posets, Integrals & Sums
Mordell-Tornheim Sums
Posets,
Integrals &
Sums
Michael E.
Hoffman
Outline
Yamamoto’s
formalism
An algebra of
2-labeled
posets
Integerlabeled posets
and nested
sums
From iterated
integrals to
nested sums
Extension to
(r + 1)-labeled
posets
An advantage of Yamamoto’s formalism is that is makes it easy
to see how various zeta functions can be written in terms of
MZVs. For example, the Mordell-Tornheim sums
T (n1 , n2 , . . . , nk ; p) =
1
mn1
m1 ,...,mk ≥1 1
· · · mknk (m1
+ · · · + mk )p
can be expressed as integrals associated with 2-labeled posets,
as shown by the example
I
t4
dt4 t4 dt3 t3 dt2
dt1
=
t3 0 1 − t2 0 1 − t1
0 t4
0
1
dt4 t4i +j
1
=
= T (2, 1; 1).
2
2 j(i + j)
t
i
j
i
4
0
1
=
i ,j≥1
Michael E. Hoffman
i ,j≥1
Posets, Integrals & Sums
Mordell-Torheim Sums cont’d
Posets,
Integrals &
Sums
Michael E.
Hoffman
Outline
Yamamoto’s
formalism
An algebra of
2-labeled
posets
Integerlabeled posets
and nested
sums
From iterated
integrals to
nested sums
Extension to
(r + 1)-labeled
posets
If we compare the shuffle product with the poset representing
Mordell-Tornheim sums, it is easy to see that
T (n1 , n2 , . . . , nk ; p) =
ζ(x p (x n1 −1 y
x n −1y · · · x n −1y )),
2
k
(2)
which gives a succinct statement of how Mordell-Tornheim
sums can be expanded into MZVs. For example,
T (2, 1; 3) = ζ(x 3 (xy
y )) = ζ(x 3(yxy + 2xy 2))
= ζ(x 3 yxy + 2x 4 y 2 ) = ζ(4, 2) + 2ζ(5, 1).
Before seeing Yamamoto’s idea, I’d been expanding
Mordell-Torhneim sums into MZVs for years without
recognizing equation (2).
Michael E. Hoffman
Posets, Integrals & Sums
Multiple Zeta-Star Values
Posets,
Integrals &
Sums
The multiple zeta-star values
ζ (k1 , . . . , kr ) =
Michael E.
Hoffman
Outline
Yamamoto’s
formalism
An algebra of
2-labeled
posets
Integerlabeled posets
and nested
sums
From iterated
integrals to
nested sums
Extension to
(r + 1)-labeled
posets
1
k1
n1 ≥n2 ≥···≥nk ≥1 n1
· · · nrkr
,
can also be represented by integrals associated with 2-labeled
posets, as in the example
1
1
dt4 t4 dt3
dt2 t2 dt1
=
=
I
0 1 − t3 t3 t2
0 1 − t1
0 t4
1 j−1
1
t4i +j−1
t4
dt4 t4 dt3 1 − t3i
− 2
dt4
=
i2
i 2j
i (i + j)
0 t4
0
0 1 − t3
=
∞
i =1
1
i2
∞ j=1
i ≥1
i ,j≥1
∞
i
1
1 1
1
−
= ζ (2, 2).
=
j 2 (i + j)2
i2
j2
Michael E. Hoffman
i =1
j=1
Posets, Integrals & Sums
Using the Properties
Posets,
Integrals &
Sums
Michael E.
Hoffman
Outline
Applying Part 2 of Yamamoto’s theorem to the representation
of T (2, 1; 1) gives
=I
+ 2I
,
I
Yamamoto’s
formalism
An algebra of
2-labeled
posets
Integerlabeled posets
and nested
sums
From iterated
integrals to
nested sums
Extension to
(r + 1)-labeled
posets
which just recovers the identity T (2, 1; 1) = ζ(2, 2) + 2ζ(3, 1)
well-known from partial fractions. But applying it to the
representation of ζ (2, 2) gives
=I
+ 4I
,
I
implying ζ (2, 2) = ζ(2, 2) + 4ζ(3, 1); comparing with the more
familiar ζ (2, 2) = ζ(2, 2) + ζ(4) gives ζ(3, 1) = 14 ζ(4).
Michael E. Hoffman
Posets, Integrals & Sums
An Algebra of 2-labeled Posets
Posets,
Integrals &
Sums
Michael E.
Hoffman
Outline
Yamamoto’s
formalism
An algebra of
2-labeled
posets
Integerlabeled posets
and nested
sums
From iterated
integrals to
nested sums
Extension to
(r + 1)-labeled
posets
We define a graded Q-algebra A as follows. For each 2-labeled
poset X , A has a generator [X ] in degree card X . We agree
that [X ] = [Y ] if X and Y are isomorphic as 2-labeled posets,
and define the product by [X ][Y ] = [X Y ]. If A0 ⊂ A is the
subalgebra generated by admissible 2-labeled posets, then
Yamamoto’s theorem implies that I : A0 → R is a
homomorphism.
Call a 2-labeled poset X lower admissible if δ(x) = 1 for every
minimal element of X . Then the subspace A1 of A generated
by lower admissible 2-labeled posets is a subalgebra, and in fact
A0 ⊂ A1 ⊂ A.
Michael E. Hoffman
Posets, Integrals & Sums
A Homomorphism From A to H
Posets,
Integrals &
Sums
Michael E.
Hoffman
Outline
Yamamoto’s
formalism
An algebra of
2-labeled
posets
Integerlabeled posets
and nested
sums
From iterated
integrals to
nested sums
Now define a Q-linear map J : A → H by sending any chain
{x1 > x2 > · · · > xn } to the monomial a(x1 )a(x2 ) · · · a(xn ),
= 0 and a(xi ) =y if δ(xi ) = 1. In
where a(xi ) = x if δ(xi )
general define J(P) as i J(ci ), where i ci is the formal sum
of all chains obtained by adding more relations to P.
Let H1 be the subspace of H generated by 1 and monomials
ending in y . Note that (H1 , ) is a commutative algebra. Then
J restricted to A1 is a homomorphism J : A1 → H1 . Similarly,
if H0 is the subspace of H1 generated by 1 and all monomials
that start in x and end in y , then (H0 , ) is an algebra and
J : A0 → H0 is a homomorphism such that I (X ) = ζ(J(X )).
Extension to
(r + 1)-labeled
posets
Michael E. Hoffman
Posets, Integrals & Sums
Combinatorics
Posets,
Integrals &
Sums
Michael E.
Hoffman
An advantage of expanding the domain of J to A1 is that we
now have interesting combinatorial examples, e.g.,
Yamamoto’s
formalism
An algebra of
2-labeled
posets
Integerlabeled posets
and nested
sums
Outline
J(•) = y ,
=y ,
2
J
= 5y ,
4
J
J
= 2y 3 ,
J
= 16y 5 ,
J
= 61y 6 ,
From iterated
integrals to
nested sums
Extension to
(r + 1)-labeled
posets
where the series 1, 1, 2, 5, 16, 61, . . . is that of Euler or
up/down numbers.
Michael E. Hoffman
Posets, Integrals & Sums
Integer-labeled Posets and Nested Sums
Posets,
Integrals &
Sums
Michael E.
Hoffman
Outline
Yamamoto’s
formalism
An algebra of
2-labeled
posets
Integerlabeled posets
and nested
sums
We can encode nested sums with a formalism quite similar to
Yamamoto’s. Let P be a finite poset with a labeling function
δ : P → Z+ . Call P admissible if δ(x) > 1 for every maximal
element of P. For admissible P, let SOP(P) be the set of
strictly order-preserving functions σ : P → Z+ , and define the
sum S(P) by
S(P) =
σ∈SOP(P) x∈P
1
.
σ(x)δ(x)
From iterated
integrals to
nested sums
If P is an admissible integer-labeled poset that is a chain, then
S(P) is a multiple zeta value; in fact
Extension to
(r + 1)-labeled
posets
S({x1 > x2 > · · · > xn }) = ζ(δ(x1 ), δ(x2 ), . . . , δ(xn )).
Michael E. Hoffman
Posets, Integrals & Sums
Properties of Nested Sums
Posets,
Integrals &
Sums
Michael E.
Hoffman
We have the following counterpart of Yamamoto’s theorem for
nested sums.
Outline
Theorem
Yamamoto’s
formalism
Let P be an admissible integer-labeled poset.
An algebra of
2-labeled
posets
Integerlabeled posets
and nested
sums
From iterated
integrals to
nested sums
1
2
If Q is another admissible integer-labeled poset,
S(P Q) = S(P)S(Q)
If a, b ∈ P are incomparable, let Pa<b be P with the
additional relation a < b, and let Pa=b be P with a and b
merged into a new element having label δ(a) + δ(b). Then
S(P) = S(Pa<b ) + S(Pb<a ) + S(Pa=b ).
Extension to
(r + 1)-labeled
posets
Michael E. Hoffman
Posets, Integrals & Sums
An Algebra of Integer-labeled Posets
Posets,
Integrals &
Sums
Michael E.
Hoffman
Outline
Yamamoto’s
formalism
An algebra of
2-labeled
posets
Integerlabeled posets
and nested
sums
Now define a graded Q-algebra B as follows. For each
integer-labeled
poset X , B has a generator [X ] ; we assign it
degree x∈X δ(x). We agree that [X ] = [Y ] if X and Y are
isomorphic as integer-labeled posets, and define the product by
[X ][Y ] = [X Y ]. If B0 ⊂ B is the subalgebra generated by
admissible integer-labeled posets, then the preceding theorem
implies that S : B0 → R is a homomorphism.
From iterated
integrals to
nested sums
Extension to
(r + 1)-labeled
posets
Michael E. Hoffman
Posets, Integrals & Sums
A Homomorphism From B to H1
Posets,
Integrals &
Sums
Michael E.
Hoffman
Outline
Yamamoto’s
formalism
An algebra of
2-labeled
posets
Integerlabeled posets
and nested
sums
From iterated
integrals to
nested sums
Extension to
(r + 1)-labeled
posets
Now define a Q-linear map K : B → H1 by sending any chain
δ(x1 )−1
δ(xn )−1 y .
{x1 > x2 > · · · > xn } to the
monomial x y · · · x
In general define K (P) as i K (ci ), where i ci is the formal
sum of all chains obtained by adding more relations to P.
Evidently K (B0 ) ⊂ H0 , where as above H0 is the subspace of
H1 generated by 1 and all monomials of the form xwy .
To make K : B → H1 a homomorphism, we must give H1 the
“quasi-shuffle” product ∗ defined recursively as follows:
w ∗ 1 = 1 ∗ w = w for all monomials w , and
(x p−1 yu) ∗ (x q−1 v ) = x p−1 y (u ∗ x q−1 yv ) + x q−1 y (x p−1 yu ∗ v )
+ x p+q−1 y (u ∗ v )
for all monomials u, v . Then S(w ) = ζ(K (w )) for w ∈ H0 .
Michael E. Hoffman
Posets, Integrals & Sums
Mapping 2-labeled Posets to Integer-labeled Posets
Posets,
Integrals &
Sums
Michael E.
Hoffman
Outline
Yamamoto’s
formalism
An algebra of
2-labeled
posets
Integerlabeled posets
and nested
sums
From iterated
integrals to
nested sums
Extension to
(r + 1)-labeled
posets
If we introduce an equivalence relation ≈ on elements of A by
declaring [X ] ≈ [Xa<b ] + [Xb<a ] for a, b ∈ X incomparable,
then A1 / ≈ is isomorphic to (H1 , ). Similarly, if we define ∼
on B by [P] ∼ [Pa<b ] + [Pb<a ] + [Pa=b ] for a, b ∈ P
incomparable, then B/ ∼ is isomorphic to (H1 , ∗). If we send
the chain in [X ] ∈ A1 / ≈ corresponding to x i1 −1 y · · · x ik −1 y to
the chain [Y ] = {y1 > · · · > yk } in B/ ∼ with δ(yj ) = ij , then
I ([X ]) = S([Y ]), provided [X ] ∈ A0 , since both sides are equal
to ζ(i1 , . . . , ik ). That is, the identity function from (H0 , ) to
(H0 , ∗) is certainly not a homomorphism, but
ζ(u v ) = ζ(u)ζ(v ) = ζ(u ∗ v ).
Michael E. Hoffman
Posets, Integrals & Sums
Extended MZVs
Posets,
Integrals &
Sums
Michael E.
Hoffman
Outline
Yamamoto’s
formalism
An algebra of
2-labeled
posets
Integerlabeled posets
and nested
sums
From iterated
integrals to
nested sums
Extension to
(r + 1)-labeled
posets
Now it is well-known that the homomorphisms
ζ : (H0 , ) → R and ζ : (H0 , ∗) → R can be extended to
homomorphisms ζ : (H1 , ) → R[T ] and ζ ∗ : (H1 , ∗) → R[T ]
by introducing an indeterminate T to represent ζ(y ). The
superscripts are necessary because these extensions clearly do
not agree: since y y = 2y 2 while y ∗ y = 2y 2 + xy , we have
ζ (y 2 ) = 12 T 2 but ζ ∗ (y 2 ) = 12 T 2 − 12 ζ(2). But it is natural to
ask if the identity function on H0 can be extended to a linear
function μ : H1 → H1 such that ζ (u) = ζ ∗ (μ(u)) for u ∈ H1 .
It appears so: and in fact, it appears μ can be defined
recursively on the filtration
H0 ⊂ H0 + y H0 ⊂ H0 + y H0 + y 2 H0 ⊂ · · · ⊂ H1
by μ(y
u) = y ∗ μ(u).
Michael E. Hoffman
Posets, Integrals & Sums
Extending Yamamoto’s Formalism
Posets,
Integrals &
Sums
Michael E.
Hoffman
Outline
Yamamoto’s
formalism
An algebra of
2-labeled
posets
Integerlabeled posets
and nested
sums
From iterated
integrals to
nested sums
Extension to
(r + 1)-labeled
posets
We note that Yamamoto’s formalism can be extended to
iterated integrals of the forms
ωα (t) =
αdt
dt
=
,
−1
1 − αt
α−1
where α is an r th root of unity. Here one must replace
2-labeled posets by (r + 1)-labeled posets, where the labels
come from the set {0, 1, , . . . , r −1 }, where is a primitive r th
root of unity. An (r + 1)-labeled poset is admissible if minimal
elements of X don’t have label 0 and maximal elements of X
don’t have label 1. Then equation (1) carries over to this case,
and one has Yamamoto’s theorem (except that one must drop
the part about generalized duality).
Michael E. Hoffman
Posets, Integrals & Sums
Nested Sums for r th Roots of Unity
Posets,
Integrals &
Sums
Michael E.
Hoffman
Outline
Yamamoto’s
formalism
An algebra of
2-labeled
posets
Integerlabeled posets
and nested
sums
From iterated
integrals to
nested sums
Extension to
(r + 1)-labeled
posets
The generalization of MZVs that appears in this case are
multiple polylogarithms evaluated at roots of unity: if we define
Lii1 ,...,ik (z1 , . . . , zk ) =
z1n1 · · · zknk
n1 >···>nk ≥1
n1i1 · · · nkik
,
then I (X ) corresponding to the chain whose δ’s are given by
the string
(ω0 )i1 −1 ωα1 · · · (ω0 )ik −1 ωαk ,
for α1 , . . . , αk ∈ {1, , 2 , . . . , r −1 } with α1 = 1 has value
Lii1 ,...,ik (α1 ,
Michael E. Hoffman
α2
αk
,...,
).
α1
αk−1
Posets, Integrals & Sums
The Case r = 2
Posets,
Integrals &
Sums
Michael E.
Hoffman
Outline
The case r = 1 is just that treated above. The case r = 2
applies to alternating MZVs like
ζ(2̄, 1, 3̄) =
Yamamoto’s
formalism
An algebra of
2-labeled
posets
Integerlabeled posets
and nested
sums
From iterated
integrals to
nested sums
Extension to
(r + 1)-labeled
posets
l>m>n≥1
(−1)l+n
l 2 mn3
and gives nice “diagrammatic” evaluations of them.
Shuffle-product formulas for generalized Mordell-Tornheim
sums of this type, e.g.,
n,m≥1
(−1)n
,
n2 m(n + m)2
(similar to equation (2) above) are easily obtained.
Michael E. Hoffman
Posets, Integrals & Sums