Combinatorics
& MZVs
Michael E.
Hoffman
Outline
Multiple Zeta
Values
The QuasiSymmetric
Functions
ζ : QSym0 →
R and its
Extension ζu
Other
Homomorphic
Images of
Quasisymmetric
Functions
Combinatorics and
Multiple Zeta Values
Michael E. Hoffman
U. S. Naval Academy
Simon Fraser University
Discrete Mathematics Seminar
8 July 2014
Factoring ζγ
Michael E. Hoffman
Combinatorics & MZVs
Outline
Combinatorics
& MZVs
Michael E.
Hoffman
1 Multiple Zeta Values
Outline
Multiple Zeta
Values
The QuasiSymmetric
Functions
2 The Quasi-Symmetric Functions
3 ζ : QSym0 → R and its Extension ζu
0
ζ : QSym →
R and its
Extension ζu
Other
Homomorphic
Images of
Quasisymmetric
Functions
4 Other Homomorphic Images of Quasi-symmetric Functions
5 Factoring ζγ
Factoring ζγ
Michael E. Hoffman
Combinatorics & MZVs
Multiple Zeta Values
Combinatorics
& MZVs
Michael E.
Hoffman
Outline
Multiple Zeta
Values
The QuasiSymmetric
Functions
ζ : QSym0 →
R and its
Extension ζu
Other
Homomorphic
Images of
Quasisymmetric
Functions
Factoring ζγ
For positive integers i1 , i2 , . . . ik with i1 > 1 (to ensure
convergence), we define the multiple zeta value (henceforth
MZV) ζ(i1 , i2 , . . . , ik ) as the sum of the k-fold series
X
1
.
ik
i1 i2
n
n
·
·
·
n
k
n1 >n2 ···>nk ≥1 1 2
We call k the depth of ζ(i1 , . . . , ik ), and i1 + · · · + ik its weight.
Of course, the MZVs of depth one are the values ζ(i) of the
Riemann zeta function at positive integers i > 1, for which
Euler proved the formula
(2π)2i
|B2i |,
2(2i)!
but he also studied depth-two MZVs. Interest in MZVs of
general depth dates from about 1990, stimulated by their
appearance in physics and knot theory.
ζ(2i) =
Michael E. Hoffman
Combinatorics & MZVs
(1)
Multiplication of MZVs
Combinatorics
& MZVs
Michael E.
Hoffman
An important point is that the product of two MZVs is a sum
of MZVs. For example,
Outline
Multiple Zeta
Values
The QuasiSymmetric
Functions
ζ : QSym0 →
R and its
Extension ζu
Other
Homomorphic
Images of
Quasisymmetric
Functions
Factoring ζγ
ζ(2, 1)ζ(2) =
X 1 X 1
=
i 2j
k2
i>j≥1
X 1
+
i 4j
i>j≥1
X
i>k>j≥1
k≥1
1
X
k>i>j≥1
k 2i 2j
X 1
1
+
+
i 2k 2j
i 2j 3
i>j≥1
+
X
i>j>k≥1
1
i 2 jk 2
= ζ(2, 2, 1) + ζ(4, 1) + ζ(2, 2, 1) + ζ(2, 3) + ζ(2, 1, 2)
= 2ζ(2, 2, 1) + ζ(2, 1, 2) + ζ(4, 1) + ζ(2, 3)
My study of this product led me to the quasi-symmetric
functions, first introduced by Ira Gessel in 1984 as a source of
generating functions for poset partitions.
Michael E. Hoffman
Combinatorics & MZVs
Symmetric and Quasi-Symmetric Functions
Combinatorics
& MZVs
Michael E.
Hoffman
Let x1 , x2 , . . . be a countable sequence of indeterminates, each
of degree 1, and let
P ⊂ Q[[x1 , x2 , . . . ]]
Outline
Multiple Zeta
Values
The QuasiSymmetric
Functions
ζ : QSym0 →
R and its
Extension ζu
Other
Homomorphic
Images of
Quasisymmetric
Functions
Factoring ζγ
be the set of formal power series in the xi of bounded degree:
P is a graded Q-algebra. Any f ∈ P is quasi-symmetric if the
coefficients in f of
xip11 xip22 · · · xipnn
and xjp11 xjp22 · · · xjpnn
agree whenever i1 < · · · < in and j1 < · · · < jn . The
quasi-symmetric functions QSym form an algebra, which
properly includes the algebra Sym of symmetric functions, e.g.,
X
xi2 xj = x12 x2 + x12 x3 + x22 x3 + · · ·
(2)
i<j
is quasi-symmetric but not symmetric.
Michael E. Hoffman
Combinatorics & MZVs
Monomial (Quasi)symmetric Functions
Combinatorics
& MZVs
Michael E.
Hoffman
Outline
Multiple Zeta
Values
The QuasiSymmetric
Functions
ζ : QSym0 →
R and its
Extension ζu
Other
Homomorphic
Images of
Quasisymmetric
Functions
Factoring ζγ
For a composition (ordered sequence of positive integers)
I = (i1 , . . . , ik ), the corresponding monomial quasi-symmetric
function MI ∈ QSym is defined by
X
MI =
xni11 xni22 · · · xnikk
n1 <n2 <···<nk
(so (2) above is M(2,1) ). Evidently {MI |I is a composition} is
an integral basis for QSym. For any composition I , let π(I ) be
the partition given by forgetting the ordering. For any partition
λ, the monomial symmetric function mλ is the sum of the MI
with π(I ) = λ, e.g.,
m21 = M(2,1) + M(1,2) .
The set {mλ |λ is a partition} is an integral basis for Sym.
Michael E. Hoffman
Combinatorics & MZVs
Other Bases for Sym
Combinatorics
& MZVs
1
Michael E.
Hoffman
ek = M(1)k = mπ((1)k ) ,
Outline
Multiple Zeta
Values
The QuasiSymmetric
Functions
2
0
ζ : QSym →
R and its
Extension ζu
Other
Homomorphic
Images of
Quasisymmetric
Functions
The elementary symmetric functions are
where (1)k is the composition consisting of k 1’s. Then
{eλ |λ is a partition} is an integral basis for Sym, where
eλ = eλ1 eλ2 · · · for λ = π(λ1 , λ2 , . . . ).
The complete symmetric functions are
X
X
hk =
MI =
mλ .
|I |=k
3
|λ|=k
Then {hλ |λ is a partition} is a integral basis for Sym.
The power-sum symmetric functions are
pk = M(k) = mk .
Factoring ζγ
Then {pλ |λ is a partition} is a rational basis for Sym.
Michael E. Hoffman
Combinatorics & MZVs
Multiplication of Monomial Quasi-Symmetric
Functions
Combinatorics
& MZVs
Michael E.
Hoffman
Outline
Two monomial quasi-symmetric functions MI and MJ multiply
according to a “quasi-shuffle” rule in which the parts of I and
J are shuffled and also combined. For example,
M(1) M(1,2) = M(1,1,2) + M(1,1,2) + M(1,2,1) + M(2,2) + M(1,3)
Multiple Zeta
Values
The QuasiSymmetric
Functions
0
ζ : QSym →
R and its
Extension ζu
Other
Homomorphic
Images of
Quasisymmetric
Functions
Factoring ζγ
= 2M(1,1,2) + M(1,2,1) + M(2,2) + M(1,3) .
In general the number of terms in the product MI MJ is
min{`(I ),`(J)} X
`(I ) + `(J) − i
i, `(I ) − i, `(J) − i
i=0
3
so, e.g., M(1,1) M(2,1) has 42 + 111
+ 22 = 13 terms:
M(1,1) M(1,2) = 3M(1,1,1,2) + 2M(1,1,2,1) + M(1,2,1,1)
+ M(1,2,2) + M(2,1,2) + M(2,2,1) + 2M(1,1,3) + M(1,1,3,1) + M(2,3) .
Michael E. Hoffman
Combinatorics & MZVs
Algebraic Structure of QSym
Combinatorics
& MZVs
Michael E.
Hoffman
Outline
Over the rationals, the algebraic structure of QSym can be
described as follows. Order compositions lexicographically:
(1) < (1, 1) < (1, 1, 1) < · · · < (1, 2) < · · ·
< (2) < (2, 1) < · · · < (3) < · · ·
Multiple Zeta
Values
The QuasiSymmetric
Functions
A composition I is Lyndon if I < K whenever I = JK for
nonempty compositions J, K .
ζ : QSym0 →
R and its
Extension ζu
Theorem (Reutenauer-Malevenuto, 1995)
Other
Homomorphic
Images of
Quasisymmetric
Functions
Factoring ζγ
QSym is a polynomial algebra on MI , I Lyndon.
Since the only Lyndon composition ending in 1 is (1) itself, the
subspace QSym0 of QSym generated by the MI such that I
ends in an integer greater than 1 is a subalgebra, and in fact
QSym = QSym0 [M(1) ].
Michael E. Hoffman
Combinatorics & MZVs
MZVs as Homomorphic Images
Combinatorics
& MZVs
Michael E.
Hoffman
Outline
Multiple Zeta
Values
The QuasiSymmetric
Functions
ζ : QSym0 →
R and its
Extension ζu
Other
Homomorphic
Images of
Quasisymmetric
Functions
Factoring ζγ
There is a homomorphism ζ : QSym0 → R given by
ζ(MI ) = ζ(ik , ik−1 , . . . , i1 )
for I = (i1 , . . . , ik ), induced by sending xj → 1j . (This is
well-defined since ik > 1 for MI ∈ QSym0 .) Note
ζ(pi ) = ζ(M(i) ) = ζ(i) for i > 1. The intersection
Sym0 = QSym0 ∩ Sym
is the subspace of Sym generated by the mλ such that all parts
of λ are 2 or more, and is in fact the subalgebra of Sym
generated by the pi with i > 1.
Michael E. Hoffman
Combinatorics & MZVs
MZVs as Homomorphic Images cont’d
Combinatorics
& MZVs
Michael E.
Hoffman
Outline
Knowing MZVs are homomorphic images has immediate
consequences. For example, we know that the power-sums pn
generate Sym rationally, so there is a polynomial Pn with
rational coefficients such that
Multiple Zeta
Values
The QuasiSymmetric
Functions
ζ : QSym0 →
R and its
Extension ζu
Other
Homomorphic
Images of
Quasisymmetric
Functions
Factoring ζγ
en = Pn (p1 , p2 . . . , pn ).
Now apply to both sides the homomorphism sending M(i1 ,...,im )
to M(ki1 ,...,kim ) to get
M(k, . . . , k ) = Pn (pk , p2k , . . . , pnk )
| {z }
n
and thus
ζ(k, . . . , k ) = Pn (ζ(k), ζ(2k), . . . , ζ(nk))
| {z }
n
Michael E. Hoffman
Combinatorics & MZVs
MZVs as Homomorphic Images cont’d
Combinatorics
& MZVs
Michael E.
Hoffman
Outline
Multiple Zeta
Values
The QuasiSymmetric
Functions
ζ : QSym0 →
R and its
Extension ζu
Other
Homomorphic
Images of
Quasisymmetric
Functions
for any k > 1. In particular, since we know from equation (1)
that ζ(k) is a rational multiple of π k if k is even, and since Pn
has degree n, we have the following result.
Theorem
If k is even, then ζ(k, . . . , k ) is a rational multiple of π kn .
| {z }
n
For particular values of k more specific results are known, e.g.,
π 2n
22n+1 π 4n
ζ(2, . . . , 2) =
, ζ(4, . . . , 4) =
,
| {z }
| {z }
(2n + 1)!
(4n + 2)!
n
n
6(2π)6n
ζ(6, . . . , 6) =
.
| {z }
(6n + 3)!
Factoring ζγ
n
Michael E. Hoffman
Combinatorics & MZVs
The Function A− : QSym → QSym
Combinatorics
& MZVs
Michael E.
Hoffman
Outline
Multiple Zeta
Values
The QuasiSymmetric
Functions
ζ : QSym0 →
R and its
Extension ζu
Other
Homomorphic
Images of
Quasisymmetric
Functions
Factoring ζγ
Consider the linear function from QSym to itself defined by
(
M(a1 ,...,ak−1 ) , if ak = 1,
A− (M(a1 ,...,ak−1 ,ak ) ) =
0,
otherwise.
(Here M∅ is interpreted as 1, so A− (M(1) ) = 1.) From the
quasi-shuffle description of multiplication in QSym the
following result can be proved.
Proposition
A− : QSym → QSym is a derivation.
Note that ker A− = QSym0 . So if we think of QSym as
QSym0 [M(1) ], then A− is differentiation by M(1) .
Michael E. Hoffman
Combinatorics & MZVs
Extending ζ to ζu : QSym → R[u]
Combinatorics
& MZVs
Michael E.
Hoffman
Outline
Multiple Zeta
Values
The QuasiSymmetric
Functions
ζ : QSym0 →
R and its
Extension ζu
Other
Homomorphic
Images of
Quasisymmetric
Functions
Factoring ζγ
Since QSym = QSym0 [M(1) ], we can extend ζ to a
homomorphism ζu : QSym → R[u] by defining ζu (w ) = ζ(w )
for w ∈ QSym0 and ζu (M(1) ) = u. In view of the Proposition
above, we have the following commutative diagram:
ζu
QSym −−−−→ R[u]



d 
A− y
du y
ζu
QSym −−−−→ R[u]
In the case u = γ ≈ 0.5572 (Euler’s constant), the
homomorphism ζu turns out to be especially useful. To explain
why we must digress a bit to introduce some generating
functions.
Michael E. Hoffman
Combinatorics & MZVs
Generating Functions
Combinatorics
& MZVs
1
Michael E.
Hoffman
Outline
The generating function of the elementary symmetric
functions is
X
Y
E (t) =
ei t i =
(1 + txi ).
i≥0
Multiple Zeta
Values
The QuasiSymmetric
Functions
2
ζ : QSym0 →
R and its
Extension ζu
Other
Homomorphic
Images of
Quasisymmetric
Functions
Factoring ζγ
That for the complete symmetric functions is
X
Y 1
= E (−t)−1 .
H(t) =
hi t i =
1 − txi
i≥0
3
i≥1
i≥1
The logarithmic derivative of H(t) is the generating
function P(t) for the power sums:
P(t) =
X d
X
d
log H(t) = −
log(1 − txi ) =
pi t i−1 .
dt
dt
i≥1
Michael E. Hoffman
Combinatorics & MZVs
i≥1
Generating Functions cont’d
Combinatorics
& MZVs
Michael E.
Hoffman
Outline
Multiple Zeta
Values
The power series expansion of ψ(1 − t), where ψ is the
logarithmic derivative of the gamma function, is
X
ψ(1 − t) = −γ −
ζ(i)t i−1 .
i≥2
The QuasiSymmetric
Functions
ζ : QSym0 →
R and its
Extension ζu
Other
Homomorphic
Images of
Quasisymmetric
Functions
Factoring ζγ
If u = γ, then ζu (p1 ) = γ and so ζu (P(t)) = −ψ(1 − t).
d
log H(t)) we have
Hence (since P(t) = dt
Theorem (Generating Function)
ζγ (H(t)) = Γ(1 − t) .
From E (t) = H(−t)−1 follows ζγ (E (t)) = Γ(1 + t)−1 .
Michael E. Hoffman
Combinatorics & MZVs
Generating Functions cont’d
Combinatorics
& MZVs
Michael E.
Hoffman
Now it can be shown analytically that the generating function
X
F (s, t) =
ζ(n + 1, (1)m−1 )s n t m
n,m≥1
Outline
Multiple Zeta
Values
The QuasiSymmetric
Functions
ζ : QSym0 →
R and its
Extension ζu
Other
Homomorphic
Images of
Quasisymmetric
Functions
Factoring ζγ
can be written
F (s, t) = 1 −
Γ(1 − t)Γ(1 − s)
.
Γ(1 − t − s)
Applying the Generating Function Theorem, this is
H(t)H(s)
F (s, t) = ζγ 1 −
.
H(t + s)
Since

H(t) = exp 


X pi t i
 = e p1 t exp 

i
i
X pi t i
i≥1
Michael E. Hoffman

i≥2
Combinatorics & MZVs
Generating Functions cont’d
Combinatorics
& MZVs
Michael E.
Hoffman
it follows that

1−
Outline
Multiple Zeta
Values
The QuasiSymmetric
Functions
ζ : QSym0 →
R and its
Extension ζu
Other
Homomorphic
Images of
Quasisymmetric
Functions
Factoring ζγ
H(t)H(s)
= 1 − exp 
H(t + s)
X
pi
ti
+
si
i≥2
− (t +
i
s)i


and hence, applying ζγ ,
X
ζ(n + 1, (1)m−1 )s n t m =
n,m≥1

1 − exp 
X
ζ(i)
ti
i≥2
+
si
− (t +
i
s)i

.
It follows that any MZV of the form ζ(n + 1, (1)m−1 ) is a
polynomial in the ζ(i), i ≥ 2, with rational coefficients, though
/ Sym for m > 1.
M((1)m−1 ,n+1) ∈
Michael E. Hoffman
Combinatorics & MZVs
Multiple t-Values
Combinatorics
& MZVs
Michael E.
Hoffman
Besides MZVs, other quantities are homomorphic images of
quasi-symmetric functions. For a sequence i1 , i2 , . . . , ik with
i1 > 1, the associated multiple t-value is
Outline
t(i1 , . . . , ik ) =
Multiple Zeta
Values
The QuasiSymmetric
Functions
ζ : QSym0 →
R and its
Extension ζu
Other
Homomorphic
Images of
Quasisymmetric
Functions
Factoring ζγ
X
i1
n1 >···>nk ≥1, ni odd n1
1
· · · nkik
Evidently
X
t(i) =
n ≥ 1 odd
1
= (1 − 2−i )ζ(i)
ni
for i ≥ 2, and from equation (1) it follows that
t(2i) =
(−1)i−1 B2i (22i−1 − 1)π 2i
.
2(2i)!
Michael E. Hoffman
Combinatorics & MZVs
.
Multiple t-Values cont’d
Combinatorics
& MZVs
Michael E.
Hoffman
Outline
Multiple Zeta
Values
The QuasiSymmetric
Functions
ζ : QSym0 →
R and its
Extension ζu
Other
Homomorphic
Images of
Quasisymmetric
Functions
1
Then sending xj → 2j−1
induces a homomorphism
θ : QSym0 → R with θ(M(i1 ,...,ik ) ) = t(i1 , . . . , ik ) By essentially
the the same argument as for MZVs, we have that t(k, . . . , k )
| {z }
n
is a rational multiple of π kn , but the multiple is different than
for MZVs, e.g.,
π 2n
π 4n
t(2, . . . , 2) = 2n
, t(4, . . . , 4) = 2n
,
| {z }
| {z }
2 (2n)!
2 (4n)!
n
n
π 6n
t(6, . . . , 6) =
.
| {z }
8n(6n − 1)!
n
Factoring ζγ
Michael E. Hoffman
Combinatorics & MZVs
Multiple t-Values cont’d
Combinatorics
& MZVs
Michael E.
Hoffman
Outline
Multiple Zeta
Values
The QuasiSymmetric
Functions
ζ : QSym0 →
R and its
Extension ζu
Other
Homomorphic
Images of
Quasisymmetric
Functions
We can also extend θ to a map from QSym to R by choosing a
value for θ(M(1) ): the most natural choice seems to be log 2.
Then for the generating function H(t) we have
1
1−t
.
θlog 2 (H(t)) = √
Γ
2
πe γt
Despite the similarities with MZVs, multiple t-values have a
flavor all their own. For example, instead of ζ(2, 1) = ζ(3) we
have
1
t(2, 1) = − t(3) + t(2) log 2.
2
Factoring ζγ
Michael E. Hoffman
Combinatorics & MZVs
Finite Multiple Harmonic Sums
Combinatorics
& MZVs
Michael E.
Hoffman
Outline
Multiple Zeta
Values
The QuasiSymmetric
Functions
ζ : QSym0 →
R and its
Extension ζu
Other
Homomorphic
Images of
Quasisymmetric
Functions
Factoring ζγ
Another quantity one can consider are the finite truncations of
MZV series:
X
1
ζ≤n (i1 , . . . , ik ) =
ik
i1
n≥n1 >···>nk ≥1 n1 · · · nk
These are images of monomial quasi-symmetric functions via
the mapping
(
1
, j ≤ n,
xj → j
0, otherwise.
There are no convergence issues here, so the homomorphism is
immediately defined on all of QSym.
Michael E. Hoffman
Combinatorics & MZVs
Hopf Algebra Characters
Combinatorics
& MZVs
Michael E.
Hoffman
Now QSym is a graded Hopf algebra, with coproduct
X
∆(MI ) =
MJ ⊗ MK
I =JK
Outline
Multiple Zeta
Values
The QuasiSymmetric
Functions
ζ : QSym0 →
R and its
Extension ζu
Other
Homomorphic
Images of
Quasisymmetric
Functions
Factoring ζγ
where the sum is over all decompositions of I as a juxtaposition
(including the cases J = ∅ and K = ∅). Also, ζγ is a real
character of the Hopf algebra QSym (that is, an algebra
homomorphism QSym → R). A character χ of QSym is even if
χ(w ) = (−1)|w | χ(w )
(3)
for homogeneous elements w of QSym, and odd if
χ(w ) = (−1)|w | χ(S(w )),
where S is the antipode of QSym.
Michael E. Hoffman
Combinatorics & MZVs
(4)
The ABS Theorem
Combinatorics
& MZVs
Michael E.
Hoffman
Outline
Multiple Zeta
Values
The QuasiSymmetric
Functions
ζ : QSym0 →
R and its
Extension ζu
Other
Homomorphic
Images of
Quasisymmetric
Functions
Factoring ζγ
Theorem (Aguiar, Bergeron and Sottille, 2006)
Any real character of a (graded connected) Hopf algebra has a
unique decomposition in the convolution algebra into an even
character times an odd one.
Thus, there is an even character ζ+ and an odd character ζ− so
that ζγ = ζ+ ζ− in the convolution algebra, i.e.,
X
ζγ (w ) =
ζ+ (w(1) )ζ− (w(2) )
(5)
(w )
using Sweedler’s notation for coproducts:
X
∆(w ) =
w(1) ⊗ w(2) .
(w )
Michael E. Hoffman
Combinatorics & MZVs
Primitives
Combinatorics
& MZVs
The power sums pi = M(i) are primitive, i.e.
Michael E.
Hoffman
Outline
Multiple Zeta
Values
The QuasiSymmetric
Functions
ζ : QSym0 →
R and its
Extension ζu
Other
Homomorphic
Images of
Quasisymmetric
Functions
Factoring ζγ
∆(pi ) = 1 ⊗ pi + pi ⊗ 1,
and so
ζ(i) = ζ(pi ) = ζ− (pi ) + ζ+ (pi ).
(6)
Equation (3) implies that ζ+ (w ) = 0 for any odd-dimensional
w , particularly w = pi , i odd. On the other hand, equation (4)
gives ζ− (pi ) = 0 for i even, since S(pi ) = −pi . Together with
equation (6), this says

(

i even,
0,
ζ(i), i even,
ζ+ (pi ) =
and ζ− (pi ) = ζ(i), i > 1 odd,

0,
i odd,

γ,
i = 1.
Michael E. Hoffman
Combinatorics & MZVs
Even Parts Theorem
Combinatorics
& MZVs
Michael E.
Hoffman
Outline
In general, computing ζ+ (MI ) and ζ− (MI ) for an arbitrary
monomial quasi-symmetric function MI is difficult, apart from
the general fact that
Multiple Zeta
Values
The QuasiSymmetric
Functions
ζ : QSym0 →
R and its
Extension ζu
Other
Homomorphic
Images of
Quasisymmetric
Functions
ζ+ (MI ) = 0
if |I | is odd.
Nevertheless, there is the following result, which can be proved
from the general theory of Aguiar-Bergeron-Sottille together
with a specific result of Aguiar-Hsiao (2004).
Theorem
If all parts of I are even, then ζ− (MI ) = 0.
Factoring ζγ
Michael E. Hoffman
Combinatorics & MZVs
Computing ζ+ and ζ− on Sym
Combinatorics
& MZVs
Michael E.
Hoffman
The situation is dramatically different if ζγ is restricted to Sym
(which is all we need for the Γ- and Γ̂-genera). The pi generate
Sym, so for every partition λ
Outline
mλ = Pλ (p1 , p2 , p3 , p4 , . . . )
Multiple Zeta
Values
The QuasiSymmetric
Functions
for some polynomial Pλ with rational coefficients. Since ζ+ and
ζ− are homomorphisms,
ζ : QSym0 →
R and its
Extension ζu
Other
Homomorphic
Images of
Quasisymmetric
Functions
Factoring ζγ
ζ− (mλ ) = Pλ (γ, 0, ζ(3), 0, . . . )
and
ζ+ (mλ ) = Pλ (0, ζ(2), 0, ζ(4), . . . ).
In view of Euler’s identity (1), the latter formula means that
ζ+ (mλ ) is a rational multiple of π |λ| when |λ| is even (and of
course ζ+ (mλ ) = 0 when |λ| is odd).
Michael E. Hoffman
Combinatorics & MZVs
Generating Functions Yet Again
Combinatorics
& MZVs
Michael E.
Hoffman
Outline
Multiple Zeta
Values
The QuasiSymmetric
Functions
ζ : QSym0 →
R and its
Extension ζu
Other
Homomorphic
Images of
Quasisymmetric
Functions
Factoring ζγ
Since the involution of Sym that exchanges the ei and the hi
leaves the odd pi fixed, it follows that ζ− (en ) = ζ− (hn ), i.e.,
H(t)
ζ−
= 1.
E (t)
On the other hand, ζ+ (E (t)) is an even function, so
ζ+ (E (t)) = ζ+ (E (−t)) = ζ+ (H(t))−1 and thus
H(t)
ζ+
= ζ+ (H(t))2 .
E (t)
Hence, using ζ+ ζ− = ζγ and the Generating Function Theorem,
H(t)
H(t)
H(t)
ζ+ (H(t))2 = ζ+
ζ−
= ζγ
=
E (t)
E (t)
E (t)
Γ(1 − t)Γ(1 + t).
Michael E. Hoffman
Combinatorics & MZVs
Factored Generating Function Theorem
Combinatorics
& MZVs
Michael E.
Hoffman
Outline
Multiple Zeta
Values
The QuasiSymmetric
Functions
ζ : QSym0 →
R and its
Extension ζu
Other
Homomorphic
Images of
Quasisymmetric
Functions
Because of the reflection formula for the gamma function, this
is
πt
.
ζ+ (H(t))2 =
sin πt
Thus we have the following result.
Theorem (Factored Generating Function)
The ABS factors of ζγ are given by
r
−1
ζ+ (E (t)) = ζ+ (H(t))
ζ− (E (t)) = ζ− (H(t)) =
Factoring ζγ
Michael E. Hoffman
=
r
sin πt
πt
sin πt
Γ(1 − t).
πt
Combinatorics & MZVs
Explicit Formula for ζ− (en )
Combinatorics
& MZVs
There is also the explicit formula for en in terms of power sums:
Michael E.
Hoffman
en =
Outline
Multiple Zeta
Values
The QuasiSymmetric
Functions
0
ζ : QSym →
R and its
Extension ζu
Other
Homomorphic
Images of
Quasisymmetric
Functions
Factoring ζγ
X
i1 +2i2 +···+nin =n
p in
(−1)n
n
(−p1 )i1 · · · −
,
i1 !i2 ! · · · in !
n
which follows from
E (t) = H(−t)
−1
Z
= exp −
−t
P(s)ds .
0
If we apply ζ− to this, the minus signs cancel nicely to give
X
ζ− (en ) =
i1 +3i3 +5i5 ···=n
Michael E. Hoffman
γ i1 ζ(3)i3 ζ(5)i5 · · ·
.
i1 !3i3 i3 !5i5 i5 ! · · ·
Combinatorics & MZVs
(7)
An Example
Combinatorics
& MZVs
Michael E.
Hoffman
Outline
Multiple Zeta
Values
The QuasiSymmetric
Functions
ζ : QSym0 →
R and its
Extension ζu
Other
Homomorphic
Images of
Quasisymmetric
Functions
Factoring ζγ
To see how to put these results together, we note that
r
sin πt
π2t 2
π4t 4
=1−
+
− ···
πt
12
1440
Then since the ei are divided powers,
∆(e4 ) = 1 ⊗ e4 + e1 ⊗ e3 + e2 ⊗ e2 + e3 ⊗ e1 + e4 ⊗ 1
and from the Factored Generating Function Theorem together
with equations (5) and (7) it follows that
ζγ (e4 ) = ζ− (e4 ) + ζ+ (e2 )ζ− (e2 ) + ζ+ (e4 )
π4
γ 4 γζ(3) π 2 γ 2
+
−
·
+
4!
3
12 2
1440
γ 4 γ 2 ζ(2) γζ(3) ζ(4)
=
−
+
+
.
24
4
3
16
=
Michael E. Hoffman
Combinatorics & MZVs
Factoring θlog 2
Combinatorics
& MZVs
Michael E.
Hoffman
Outline
Multiple Zeta
Values
The QuasiSymmetric
Functions
0
ζ : QSym →
R and its
Extension ζu
Other
Homomorphic
Images of
Quasisymmetric
Functions
Recall that θlog 2 : QSym → R is the homomorphism that sends
M(1) to log 2 and M(i1 ,...,ik ) to the multiple t-value t(ik , . . . , i1 )
when i1 > 1. Just as ζγ can be factored, so can θlog 2 . In this
case the result is as follows.
Theorem
The ABS factorization θlog 2 = θ+ θ− satisfies
r
πt
θ+ (H(t)) = sec
= θ+ (E (t))−1 .
2
Factoring ζγ
Michael E. Hoffman
Combinatorics & MZVs