Odd Relatives
of MZVs
ME Hoffman
Outline
Introduction
QuasiSymmetric
Functions
MtVs of
Repeated and
Even
Arguments
Iterated
Integrals and
Alternating
MZVs
Calculations
and
Conjectures
Odd Relatives of
Multiple Zeta Values
Michael E. Hoffman
USNA Mathematics Colloquium
6 April 2016
Depth 2
Results
ME Hoffman
Odd Relatives of MZVs
Outline
Odd Relatives
of MZVs
ME Hoffman
1 Introduction
Outline
Introduction
QuasiSymmetric
Functions
MtVs of
Repeated and
Even
Arguments
2 MtVs as Images of Quasi-Symmetric Functions
3 MtVs of Repeated and Even Arguments
4 Iterated Integrals and Alternating MZVs
Iterated
Integrals and
Alternating
MZVs
5 Calculations and Conjectures
Calculations
and
Conjectures
6 Depth 2 Results
Depth 2
Results
ME Hoffman
Odd Relatives of MZVs
Introduction
Odd Relatives
of MZVs
ME Hoffman
For positive integers a1 , . . . , ak with a1 > 1 we define the
corresponding multiple t-value (MtV) by
Outline
Introduction
t(a1 , a2 , . . . , ak ) =
QuasiSymmetric
Functions
MtVs of
Repeated and
Even
Arguments
Iterated
Integrals and
Alternating
MZVs
Calculations
and
Conjectures
Depth 2
Results
X
1
.
· · · nkak
n a1 n a2
n1 >n2 >···>nk ≥1, ni odd 1 2
Except for the specification that the ni are odd, this is just the
usual definition of the multiple zeta value ζ(a1 , . . . , ak ). We
say t(a1 , . . . , ak ) has depth k and weight a1 + · · · + ak . MtVs
of depth 1 are simply related to values of the Riemann zeta
function:
X
X 1
1
=
ζ(a)−
= 1 − 2−a ζ(a). (1)
t(a) =
a
a
n
n
n > 0 even
n > 0 odd
ME Hoffman
Odd Relatives of MZVs
History
Odd Relatives
of MZVs
ME Hoffman
Outline
Introduction
QuasiSymmetric
Functions
MtVs of
Repeated and
Even
Arguments
Iterated
Integrals and
Alternating
MZVs
Calculations
and
Conjectures
While multiple zeta values (for depth ≤ 2) appear in the
correspondence between Euler and Goldbach in the 1740s,
t-values can only claim a history of 110 years. In 1906 N.
Nielsen showed that
n−1
X
t(2i)t(2n − 2i) =
i=1
2n − 1
,
2
which parallels the result Euler gave for zeta values:
n−1
X
ζ(2i)ζ(2n − 2i) =
i=1
2n + 1
.
2
Depth 2
Results
ME Hoffman
Odd Relatives of MZVs
MtVs vs. MZVs
Odd Relatives
of MZVs
ME Hoffman
Outline
It turns out that the theory of multiple t-values is in some ways
directly parallel to that for multiple zeta values, and in other
ways completely different. The MtVs share with the MZVs the
“harmonic algebra” (AKA “stuffle”) multiplication, e.g., just as
Introduction
QuasiSymmetric
Functions
ζ(2)ζ(3, 1) = ζ(2, 3, 1) + ζ(3, 2, 1) + ζ(3, 1, 2) + ζ(5, 1) + ζ(3, 3)
MtVs of
Repeated and
Even
Arguments
we have
Iterated
Integrals and
Alternating
MZVs
Calculations
and
Conjectures
Depth 2
Results
t(2)t(3, 1) = t(2, 3, 1) + t(3, 2, 1) + t(3, 1, 2) + t(5, 1) + t(3, 3).
From this (and equation (1)) follows, e.g.,
π 2n
,
t(2, . . . , 2) =
| {z }
2(2n)!
n
ME Hoffman
π 2n
cf. ζ(2, . . . , 2) =
.
| {z }
(2n + 1)!
n
Odd Relatives of MZVs
MtVs vs. MZVs cont’d
Odd Relatives
of MZVs
Similarly one has
ME Hoffman
Outline
Introduction
QuasiSymmetric
Functions
MtVs of
Repeated and
Even
Arguments
Iterated
Integrals and
Alternating
MZVs
Calculations
and
Conjectures
Depth 2
Results
π 4n
t(4, . . . , 4) = 2n
,
| {z }
2 (4n)!
n
3π 6n
t(6, . . . , 6) =
,
| {z }
4(6n)!
n
which directly parallel
22n+1 π 4n
ζ(4, . . . , 4) =
,
| {z }
(4n + 2)!
n
6π 6n
ζ(6, . . . , 6) =
.
| {z }
(6n + 3)!
n
But in other ways MtVs differ markedly from MZVs. For MZVs
one has the sum theorem: the sum of all MZVs of weight n
and depth k is ζ(n), regardless of k. There seems to be
nothing comparable for MtVs.
ME Hoffman
Odd Relatives of MZVs
MtVs vs. MZVs cont’d
Odd Relatives
of MZVs
ME Hoffman
Outline
Introduction
QuasiSymmetric
Functions
MtVs of
Repeated and
Even
Arguments
Iterated
Integrals and
Alternating
MZVs
Calculations
and
Conjectures
Depth 2
Results
Also absent from the MtVs is an analogue of the MZV duality;
while one has, for example, the relations ζ(2, 1, 1) = ζ(4) and
ζ(3, 1, 2) = ζ(2, 3, 1), nothing like this exists for MtVs. Duality
of MZVs comes from an iterated integral representation: for
example,
Z 1
Z
Z
Z x3
Z
dx2 x2 dx1
dx5 x5 dx4 x4 dx3
ζ(3, 2) =
x4 0 1 − x3 0 x2 0 1 − x1
0
0 x5
and the change of variable
(x1 , x2 , x3 , x4 , x5 ) → (1 − x5 , 1 − x4 , 1 − x3 , 1 − x2 , 1 − x1 )
leads to the duality relation ζ(3, 2) = ζ(2, 2, 1). The iterated
integral representation of MZVs leads to a second algebra
structure (shuffle product).
ME Hoffman
Odd Relatives of MZVs
MtVs vs. MZVs cont’d
Odd Relatives
of MZVs
ME Hoffman
Outline
Introduction
QuasiSymmetric
Functions
MtVs of
Repeated and
Even
Arguments
Iterated
Integrals and
Alternating
MZVs
Calculations
and
Conjectures
Depth 2
Results
For MtVs there is an iterated integral representation, but it’s
not nearly as nice; for example,
Z
Z
Z x3
Z
Z 1
dx2 x2 dx1
dx5 x5 dx4 x4 x3 dx3
.
t(3, 2) =
x4 0 1 − x32 0 x2 0 1 − x12
0
0 x5
The change of variable above doesn’t take an iterated integral
of this form to another, and integrals of this form are not
closed under the shuffle product of forms. Already one sees the
consequences in weight 3: while for MZVs one has the first
instance of duality
ζ(2, 1) = ζ(3)
(discovered by Euler and many others since), for MtVs one has
1
t(2, 1) = − t(3) + t(2) log 2.
2
ME Hoffman
Odd Relatives of MZVs
MtVs and alternating MZVs
Odd Relatives
of MZVs
ME Hoffman
Outline
Nevertheless, the iterated integral represetation for MtVs does
lead to a formula expressing any MtV as a sum of alternating
MZVs, e.g.,
Introduction
QuasiSymmetric
Functions
MtVs of
Repeated and
Even
Arguments
Iterated
Integrals and
Alternating
MZVs
Calculations
and
Conjectures
ζ(3̄, 2) =
(−1)n1
,
3 n2
n
1
2
≥1
X
n1 >n2
ζ(3, 2̄) =
(−1)n2
.
3 n2
n
1
2
≥1
X
n1 >n2
Alternating or “colored” MZVs have been studied, at least by
physicists, since they began to turn up in quantum field theory
in the late 1980s. Thanks to the Multiple Zeta Value Data
Mine project of Blümlein, Broadhurst and Vermaseren,
extensive tables expressing alternating MZVs in terms of a
(provisional) basis exist.
Depth 2
Results
ME Hoffman
Odd Relatives of MZVs
A conjecture
Odd Relatives
of MZVs
ME Hoffman
We have used these expressions to write out the t-values
through weight 7. Thus, e.g.,
Outline
Introduction
QuasiSymmetric
Functions
MtVs of
Repeated and
Even
Arguments
Iterated
Integrals and
Alternating
MZVs
Calculations
and
Conjectures
Depth 2
Results
t(2, 1, 3) =
27
5
1
t(6) − t(3)2 + t(2)ζ(3̄, 1).
112
28
2
Calculations using these tables have led us to the following
conjecture.
Conjecture
The dimension of the rational vector space spanned by MtVs of
weight n ≥ 2 is Fn , the nth Fibonacci number.
The corresponding conjecture for MZVs reads the same, but
with Fn replaced by Pn , the nth Padovan number (i.e., P0 = 1,
P1 = 0, P2 = 1, Pn = Pn−2 + Pn−3 ).
ME Hoffman
Odd Relatives of MZVs
Relative growth
Odd Relatives
of MZVs
ME Hoffman
Outline
Introduction
QuasiSymmetric
Functions
MtVs of
Repeated and
Even
Arguments
Iterated
Integrals and
Alternating
MZVs
Calculations
and
Conjectures
The Padovan number Pn is known to be an upper bound on
the dimension of the rational vector space generated by the
MZVs of weight n. If we denote by MZVn and MtVn ,
respectively, the rational vector spaces generated by weight-n
MZVs and MtVs, we can judge their relative growth by
comparing Pn and Fn .
n
Pn ≥ dimQ MZVn
Fn = dimQ MtVn ?
2
1
1
3
1
2
4
1
3
5
2
5
6
2
8
7
3
13
Evidently there are a lot more relations among MZVs than
among MtVs.
Depth 2
Results
ME Hoffman
Odd Relatives of MZVs
MtVs as multiple Hurwitz zeta functions
Odd Relatives
of MZVs
ME Hoffman
Outline
Introduction
QuasiSymmetric
Functions
MtVs of
Repeated and
Even
Arguments
Iterated
Integrals and
Alternating
MZVs
Calculations
and
Conjectures
Several results about multiple t-values are scattered about the
literature, in a variety of different notations. Sometimes they
are discussed in terms of multiple Hurwitz zeta values, defined
by
ζ(a1 , . . . , ak ; p1 , . . . , pk ) =
X
n1 >···>nk ≥1
(n1 + p1
) a1
1
.
· · · (nk + pk )ak
Evidently
1
1
t(a1 , . . . , ak ) = 2−a1 −···−ak ζ(a1 , . . . , ak ; − , . . . , − ).
2
2
Depth 2
Results
ME Hoffman
Odd Relatives of MZVs
Quasi-symmetric functions
Odd Relatives
of MZVs
ME Hoffman
Outline
Introduction
QuasiSymmetric
Functions
MtVs of
Repeated and
Even
Arguments
Iterated
Integrals and
Alternating
MZVs
Calculations
and
Conjectures
Consider the algebra Q[[x1 , x2 , . . . ]] of formal power series in a
countable set of commuting generators x1 , x2 , . . . . This has a
graded subalgebra B consisting of those series of bounded
degree (where each xi has degree 1). An element of f ∈ B is a
quasi-symmetric function if
coefficient of xia11 xia22 · · · xiakk in f =
coefficient of x1a1 x2a2 · · · xkak in f
whenever i1 < i2 < · · · < ik . The set of such f is a subalgebra
QSym of B, called the quasi-symmetric functions.
Depth 2
Results
ME Hoffman
Odd Relatives of MZVs
Quasi-symmetric functions cont’d
Odd Relatives
of MZVs
ME Hoffman
Outline
Introduction
QuasiSymmetric
Functions
MtVs of
Repeated and
Even
Arguments
Iterated
Integrals and
Alternating
MZVs
Calculations
and
Conjectures
Depth 2
Results
As a vector space, QSym has the basis consisting of monomial
quasi-symmetric functions M(a1 ,...,ak ) , where (a1 , . . . , ak ) is a
composition (ordered sequence) of positive integers and
X a
M(a1 ,...,ak ) =
xi11 · · · xiakk .
i1 <···<ik
QSym has as a subalgebra the symmetric functions Sym; the
symmetric functions have the vector space basis
X
xia11 · · · xiakk
ma1 ,...,ak =
i1 , . . . , ik distinct
indexed by partitions (unordered sequences) of positive
integers. Note that, e.g.,
m2,2,1 = M(2,2,1) + M(2,1,2) + M(1,2,2) .
ME Hoffman
Odd Relatives of MZVs
Quasi-symmetric functions cont’d
Odd Relatives
of MZVs
ME Hoffman
Outline
In fact, QSym is a polynomial algebra on certain of the MI . We
order compositions lexicographically, i.e,
Introduction
QuasiSymmetric
Functions
MtVs of
Repeated and
Even
Arguments
Iterated
Integrals and
Alternating
MZVs
Calculations
and
Conjectures
(1) < (1, 1) < (1, 2) < (1, 2, 1) < · · ·
and call a composition I Lyndon if I < K for any proper
factorization I = JK : e.g., (1, 3) is Lyndon but (1, 1) and
(1, 2, 1) aren’t. Then QSym is the polynomial algebra on the
MI with I Lyndon (Malvenuto and Reutenauer, 1995). Note
that the only Lyndon composition ending in 1 is (1). Let
QSym0 be the subalgebra of QSym generated by MI for
I 6= (1) Lyndon, so that QSym = QSym0 [M(1) ].
Depth 2
Results
ME Hoffman
Odd Relatives of MZVs
Quasi-symmetric functions and MtVs
Odd Relatives
of MZVs
The following result is well-known (Hoffman 1997).
ME Hoffman
Theorem
Outline
There is a homomorphism ζ : QSym0 → R sending 1 to 1 and
M(a1 ,a2 ,...,ak ) to ζ(a1 , a2 , . . . ak ) for a1 ≥ 2.
Introduction
QuasiSymmetric
Functions
MtVs of
Repeated and
Even
Arguments
Iterated
Integrals and
Alternating
MZVs
Calculations
and
Conjectures
Depth 2
Results
There is an exact analogue for MtVs.
Theorem
There is a homomorphism θ : QSym0 → R sending 1 to 1 and
M(a1 ,a2 ,...,ak ) to t(a1 , a2 , . . . , ak ) for a1 ≥ 2.
It follows that if any linear combination of t-values is the image
of a symmetric function, then it is actually expressible as a
rational polynomial in the depth 1 t-values, e.g.,
t(2, 3) + t(3, 2) = t(2)t(3) − t(5).
ME Hoffman
Odd Relatives of MZVs
MtVs of repeated arguments
Odd Relatives
of MZVs
ME Hoffman
Outline
In particular, any t-value of repeated arguments is a rational
polynomial in the t(i). If we write {k}n for k repeated n times,
then we have the following.
Introduction
Theorem
QuasiSymmetric
Functions
For integer k ≥ 2, let Zk (x) be the generating function
MtVs of
Repeated and
Even
Arguments
Iterated
Integrals and
Alternating
MZVs
Calculations
and
Conjectures
Depth 2
Results
Zk (x) = 1 +
∞
X
ζ({k}i )x ik .
i=1
Then the corresponding generating function for multiple
t-values is
∞
X
Zk (x)
1+
t({k}i )x ik =
.
Zk ( x2 )
i=1
ME Hoffman
Odd Relatives of MZVs
MtVs of repeated arguments cont’d
Odd Relatives
of MZVs
ME Hoffman
Outline
Introduction
Now the generating functions Zk (x) were studied in some detail
back in the 1990’s: see Broadhurst, Borwein and Bradley 1997.
From their results and the preceding theorem we have, e.g.,
QuasiSymmetric
Functions
MtVs of
Repeated and
Even
Arguments
Iterated
Integrals and
Alternating
MZVs
Calculations
and
Conjectures
t({4}n ) =
π 4n
π 6n
,
t({6}
)
=
,
n
22n (4n)!
(8n)(6n − 1)!
π 10n (L10n + 1)
t({10}n ) =
,
(32n)(10n − 1)!
where in the last formula Ln is the nth Lucas number, i.e.,
L0 = 2, L1 = 1 and Ln = Ln−1 + Ln−2 .
Depth 2
Results
ME Hoffman
Odd Relatives of MZVs
MtVs of even arguments
Odd Relatives
of MZVs
ME Hoffman
Outline
Introduction
QuasiSymmetric
Functions
MtVs of
Repeated and
Even
Arguments
Iterated
Integrals and
Alternating
MZVs
Calculations
and
Conjectures
Depth 2
Results
Another area where results on MtVs closely parallel those on
MZVs is in sums of elements of purely even arguments. If for
k ≤ n we let T (2n, k) be the sum of all MtVs of even
arguments having depth k and weight 2n, then Zhao 2015
showed that
c
b k−1
2
X
(−1)j π 2j t(2n − 2j) 2k − 2j − 2
,
T (2n, k) =
k −1
22k−2 (2j)!k
j=0
following my result
c
b k−1
2
X (−1)j π 2j ζ(2n − 2j) 2k − 2j − 1
E (2n, k) =
22n−2j−2 (2j + 1)!
k
j=0
for E (2n, k) the sum of all MZVs of even arguments having
depth k and weight 2n.
ME Hoffman
Odd Relatives of MZVs
Generating functions
Odd Relatives
of MZVs
ME Hoffman
Outline
Introduction
QuasiSymmetric
Functions
MtVs of
Repeated and
Even
Arguments
Iterated
Integrals and
Alternating
MZVs
Calculations
and
Conjectures
Depth 2
Results
Both formulas are proved from corresponding generating
functions: we have the parallel results
p
∞ X
n
cos π2 (1 − s)t
X
√ T(t, s) = 1 +
T (2n, k)t n s k =
cos π2 t
n=1 k=1
and
∞ X
n
X
p
sin(π (1 − s)t)
√ .
E (2n, k)t s = √
E(t, s) = 1 +
1 − s sin(π t)
n=1 k=1
n k
These in turn follow from taking images under appropriate
homomorphisms of the generating function
ME Hoffman
Odd Relatives of MZVs
Generating functions cont’d
Odd Relatives
of MZVs
ME Hoffman
Outline
Introduction
QuasiSymmetric
Functions
MtVs of
Repeated and
Even
Arguments
Iterated
Integrals and
Alternating
MZVs
Calculations
and
Conjectures
Depth 2
Results
F(t, s) = 1 +
∞ X
n
X
Nn,k t n s k ∈ QSym[[t, s]],
n=1 k=1
where Nn,k is the sum of all monomial symmetric functions
corresponding of partitions of n with k parts. In fact
E(t, s) = ζP2 F(t, s)
and T(t, s) = θP2 F(t, s)
where P2 : QSym → QSym sends each xi to xi2 . The formulas
on the preceding slide then follow from
F(t, s) = H(t)E ((s − 1)t),
where H(t) and E (t) are respectively the generating functions
for the complete and elementary symmetric functions.
ME Hoffman
Odd Relatives of MZVs
Iterated integral representations
Odd Relatives
of MZVs
ME Hoffman
Outline
Introduction
QuasiSymmetric
Functions
MtVs of
Repeated and
Even
Arguments
Iterated
Integrals and
Alternating
MZVs
Calculations
and
Conjectures
Depth 2
Results
Call a sequence of (a1 , . . . , ak ) of positive integers admissible if
a1 > 1. Now if
ω0 =
dt
dt
dt
, ω1 =
, ω−1 =
,
t
1−t
1+t
then there is the well-known representation of MZVs
Z 1
ζ(a1 , . . . , ak ) =
ω0a1 −1 ω1 · · · ω0ak −1 ω1
0
for any admissible sequence (a1 , . . . , ak ). This can be extended
as follows. Let P be the set of positive integers,
P̃ = {. . . , 3̄, 2̄, 1̄, 1, 2, 3, . . . }. The set {−1, 1} acts on P̃ as
follows: 1 ◦ n = n, 1 ◦ n̄ = n̄, (−1) ◦ n = n̄, (−1) ◦ n̄ = n. For
a ∈ P̃, let sgn(a) = 1 if a ∈ P and −1 otherwise. Note
sgn(a) ◦ a ∈ P.
ME Hoffman
Odd Relatives of MZVs
Alternating MZVs
Odd Relatives
of MZVs
ME Hoffman
For b1 , . . . , bk ∈ P̃ with b1 6= 1, define the alternating MZV by
Outline
Iterated
Integrals and
Alternating
MZVs
Calculations
and
Conjectures
sgn(b1 )n1 · · · sgn(bk )nk
sgn(b1 )◦b1
n1 >···>nk ≥1
QuasiSymmetric
Functions
MtVs of
Repeated and
Even
Arguments
X
ζ(b1 , . . . , bk ) =
Introduction
n1
sgn(bk )◦bk
.
· · · nk
Now let (a1 , . . . , ak ) be admissible. If 1 , . . . , k ∈ {−1, 1},
then
Z
0
1
ω0a1 −1 ω1 · · · ω0ak −1 ωk =
1 · · · k ζ(1 ◦ a1 , 1 2 ◦ a2 , 2 3 ◦ a3 , . . . , k−1 k ◦ ak )
Depth 2
Results
ME Hoffman
Odd Relatives of MZVs
Iterated integrals for MtVs
Odd Relatives
of MZVs
ME Hoffman
For multiple t-values there is also an iterated integral
representation, but it is not so simple.
Outline
Introduction
Proposition
QuasiSymmetric
Functions
For any admissible sequence (a1 , . . . , ak ),
Z
MtVs of
Repeated and
Even
Arguments
Iterated
Integrals and
Alternating
MZVs
Calculations
and
Conjectures
1
t(a1 , . . . , ak ) =
0
a
ω0a1 −1 Ωa · · · ω0k−1
−1
Ωa ω0ak −1 Ωb
where
Ωa =
1
tdt
= (ω1 − ω−1 ),
2
1−t
2
Ωb =
dt
1
= (ω1 + ω−1 ).
2
1−t
2
Depth 2
Results
ME Hoffman
Odd Relatives of MZVs
MtVs and alternating MZVs
Odd Relatives
of MZVs
ME Hoffman
Expanding out the preceding result and using the extended
integral representation of alternating MZVs gives the following.
Outline
Introduction
QuasiSymmetric
Functions
MtVs of
Repeated and
Even
Arguments
Iterated
Integrals and
Alternating
MZVs
Calculations
and
Conjectures
Depth 2
Results
Theorem
For an admissible sequence (a1 , . . . , ak ), the multiple t-value
t(a1 , . . . , ak ) is given by
1 X
1 · · · k ζ(1 ◦ a1 , . . . , k ◦ ak ),
2k ,...,
1
k
where the sum is over the 2k k-tuples (1 , . . . , k ) with
i ∈ {−1, 1}.
This expresses t(a1 , . . . , ak ) as a sum of 2k alternating MZVs.
ME Hoffman
Odd Relatives of MZVs
MtVs and alternating MZVs cont’d
Odd Relatives
of MZVs
ME Hoffman
Outline
But it’s possible to do better. For admissible (a1 , . . . , ak ) and
p ≤ k, let Lp ζ(a1 , . . . , ak ) be the sum of the kp alternating
MZVs in which (−1)◦ is applied to exactly p of the arguments.
Introduction
QuasiSymmetric
Functions
MtVs of
Repeated and
Even
Arguments
Iterated
Integrals and
Alternating
MZVs
Calculations
and
Conjectures
Depth 2
Results
Theorem
For an admissible sequence (a1 , . . . , ak ), the multiple t-value
t(a1 , . . . , ak ) is given by
1
2a1 +···+ak
ζ(a1 , . . . , ak ) −
1
2k−1
X
Lp ζ(a1 , . . . , ak ).
p≤k odd
This expresses t(a1 , . . . , ak ) as a sum of 2k−1 + 1 alternating
MZVs.
ME Hoffman
Odd Relatives of MZVs
MtVs and alternating MZVs cont’d
Odd Relatives
of MZVs
Even for double t-values this is an improvement; it replaces
ME Hoffman
t(a, b) =
Outline
1
ζ(a, b) − ζ(a, b̄) − ζ(ā, b) + ζ(ā, b̄)
4
Introduction
QuasiSymmetric
Functions
MtVs of
Repeated and
Even
Arguments
Iterated
Integrals and
Alternating
MZVs
Calculations
and
Conjectures
Depth 2
Results
with
t(a, b) =
1
2
ζ(a, b) −
a+b
1
ζ(a, b̄) + ζ(ā, b) .
2
For triple t-values we have
t(a, b, c) =
−
1
2a+b+c
ζ(a, b, c)
1
ζ(a, b, c̄) + ζ(a, b̄, c) + ζ(ā, b, c) + ζ(ā, b̄, c̄) .
4
ME Hoffman
Odd Relatives of MZVs
The Multiple Zeta Value Data Mine
Odd Relatives
of MZVs
ME Hoffman
Using the preceding result, we can expand any MtV into a sum
of alternating MZVs. For example,
Outline
Introduction
QuasiSymmetric
Functions
MtVs of
Repeated and
Even
Arguments
Iterated
Integrals and
Alternating
MZVs
Calculations
and
Conjectures
Depth 2
Results
t(2, 3, 1) =
1
1
ζ(2, 3, 1) − ζ(2̄, 3, 1) + ζ(2, 3̄, 1) + ζ(2, 3, 1̄)
64
4
+ ζ(2̄, 3̄, 1̄) .
Now we can use tables from the Multiple Zeta Value Data
Mine project of Blümlein, Broadhurst and Vermaseren to
expand each of the alternating MZVs in this sum, resulting in
t(2, 3, 1) = −
1
2
3
1
t(6) −
t(3)2 − t(2)ζ(3̄, 1) + ζ(5̄, 1)
21
196
2
4
1
4
− t(5) log 2 + t(2)t(3) log 2.
2
7
ME Hoffman
Odd Relatives of MZVs
Tables and conjectures
Odd Relatives
of MZVs
ME Hoffman
Similarly we have obtained formulas for all multiple t-values of
weight ≤ 7. In our tables we have
4
1
t(2, 3) = − t(5) + t(2)t(3).
2
7
Outline
Introduction
QuasiSymmetric
Functions
MtVs of
Repeated and
Even
Arguments
Iterated
Integrals and
Alternating
MZVs
Calculations
and
Conjectures
Depth 2
Results
This and many similar instances have led us to the following
conjecture.
Conjecture
Every multiple t-value admits a representation as a polynomial
in the t(i), i ≥ 2, log 2, and selected alternating MZVs in such
a way that
∂t(s, 1)
= t(s).
∂ log 2
for any admissible sequence s.
ME Hoffman
Odd Relatives of MZVs
Derivation Conjecture
Odd Relatives
of MZVs
ME Hoffman
Outline
Consider the function A− : QSym → QSym given by
(
M(i1 ,...,ik−1 ) , if ik = 1,
A− (M(i1 ,...,ik ) ) =
0,
otherwise.
Introduction
QuasiSymmetric
Functions
MtVs of
Repeated and
Even
Arguments
Iterated
Integrals and
Alternating
MZVs
Calculations
and
Conjectures
Depth 2
Results
Then A− is a derivation of QSym, and it restricts to a
derivation of QSym0 . The conjecture above can be restated as
follows.
Conjecture
There is a derivation d on MtV = θ(QSym0 ) such that
θA− = dθ.
Note that if there were a derivation ∂ on MZV = ζ(QSym0 )
such that ζA− = ∂ζ, then we’d have the contradiction
0 = ζ(A− M3 ) = ∂ζ(3) = ∂ζ(2, 1) = ζ(A− M(2,1) ) = ζ(2).
ME Hoffman
Odd Relatives of MZVs
The Fibonacci Conjecture
Odd Relatives
of MZVs
ME Hoffman
Outline
Introduction
QuasiSymmetric
Functions
MtVs of
Repeated and
Even
Arguments
Iterated
Integrals and
Alternating
MZVs
Calculations
and
Conjectures
The second conjecture, which we have already mentioned, is
based on computations of the rank of the set of multiple
t-values of a given weight. Based on our tables through weight
7, we have computed the ranks (starting in weight 2) as
1, 2, 3, 5, 8, 13
which leads to the following “Fibonacci Conjecture.”
Conjecture
If MtVn is the rational vector subspace of R spanned by
weight-n multiple t-values, then dimQ MtVn = Fn , the nth
Fibonacci number.
Depth 2
Results
ME Hoffman
Odd Relatives of MZVs
Fibonacci Conjecture cont’d
Odd Relatives
of MZVs
ME Hoffman
Outline
Introduction
QuasiSymmetric
Functions
MtVs of
Repeated and
Even
Arguments
Iterated
Integrals and
Alternating
MZVs
Calculations
and
Conjectures
Of course the existing evidence for the Fibonacci Conjecture is
rather thin. The Multiple Zeta Value Data Mine actually has
publically available files on alterating MZVs for weights
through 14, so there is plenty of testing yet to do even without
extending their work. But this gets laborious: in weight 7 there
are 32 admissible sequences, and the sum for t(2, 1, 1, 1, 1, 1)
has 33 terms. Blümlein, Broadhurst and Vermaseren are all
particle physicists, with plenty of experience with large-scale
calculations, so they probably have helpful ideas on how to
manage the higher weights.
Depth 2
Results
ME Hoffman
Odd Relatives of MZVs
Explicit formulas in depth 2
Odd Relatives
of MZVs
ME Hoffman
Outline
Introduction
QuasiSymmetric
Functions
MtVs of
Repeated and
Even
Arguments
Iterated
Integrals and
Alternating
MZVs
Calculations
and
Conjectures
Depth 2
Results
One place where the theories of MZVs and MtVs are closely
parallel is in depth 2. It is convenient to define
2i − 2
2i − 2
N(a, b, i) =
+
,
a−1
b−1
where pj = 0 if j > p. It is well-known (and implicit in Euler’s
1776 paper) that
(−1)a
a+b
ζ(a, b) =
− 1 ζ(2n + 1)+
2
a
(
P
ζ(a)ζ(b) − ni=2 N(a, b, i)ζ(2i − 1)ζ(2n − 2i + 2), a even,
Pn
i=2 N(a, b, i)ζ(2i − 1)ζ(2n − 2i + 2), otherwise.
if a + b = 2n + 1 and a, b ≥ 2 (No general formula in terms of
single zeta values is known for ζ(a, b) if a + b is even.)
ME Hoffman
Odd Relatives of MZVs
Explicit formulas in depth 2 cont’d
Odd Relatives
of MZVs
ME Hoffman
Outline
For ζ(2n, 1) one has another formula
2n−1
1 X
ζ(2n, 1) = nζ(2n + 1) −
ζ(i)ζ(2n + 1 − i).
2
i=2
Introduction
QuasiSymmetric
Functions
MtVs of
Repeated and
Even
Arguments
Iterated
Integrals and
Alternating
MZVs
Calculations
and
Conjectures
Depth 2
Results
Basu 2008 gave the similar formula
1
t(a, b) = − t(2n + 1)+
2
(
P
t(a)t(b) − ni=2 N(a, b, i) t(2i−2)t(2n−2i+2)
, a even,
22i−1 −1
Pn
t(2i−1)t(2n−2i+2)
, otherwise,
i=2 N(a, b, i)
22i−1 −1
for a + b = 2n + 1. Basu actually stipulates that a, b ≥ 2, but
if we take a = 2n, b = 1 and set t(1) = log 2 the formula is
still valid (as follows from Jordan 1973 or Nakamura and
Tasaka 2013).
ME Hoffman
Odd Relatives of MZVs
Explicit formulas in depth 2 cont’d
Odd Relatives
of MZVs
ME Hoffman
Outline
Introduction
QuasiSymmetric
Functions
MtVs of
Repeated and
Even
Arguments
Iterated
Integrals and
Alternating
MZVs
Calculations
and
Conjectures
The proofs used by Basu and Nakamura-Tasaka are elementary
but different. I have been able to prove the a = 2n, b = 1 case,
that is,
1
t(2n, 1) = − t(2n + 1) + t(2n) log 2
2
n
X
t(2i − 1)t(2n − 2i + 2)
−
,
22i−1 − 1
i=2
using yet another elementary method. I believe this can be
extended to the general case a + b = 2n + 1, though I haven’t
yet done this. My method involves modified Tornheim sums.
Depth 2
Results
ME Hoffman
Odd Relatives of MZVs
Tornheim sums
Odd Relatives
of MZVs
ME Hoffman
The depth 2 formulas for the MZV case can all be rather neatly
proved using sums invented in Tornheim 1950:
Outline
Introduction
T (a, b, c) =
QuasiSymmetric
Functions
MtVs of
Repeated and
Even
Arguments
Iterated
Integrals and
Alternating
MZVs
Calculations
and
Conjectures
∞ X
∞
X
i=1 j=1
1
.
+ j)c
i a j b (i
They have the properties T (a, b, c) = T (b, a, c),
T (a, 0, c) = ζ(c, a) for c > 1, T (a, b, 0) = ζ(a)ζ(b) for
a, b ≥ 2, and
T (a, b, c) = T (a − 1, b, c + 1) + T (a, b − 1, c + 1)
for a, b ≥ 1.
Depth 2
Results
ME Hoffman
Odd Relatives of MZVs
Modified Tornheim sums
Odd Relatives
of MZVs
ME Hoffman
For the t-values one can use a modified Tornheim sum
Outline
Introduction
QuasiSymmetric
Functions
MtVs of
Repeated and
Even
Arguments
Iterated
Integrals and
Alternating
MZVs
Calculations
and
Conjectures
M(a, b, c) =
∞ X
∞
X
i=1 j=0
1
(2i)a (2j
+
1)b (2i
+ 2j + 1)c
One has M(0, b, c) = t(c, b) for c > 1,
M(a, b, 0) = 2−a ζ(a)t(b) for a, b ≥ 2 and
M(a, b, c) = M(a − 1, b, c + 1) + M(a, b − 1, c + 1)
for a, b ≥ 1, but the symmetry in the first two arguments is lost
and M(a, 0, c) is not particularly simple. Nevertheless, one can
mimic many of the arguments used for Tornheim sums.
Depth 2
Results
ME Hoffman
Odd Relatives of MZVs
Tornheim and modified Tornheim sums
Odd Relatives
of MZVs
ME Hoffman
Outline
Here is an interesting point of difference between T and M. By
partial fractions one has
Introduction
∞
∞ X
1
1 X 1
−
,
T (1, p, 1) =
j p+1
i
i +j
QuasiSymmetric
Functions
MtVs of
Repeated and
Even
Arguments
Iterated
Integrals and
Alternating
MZVs
Calculations
and
Conjectures
j=1
i=1
and the inner sum telescopes to give
∞
X
1
1
1 + ··· +
= ζ(p + 2) + ζ(p + 1, 1).
j p+1
j
j=1
Depth 2
Results
ME Hoffman
Odd Relatives of MZVs
Tornheim and modified Tornheim sums cont’d
Odd Relatives
of MZVs
ME Hoffman
But in the corresponding sum
Outline
Introduction
M(1, p, 1) =
QuasiSymmetric
Functions
MtVs of
Repeated and
Even
Arguments
Iterated
Integrals and
Alternating
MZVs
Calculations
and
Conjectures
∞
X
j=0
∞ X
1
1
1
−
(2j + 1)p+1
2i
2i + 2j + 1
i=1
the inner sum does not telescope, and instead we have
∞
X
j=0
1
1
1
+
·
·
·
+
−
log
2
+
1
+
(2j + 1)p+1
3
2j + 1
= t(p + 2) + t(p + 1, 1) − t(p + 1) log 2.
Depth 2
Results
ME Hoffman
Odd Relatives of MZVs
A formula for t(2n + 1, 1)
Odd Relatives
of MZVs
ME Hoffman
Outline
Introduction
QuasiSymmetric
Functions
MtVs of
Repeated and
Even
Arguments
Iterated
Integrals and
Alternating
MZVs
Calculations
and
Conjectures
The argument for t(2n, 1) can be modified slightly to give a
formula for t(2n + 1, 1) (but one must pay the price of
including an alternating MZV):
1
t(2n + 1, 1) = t(2n + 1) log 2 − ζ(2n + 1, 1)+
2
(2n + 1)(1 − 22n+1 ) − 22n+2
t(2n + 2)+
4(22n+2 − 1)
n−1 2n+1
X
2
− 22i+1 − 22n−2i+1 + 1
t(2i + 1)t(2n − 2i + 1).
2(22i+1 − 1)(22n−2i+1 − 1)
i=1
Depth 2
Results
ME Hoffman
Odd Relatives of MZVs