Sequential
Zeta Values
ME Hoffman
Outline
Sequential Zeta Values
Introduction
Proof of the
Sum Theorem
for H-series
Michael E. Hoffman
Iterated
Integrals
Sequential
Zeta Values
Sum
Conjecture for
SZVs
Product
Structure
U. S. Naval Academy
Number Theory Talk
Max-Planck-Institut für Mathematik, Bonn
17 June 2015
ME Hoffman
Sequential Zeta Values
Outline
Sequential
Zeta Values
ME Hoffman
1 Introduction
Outline
Introduction
Proof of the
Sum Theorem
for H-series
2 Proof of the Sum Theorem for H-series
3 Iterated Integrals
Iterated
Integrals
Sequential
Zeta Values
Sum
Conjecture for
SZVs
Product
Structure
4 Sequential Zeta Values
5 Sum Conjecture for SZVs
6 Product Structure
ME Hoffman
Sequential Zeta Values
Introduction
Sequential
Zeta Values
ME Hoffman
For positive integers a1 , . . . , ak with a1 > 1 we define the
corresponding multiple zeta value (MZV) by
Outline
Introduction
Proof of the
Sum Theorem
for H-series
Iterated
Integrals
Sequential
Zeta Values
Sum
Conjecture for
SZVs
Product
Structure
ζ(a1 , a2 , . . . , ak ) =
X
1
.
· · · nkak
n a1 n a2
n1 >n2 >···>nk ≥1 1 2
(1)
One calls k the depth and a1 + · · · + ak the weight. Euler
already studied the cases of depth 1 and depth 2, but arguably
the present era of MZVs of general depth began with the proof
of the “sum theorem”
X
ζ(a1 , . . . , ak ) = ζ(n).
(2)
a1 +···+ak =n, a1 >1, ai ≥1
This was proved by Euler for depth 2, by C. Moen for depth 3,
and by A. Granville and D. Zagier for general depth.
ME Hoffman
Sequential Zeta Values
Introduction cont’d
Sequential
Zeta Values
ME Hoffman
Outline
Recently Moen and I proved a rather different sum theorem.
Define, for nonnegative integers a1 , . . . , ak with
a1 + · · · + ak ≥ 2, the series
Introduction
Proof of the
Sum Theorem
for H-series
Iterated
Integrals
Sequential
Zeta Values
Sum
Conjecture for
SZVs
Product
Structure
ζ(a1 |a2 | · · · |ak ) =
∞
X
n=1
(n + k −
1)a1 (n
1
. (3)
+ k − 2)a2 · · · nak
(In our paper these are called “H-series”). Note that this is a
single sum, in contrast to the k-fold sum (1). Then our result
(Integers, 2014) is
X
ζ(a1 |a2 | · · · |ak ) = kζ(n)
(4)
a1 +···+ak =n,ai ≥0
for n ≥ 2.
ME Hoffman
Sequential Zeta Values
Introduction cont’d
Sequential
Zeta Values
ME Hoffman
Outline
Introduction
Proof of the
Sum Theorem
for H-series
Iterated
Integrals
Sequential
Zeta Values
Sum
Conjecture for
SZVs
Product
Structure
The H-series (3), unlike MZVs, don’t have the property that
the product of two such series is a finite sum of series of the
same kind. After some experimentation I arrived at a definition
of “sequential zeta values” (SZVs) which has the property that
the product of two SZVs is a finite sum of SZVs. Further, there
is a plausible “sum conjecture” for SZVs that includes the
theorems (2) and (4) as special cases. This gives one hope that
SVZs are interesting mathematical objects. But so far we don’t
have plausible number-theoretic or geometric interpretations for
SZVs (though most can be expressed as iterated integrals).
ME Hoffman
Sequential Zeta Values
Lemmas about H-series
Sequential
Zeta Values
ME Hoffman
Outline
Introduction
Proof of the
Sum Theorem
for H-series
Iterated
Integrals
Sequential
Zeta Values
Sum
Conjecture for
SZVs
Product
Structure
We begin with a look at the proof of the sum theorem (4) as
given in our paper; it relies on a chain of lemmas about the
H-series. The first, which is entirely trivial, is that
m−1
X
ζ(a1 | · · · |ai−1 |k|ai+1 | · · · |aj−1 |m − k|aj+1 | · · · |an ) =
k=1
1
[ζ(a1 | · · · |ai−1 |0|ai+1 | · · · |aj−1 |m − 1|aj+1 | · · · |an )
j −i
− ζ(a1 | · · · |ai−1 |m − 1|ai+1 | · · · |aj−1 |0|aj+1 | · · · |an )]
for 1 ≤ i < j ≤ n.
ME Hoffman
Sequential Zeta Values
Lemmas about H-series cont’d
Sequential
Zeta Values
ME Hoffman
Outline
Introduction
Proof of the
Sum Theorem
for H-series
Iterated
Integrals
Sequential
Zeta Values
Sum
Conjecture for
SZVs
Product
Structure
From this it follows that
X
ζ(a1 |a2 | · · · |an ) =
ai0 +···+aik =m
(m−k)
(−1)j−1 Hi0 ,ij−1
k
X
j=1
(ij − i0 ) · · · (ij − ij−1 )(ij+1 − ij ) · · · (ik − ij )
for any fixed sequence 1 ≤ i0 < i1 < · · · < ik ≤ n, where
(m)
Hp,q
q
X
1
=
.
jm
j=p
ME Hoffman
Sequential Zeta Values
(5)
Lemmas about H-series cont’d
Sequential
Zeta Values
ME Hoffman
Outline
Introduction
From the formula (5) it follows that the sum C (k, n; m) of all
ζ(a1 |a2 | · · · |an ) with with exactly k + 1 of the ai nonzero and
a1 + · · · + an = m can be written in the form
Proof of the
Sum Theorem
for H-series
n−1 (n)
X
ck,j
Iterated
Integrals
j=1
Sequential
Zeta Values
Sum
Conjecture for
SZVs
Product
Structure
j m−k
(n)
for ck,j ∈ Q. It is an easy observation that the rational
(n)
numbers ck,j have the symmetry/antisymmetry property
(n)
(n)
ck,n−j = (−1)k−1 ck,j .
ME Hoffman
Sequential Zeta Values
Lemmas about H-series cont’d
Sequential
Zeta Values
ME Hoffman
Outline
Introduction
Proof of the
Sum Theorem
for H-series
Iterated
Integrals
Sequential
Zeta Values
Sum
Conjecture for
SZVs
Product
Structure
It is also easy to show from equation (5) and the
symmetry/antisymmetry property that
n
1
(n)
k−1 (n)
,
ck,1 = (−1) ck,n−1 =
(n − 1)! k + 1
where kn is the Stirling number of the first kind, i.e., the
number of permutations of {1, 2, . . . , n} with exactly k disjoint
cycles. The last (rather tricky) lemma is that
(n)
ck,j
=
j X
q
X
(−1)p−1
q=1 p=1
qn+1−q p
k+2−p
(q − 1)!(n − q)!
for 1 ≤ k, j ≤ n − 1.
ME Hoffman
Sequential Zeta Values
Proof of the Sum Theorem for H-series
Sequential
Zeta Values
Now
ME Hoffman
X
ζ(a1 |a2 | · · · |an ) =
Outline
Introduction
Proof of the
Sum Theorem
for H-series
Iterated
Integrals
Sequential
Zeta Values
Sum
Conjecture for
SZVs
Product
Structure
a1 +···+an =m, ai ≥0
n−1
X
C (k, n; m);
k=0
recall C (k, n; m) is the sum of those terms with nonzero entries
in exactly k + 1 positions. We have
C (0, n; m) = ζ(0| · · · |0|m)+ζ(0| · · · |m|0)+· · ·+ζ(m|0| · · · |0)
1
1
= ζ(m) + ζ(m) − 1 + · · · + ζ(m) − 1 − m − · · · −
2
(n − 1)m
n−1
X
n−j
= nζ(m) −
jm
j=1
ME Hoffman
Sequential Zeta Values
Proof of the Sum Theorem for H-series cont’d
Sequential
Zeta Values
ME Hoffman
Outline
Introduction
Proof of the
Sum Theorem
for H-series
That the remaining terms C (k, n; m) cancel the negative terms
above, i.e., that
n−1
X
C (k, n; m) =
k=1 j=1
k=1
Iterated
Integrals
Sequential
Zeta Values
Sum
Conjecture for
SZVs
Product
Structure
n−1 X
n−1 (n)
X
ck,j
j m−k
=
(n)
n−1
X
n−j
j=1
jm
,
follows from the formula expressing ck,j in terms of Stirling
numbers. It follows that
X
ζ(a1 |a2 | · · · |an ) = nζ(m).
a1 +···+an =m, ai ≥0
ME Hoffman
Sequential Zeta Values
Iterated Integral Representation
Sequential
Zeta Values
ME Hoffman
Outline
Introduction
Proof of the
Sum Theorem
for H-series
Iterated
Integrals
Sequential
Zeta Values
Sum
Conjecture for
SZVs
Product
Structure
For the multiple zeta value ζ(a1 , . . . , ak ) we have the
well-known iterated integral representation
Z 1
ζ(a1 , a2 , . . . , ak ) =
ω0a1 −1 ω1 ω0a2 −1 ω1 · · · ω0ak −1 ω1
0
where
dt
dt
, ω1 =
.
t
1−t
Provided all the ai are positive, there is a similar representation
for the H-series ζ(a1 | · · · |ak ), i.e.,
ω0 =
Z
ζ(a1 |a2 | · · · |ak ) =
0
ME Hoffman
1
ω0a1 −1 dtω0a2 −1 dt · · · ω0ak −1 ω1 .
Sequential Zeta Values
Iterated Integrals cont’d
Sequential
Zeta Values
ME Hoffman
Outline
Introduction
Proof of the
Sum Theorem
for H-series
Iterated
Integrals
Sequential
Zeta Values
Sum
Conjecture for
SZVs
Product
Structure
We can also represent H-series in the which the sequence has
zeroes in positions other than the first or last as iterated
integrals, e.g.,
Z 1
ζ(2|0|1) =
ω0 tdtω1
0
Z 1
ζ(1|0|0|1|1) =
t 2 dtdtω1
0
Z 1
ζ(1|0|2|0|0|2) =
tdtω0 t 2 dtω0 ω1 .
0
(Initial zeroes don’t affect the value of the series, but trailing
zeroes do.)
ME Hoffman
Sequential Zeta Values
Definition of SZVs
Sequential
Zeta Values
ME Hoffman
Outline
Introduction
Proof of the
Sum Theorem
for H-series
Iterated
Integrals
Sequential
Zeta Values
Sum
Conjecture for
SZVs
Product
Structure
The H-series do not form a ring. But there is a larger class of
series that does form a ring, and includes both MZVs and
H-series as special cases. We call these sequential zeta values.
Let J1 , J2 , . . . , Jk be sequences of nonnegative integers, say
Ji = (ai,1 |ai,2 | · · · |ai,li )
and set `(Ji ) = li , |Ji | = ai,1 + · · · + ai,li . Then the sequential
zeta value ζ(J1 , J2 , . . . , Jk ) is
1
X
n1 >l1 n2 >l2 ···>lk−1 nk >lk 0
Qk
i=1
Qai,j
j=1 (n
− j + 1)ai,j
P
where a >k b means a − b ≥ k. We call |J| = ki=1 |Ji | the
weight of the SZV, and (l1 , l2 , . . . , lk ) its shape.
ME Hoffman
Sequential Zeta Values
Convergence of SZVs
Sequential
Zeta Values
It is immediate from the definition that
ME Hoffman
Outline
Introduction
Proof of the
Sum Theorem
for H-series
Iterated
Integrals
Sequential
Zeta Values
Sum
Conjecture for
SZVs
Product
Structure
ζ(J1 , J2 . . . , Jk ) ≤ ζ(|J1 |, |J2 |, . . . , |Jk |).
Hence ζ(J1 , J2 , . . . , Jk ) converges provided |J1 | > 1 and
|Ji | ≥ 1 for i ≥ 2. In fact, the following is true.
Proposition
The SZV ζ(J1 , J2 , . . . , Jk ) converges provided |J1 | > 1, |Ji | ≥ 1
for 1 < i < k, and |J1 | + |J2 | + · · · + |Jk | > k.
The only case requiring examination is if |Jk | = 0. In this case
ζ(J1 , . . . , Jk ) can be written
X
nk−1 − lk
,
Qk−1 ai,1
ai,2 · · · (n − l + 1)ai ,li
n
(n
−
1)
i
i
i
i=1 i
n1 >l ···nk−1 >l
>lk
1
k−1
ME Hoffman
Sequential Zeta Values
Convergence of SZVs cont’d
Sequential
Zeta Values
which is
ME Hoffman
Outline
ζ(J1 , . . . , ak−1,1 | · · · |ak−1,lk−1 − 1) − ζ(J1 , . . . , Jk−1 ).
Introduction
Proof of the
Sum Theorem
for H-series
Iterated
Integrals
Sequential
Zeta Values
Sum
Conjecture for
SZVs
Product
Structure
This is a sum of convergent SZVs if |Jk−1 | > 1. Otherwise the
first term has last sequence 0, and we can iterate. The only
case that would produce a non-convergent SZV would be
|J1 | = 2, |J2 | = · · · = |Jk−1 | = 1, and |Jk | = 0, but this is
excluded by the hypothesis.
We note that a leading 0 in J1 can simply be omitted without
affecting the value of the series; so henceforth we assume that
J1 starts with a nonzero integer. If also |J1 | > 1, Ji has no
trailing 0 for i < k, |Ji | ≥ 1 for 1 < i < k, and
|J1 | + · · · + |Jk | > k, we call (J1 , . . . , Jk ) admissible.
ME Hoffman
Sequential Zeta Values
Properties of SZVs
Sequential
Zeta Values
Introduction
SZVs have some properties from partial fractions. Since
1
1
1
1
=
−
(n − p)(n − q)
q−p n−p n−q
Proof of the
Sum Theorem
for H-series
it follows that
ME Hoffman
Outline
Iterated
Integrals
Sequential
Zeta Values
Sum
Conjecture for
SZVs
Product
Structure
ζ(J1 , . . . , ai,1 | · · · |ai,q | · · · |ai,p | · · · |ai,li , . . . , Jk ) =
1
ζ(J1 , . . . , ai,1 | · · · |ai,p − 1| · · · |ai,q | · · · |ai,li , . . . , Jk )
q−p
1
−
ζ(J1 , . . . , ai,1 | · · · |ai,p | · · · |ai,q − 1| · · · |ai,li , . . . , Jk )
q−p
whenever ai,p , ai,q > 0.
ME Hoffman
Sequential Zeta Values
Properties of SZVs cont’d
Sequential
Zeta Values
ME Hoffman
Recall that we assume J1 has no leading 0. A trailing 0 in Jk is
rather complicated to describe in general; one has, e.g.,
Outline
Introduction
Proof of the
Sum Theorem
for H-series
Iterated
Integrals
Sequential
Zeta Values
Sum
Conjecture for
SZVs
Product
Structure
1
X
ζ(2, 1, 1|0) =
i>j>k≥2
i 2 jk
=
X
i>j>k≥1
1
i 2 jk
−
X 1
=
i 2j
i>j≥2
X 1
X 1
ζ(2, 1, 1) −
+
= ζ(2, 1, 1) − ζ(2, 1) + ζ(2) − 1.
i 2j
i2
i>j≥1
i≥2
All other leading and trailing zeroes can be disposed of via
ζ(J1 , . . . , Ji |0, Ji+1 , . . . , Jk ) = ζ(J1 , . . . , Ji , 0|Ji+1 , . . . , Jk ) =
ζ(J1 , . . . , Ji , Ji+1 , . . . , Jk ) − ζ(J1 , . . . , Ji |Ji+1 , . . . , Jk ).
ME Hoffman
Sequential Zeta Values
Strings of 1’s
Sequential
Zeta Values
ME Hoffman
Outline
Introduction
Proof of the
Sum Theorem
for H-series
Iterated
Integrals
Sequential
Zeta Values
Sum
Conjecture for
SZVs
Product
Structure
By the preceding results we have for p > 1
1
ζ(1| · · · |1 |J2 , J3 , . . . , Jk ).
ζ(1| · · · |1, J2 , . . . , Jk ) =
| {z }
p − 1 | {z }
p
p−1
Using this one can easily give a formula for ζ(J1 , . . . , Jk ) when
each Ji is a string of 1’s:
1
(|J1 | − 1)(|J1 | + |J2 | − 2) · · · (|J1 | + · · · + |Jk | − k)
1
×
.
(|J1 | + · · · + |Jk | − k)!
ME Hoffman
Sequential Zeta Values
Other Strings
Sequential
Zeta Values
ME Hoffman
Outline
Introduction
Proof of the
Sum Theorem
for H-series
A similar result is
ζ(1| 0| · · · |0 |1, J2 , . . . , Jk ) =
| {z }
p
p
1 X
ζ(1| 0| · · · |0 |J2 , J3 , . . . , Jk ).
| {z }
p+1
Iterated
Integrals
j=0
Sequential
Zeta Values
Sum
Conjecture for
SZVs
Product
Structure
j
For example,
ζ(1|0|0|1, 2, 1) =
1
[ζ(1|2, 1) + ζ(1|0|2, 1) + ζ(1|0|0|2, 1)] .
3
ME Hoffman
Sequential Zeta Values
Iterated Integrals Again
Sequential
Zeta Values
ME Hoffman
Outline
Introduction
Proof of the
Sum Theorem
for H-series
Iterated
Integrals
Sequential
Zeta Values
Sum
Conjecture for
SZVs
Provided that the last entry of Jk is nonzero, the SZV
ζ(J1 , . . . , Jk ) can be represented as an iterated integral. For
example,
Z 1
X
1
ζ(3|1, 2) =
=
ω02 dtω1 ω0 ω1
(i + j + 1)3 (i + j)j 2
0
i,j≥1
Z 1
X
1
=
ω0 ω1 tdtω1
ζ(2, 1|0|1) =
(i + j + 2)2 (i + 2)i
0
i,j≥1
Product
Structure
ME Hoffman
Sequential Zeta Values
Sum Conjecture
Sequential
Zeta Values
ME Hoffman
Outline
Introduction
Proof of the
Sum Theorem
for H-series
Iterated
Integrals
Sequential
Zeta Values
Sum
Conjecture for
SZVs
Product
Structure
An MZV is a sequential zeta value of shape (1, 1, . . . , 1), while
an H-series is a sequential zeta value of shape (k). It turns out
that the sum theorems (2) and (4) can be put into a common
form for SZVs. Recall that a sequence (J1 , . . . , Jk ) is
admissible if J1 does not start with 0, |J1 | > 1, Ji has no
trailing zero if i < k, |Ji | ≥ 1 for 1 < i < k, and
|J1 | + |J2 | + · · · + |Jk | > k. For a given sequence S, let Am (S)
be the set of admissible sequences of shape S and weight m;
the length `(S) of a shape S = (l1 , . . . , lk ) is k.
Conjecture
For m > `(S),
X
ζ(p) = ζ(m − `(S) + 1).
p∈Am (S)
ME Hoffman
Sequential Zeta Values
Sum Conjecture cont’d
Sequential
Zeta Values
ME Hoffman
Outline
Introduction
Proof of the
Sum Theorem
for H-series
Iterated
Integrals
Sequential
Zeta Values
Sum
Conjecture for
SZVs
Product
Structure
It is simple to put the sum theorem (4) for H-series into this
form: in the identity
X
ζ(a1 | · · · |ak ) = kζ(n),
a1 +···+ak =n, ai ≥0
the inadmissible terms on the right-hand side are those with
a1 = 0. These add up to
X
ζ(0|a2 | · · · |ak ) = (k − 1)ζ(n)
a2 +···+ak =n, ai ≥0
since ζ(0|a2 | · · · |ak ) = ζ(a2 | · · · |ak ); hence
X
ζ(a1 | . . . |ak ) = ζ(n).
a1 +···+ak =n, a1 ≥1, ai ≥0
ME Hoffman
Sequential Zeta Values
Sum Conjecture cont’d
Sequential
Zeta Values
ME Hoffman
To put the MZV sum theorem (2) in this form is a little more
involved; to avoid notational complications we treat the special
case depth 3, weight 5. The MZV sum theorem says
Outline
Introduction
Proof of the
Sum Theorem
for H-series
Iterated
Integrals
Sequential
Zeta Values
Sum
Conjecture for
SZVs
Product
Structure
ζ(3, 1, 1) + ζ(2, 1, 2) + ζ(2, 2, 1) = ζ(5)
(6)
There are additional admissible triple sums
ζ(3, 2, 0) = ζ(3, 1) − ζ(3, 2)
ζ(2, 3, 0) = ζ(2, 2) − ζ(2, 3)
ζ(4, 1, 0) = ζ(3) − ζ(4) − ζ(4, 1).
When these are added to the left-hand side of equation (6), the
result is
ζ(3, 1, 1) + ζ(2, 1, 2) + ζ(2, 2, 1) + ζ(3, 1) + ζ(2, 2)
− ζ(3, 2) − ζ(2, 3) − ζ(4, 1) − ζ(4) + ζ(3) = ζ(5 − 3 + 1).
ME Hoffman
Sequential Zeta Values
Known Cases of Sum Conjecture
Sequential
Zeta Values
ME Hoffman
Outline
Introduction
Proof of the
Sum Theorem
for H-series
Iterated
Integrals
Sequential
Zeta Values
Sum
Conjecture for
SZVs
Product
Structure
So the sum conjecture holds for shape (k) and for shape
(1, . . . , 1). So far we have only been able to prove that the sum
conjecture holds for a few other shapes. These include all
shapes of the form (k, m) for k ≥ 2. Shapes of the form (1, m)
seem to be harder; the conjecture has been proved for shapes
(1, 2) and (1, 3). We give the proof for shape (1, 2).
Proposition
If m ≥ 3, then
X
ζ(p) = ζ(m − 1).
p∈Am (1,2)
ME Hoffman
Sequential Zeta Values
Known Cases of Sum Conjecture cont’d
Sequential
Zeta Values
The left-hand side is
ME Hoffman
Outline
ζ(m, 0|0) +
Introduction
X
i+j=m, i≥2, j≥1
Proof of the
Sum Theorem
for H-series
+
Iterated
Integrals
Sequential
Zeta Values
Sum
Conjecture for
SZVs
Product
Structure
[ζ(i, j|0) + ζ(i, 0|j)]
m−2
X m−1−i
X
i=2
ζ(i, j|m − i − j). (7)
j=1
Now
ζ(m, 0|0) = ζ(m − 1) − 2ζ(m) − 1
ζ(i, j|0) + ζ(i, 0|j) = 2ζ(i, j) − ζ(i|j) − ζ(i) + 1
so that the first term and first sum of (7) give
ME Hoffman
Sequential Zeta Values
Known Cases of Sum Conjecture cont’d
Sequential
Zeta Values
ME Hoffman
Outline
ζ(m − 1) − 2ζ(m) − 1 +
Proof of the
Sum Theorem
for H-series
Sequential
Zeta Values
Sum
Conjecture for
SZVs
Product
Structure
[2ζ(i, m − i) − ζ(i|m − i) − ζ(i) + 1]
i=2
Introduction
Iterated
Integrals
m−1
X
= ζ(m − 1) + m − 2 −
m−2
X
ζ(i) − ζ(1|m − 2)
i=2
using the sum theorem of MZVs. The second sum in (7) gives
m−2
X
[ζ(i) − ζ(i|m − i − 1) + ζ(i) − 1] =
i=2
m−2
X
ζ(i) − ζ(1|m − 3) + ζ(m − 2) − (m − 2)
i=2
and the conclusion follows.
ME Hoffman
Sequential Zeta Values
Product of SZVs
Sequential
Zeta Values
ME Hoffman
Outline
Introduction
Proof of the
Sum Theorem
for H-series
Iterated
Integrals
Sequential
Zeta Values
Sum
Conjecture for
SZVs
Product
Structure
The product of two SZVs is a finite sum of SZVs, but the
product structure (unlike that for MZVs) does not fall into the
framework of quasi-shuffle products. As with the product of
MZVs, all shuffles appear in the product, together with terms
in which sequences combine; a new feature is that a single
sequence in one factor can “paste together” two or more
sequences in the other. Here are some examples.
ζ(2)ζ(2|1) = ζ(2, 2|1) + ζ(2|1, 2) + ζ(4|1) + ζ(2|3)
ζ(2, 1)ζ(2|1) = ζ(2, 1, 2|1) + ζ(2, 2|1, 1) + ζ(2|1, 2, 1)
+ζ(2, 3|1) + ζ(2, 2|2) + ζ(4|1, 1) + ζ(2|3, 1) + ζ(4|2)
ζ(2|1)2 = 2ζ(2|1, 2|1) + 2ζ(2|3|1) + ζ(4|2)
ME Hoffman
Sequential Zeta Values
Product Structure cont’d
Sequential
Zeta Values
Here is another example.
ME Hoffman
Outline
ζ(1|1)ζ(2|1) = ζ(2|1, 1|1) + ζ(1|1, 2|1) + ζ(1|3|1)
+ ζ(2|2|1) + ζ(3|2).
Introduction
Proof of the
Sum Theorem
for H-series
Iterated
Integrals
Since ζ(1|1) = 1, the left-hand side is simply ζ(2|1) = 2 − ζ(2).
That right-hand side equals this follows from earlier results:
Sequential
Zeta Values
ζ(2|1, 1|1) = ζ(1|1|1) + ζ(2|1) − ζ(2) + 1 + ζ(2|1|1)
Sum
Conjecture for
SZVs
ζ(1|1, 2|1) = ζ(2|1) − ζ(1|1|1)
Product
Structure
ζ(1|3|1) = ζ(3|1) − ζ(1|2|1)
ζ(2|2|1) = ζ(1|2|1) − ζ(2|1|1)
ζ(3|2) = ζ(2) − 1 − ζ(2|1) − ζ(3|1)
ME Hoffman
Sequential Zeta Values