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TRANSCRIPT
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SequentialZeta Values
ME Hoffman
Outline
Introduction
Proof of theSum Theoremfor H-series
IteratedIntegrals
SequentialZeta Values
SumConjecture forSZVs
ProductStructure
Sequential Zeta Values
Michael E. Hoffman
U. S. Naval Academy
Number Theory TalkMax-Planck-Institut für Mathematik, Bonn
17 June 2015
ME Hoffman Sequential Zeta Values
-
SequentialZeta Values
ME Hoffman
Outline
Introduction
Proof of theSum Theoremfor H-series
IteratedIntegrals
SequentialZeta Values
SumConjecture forSZVs
ProductStructure
Outline
1 Introduction
2 Proof of the Sum Theorem for H-series
3 Iterated Integrals
4 Sequential Zeta Values
5 Sum Conjecture for SZVs
6 Product Structure
ME Hoffman Sequential Zeta Values
-
SequentialZeta Values
ME Hoffman
Outline
Introduction
Proof of theSum Theoremfor H-series
IteratedIntegrals
SequentialZeta Values
SumConjecture forSZVs
ProductStructure
Introduction
For positive integers a1, . . . , ak with a1 > 1 we define thecorresponding multiple zeta value (MZV) by
ζ(a1, a2, . . . , ak) =∑
n1>n2>···>nk≥1
1
na11 na22 · · · n
akk
. (1)
One calls k the depth and a1 + · · ·+ ak the weight. Euleralready studied the cases of depth 1 and depth 2, but arguablythe present era of MZVs of general depth began with the proofof the “sum theorem”∑
a1+···+ak=n, a1>1, ai≥1ζ(a1, . . . , ak) = ζ(n). (2)
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
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SequentialZeta Values
ME Hoffman
Outline
Introduction
Proof of theSum Theoremfor H-series
IteratedIntegrals
SequentialZeta Values
SumConjecture forSZVs
ProductStructure
Introduction cont’d
Recently Moen and I proved a rather different sum theorem.Define, for nonnegative integers a1, . . . , ak witha1 + · · ·+ ak ≥ 2, the series
ζ(a1|a2| · · · |ak) =∞∑n=1
1
(n + k − 1)a1(n + k − 2)a2 · · · nak. (3)
(In our paper these are called “H-series”). Note that this is asingle sum, in contrast to the k-fold sum (1). Then our result(Integers, 2014) is∑
a1+···+ak=n,ai≥0ζ(a1|a2| · · · |ak) = kζ(n) (4)
for n ≥ 2.
ME Hoffman Sequential Zeta Values
-
SequentialZeta Values
ME Hoffman
Outline
Introduction
Proof of theSum Theoremfor H-series
IteratedIntegrals
SequentialZeta Values
SumConjecture forSZVs
ProductStructure
Introduction cont’d
The H-series (3), unlike MZVs, don’t have the property thatthe product of two such series is a finite sum of series of thesame kind. After some experimentation I arrived at a definitionof “sequential zeta values” (SZVs) which has the property thatthe product of two SZVs is a finite sum of SZVs. Further, thereis a plausible “sum conjecture” for SZVs that includes thetheorems (2) and (4) as special cases. This gives one hope thatSVZs are interesting mathematical objects. But so far we don’thave plausible number-theoretic or geometric interpretations forSZVs (though most can be expressed as iterated integrals).
ME Hoffman Sequential Zeta Values
-
SequentialZeta Values
ME Hoffman
Outline
Introduction
Proof of theSum Theoremfor H-series
IteratedIntegrals
SequentialZeta Values
SumConjecture forSZVs
ProductStructure
Lemmas about H-series
We begin with a look at the proof of the sum theorem (4) asgiven in our paper; it relies on a chain of lemmas about theH-series. The first, which is entirely trivial, is that
m−1∑k=1
ζ(a1| · · · |ai−1|k |ai+1| · · · |aj−1|m − k |aj+1| · · · |an) =
1
j − i[ζ(a1| · · · |ai−1|0|ai+1| · · · |aj−1|m − 1|aj+1| · · · |an)
− ζ(a1| · · · |ai−1|m − 1|ai+1| · · · |aj−1|0|aj+1| · · · |an)]
for 1 ≤ i < j ≤ n.
ME Hoffman Sequential Zeta Values
-
SequentialZeta Values
ME Hoffman
Outline
Introduction
Proof of theSum Theoremfor H-series
IteratedIntegrals
SequentialZeta Values
SumConjecture forSZVs
ProductStructure
Lemmas about H-series cont’d
From this it follows that∑ai0+···+aik=m
ζ(a1|a2| · · · |an) =
k∑j=1
(−1)j−1H(m−k)i0,ij−1(ij − i0) · · · (ij − ij−1)(ij+1 − ij) · · · (ik − ij)
(5)
for any fixed sequence 1 ≤ i0 < i1 < · · · < ik ≤ n, where
H(m)p,q =
q∑j=p
1
jm.
ME Hoffman Sequential Zeta Values
-
SequentialZeta Values
ME Hoffman
Outline
Introduction
Proof of theSum Theoremfor H-series
IteratedIntegrals
SequentialZeta Values
SumConjecture forSZVs
ProductStructure
Lemmas about H-series cont’d
From the formula (5) it follows that the sum C (k, n;m) of allζ(a1|a2| · · · |an) with with exactly k + 1 of the ai nonzero anda1 + · · ·+ an = m can be written in the form
n−1∑j=1
c(n)k,j
jm−k
for c(n)k,j ∈ Q. It is an easy observation that the rational
numbers c(n)k,j have the symmetry/antisymmetry property
c(n)k,n−j = (−1)
k−1c(n)k,j .
ME Hoffman Sequential Zeta Values
-
SequentialZeta Values
ME Hoffman
Outline
Introduction
Proof of theSum Theoremfor H-series
IteratedIntegrals
SequentialZeta Values
SumConjecture forSZVs
ProductStructure
Lemmas about H-series cont’d
It is also easy to show from equation (5) and thesymmetry/antisymmetry property that
c(n)k,1 = (−1)
k−1c(n)k,n−1 =
1
(n − 1)!
[n
k + 1
],
where[nk
]is the Stirling number of the first kind, i.e., the
number of permutations of {1, 2, . . . , n} with exactly k disjointcycles. The last (rather tricky) lemma is that
c(n)k,j =
j∑q=1
q∑p=1
(−1)p−1[qp
][n+1−qk+2−p
](q − 1)!(n − q)!
for 1 ≤ k, j ≤ n − 1.
ME Hoffman Sequential Zeta Values
-
SequentialZeta Values
ME Hoffman
Outline
Introduction
Proof of theSum Theoremfor H-series
IteratedIntegrals
SequentialZeta Values
SumConjecture forSZVs
ProductStructure
Proof of the Sum Theorem for H-series
Now
∑a1+···+an=m, ai≥0
ζ(a1|a2| · · · |an) =n−1∑k=0
C (k, n;m);
recall C (k , n;m) is the sum of those terms with nonzero entriesin exactly k + 1 positions. We have
C (0, n;m) = ζ(0| · · · |0|m)+ζ(0| · · · |m|0)+· · ·+ζ(m|0| · · · |0)
= ζ(m) + ζ(m)− 1 + · · ·+ ζ(m)− 1− 12m− · · · − 1
(n − 1)m
= nζ(m)−n−1∑j=1
n − jjm
ME Hoffman Sequential Zeta Values
-
SequentialZeta Values
ME Hoffman
Outline
Introduction
Proof of theSum Theoremfor H-series
IteratedIntegrals
SequentialZeta Values
SumConjecture forSZVs
ProductStructure
Proof of the Sum Theorem for H-series cont’d
That the remaining terms C (k, n;m) cancel the negative termsabove, i.e., that
n−1∑k=1
C (k , n;m) =n−1∑k=1
n−1∑j=1
c(n)k,j
jm−k=
n−1∑j=1
n − jjm
,
follows from the formula expressing c(n)k,j in terms of Stirling
numbers. It follows that∑a1+···+an=m, ai≥0
ζ(a1|a2| · · · |an) = nζ(m).
ME Hoffman Sequential Zeta Values
-
SequentialZeta Values
ME Hoffman
Outline
Introduction
Proof of theSum Theoremfor H-series
IteratedIntegrals
SequentialZeta Values
SumConjecture forSZVs
ProductStructure
Iterated Integral Representation
For the multiple zeta value ζ(a1, . . . , ak) we have thewell-known iterated integral representation
ζ(a1, a2, . . . , ak) =
∫ 10ωa1−10 ω1ω
a2−10 ω1 · · ·ω
ak−10 ω1
where
ω0 =dt
t, ω1 =
dt
1− t.
Provided all the ai are positive, there is a similar representationfor the H-series ζ(a1| · · · |ak), i.e.,
ζ(a1|a2| · · · |ak) =∫ 10ωa1−10 dtω
a2−10 dt · · ·ω
ak−10 ω1.
ME Hoffman Sequential Zeta Values
-
SequentialZeta Values
ME Hoffman
Outline
Introduction
Proof of theSum Theoremfor H-series
IteratedIntegrals
SequentialZeta Values
SumConjecture forSZVs
ProductStructure
Iterated Integrals cont’d
We can also represent H-series in the which the sequence haszeroes in positions other than the first or last as iteratedintegrals, e.g.,
ζ(2|0|1) =∫ 10ω0tdtω1
ζ(1|0|0|1|1) =∫ 10
t2dtdtω1
ζ(1|0|2|0|0|2) =∫ 10
tdtω0t2dtω0ω1.
(Initial zeroes don’t affect the value of the series, but trailingzeroes do.)
ME Hoffman Sequential Zeta Values
-
SequentialZeta Values
ME Hoffman
Outline
Introduction
Proof of theSum Theoremfor H-series
IteratedIntegrals
SequentialZeta Values
SumConjecture forSZVs
ProductStructure
Definition of SZVs
The H-series do not form a ring. But there is a larger class ofseries that does form a ring, and includes both MZVs andH-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 sequentialzeta value ζ(J1, J2, . . . , Jk) is∑
n1>l1n2>l2 ···>lk−1nk>lk 0
1∏ki=1
∏ai,jj=1(n − j + 1)ai,j
where a >k b means a− b ≥ k . We call |J| =∑k
i=1 |Ji | theweight of the SZV, and (l1, l2, . . . , lk) its shape.
ME Hoffman Sequential Zeta Values
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SequentialZeta Values
ME Hoffman
Outline
Introduction
Proof of theSum Theoremfor H-series
IteratedIntegrals
SequentialZeta Values
SumConjecture forSZVs
ProductStructure
Convergence of SZVs
It is immediate from the definition that
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 | ≥ 1for 1 < i < k, and |J1|+ |J2|+ · · ·+ |Jk | > k.
The only case requiring examination is if |Jk | = 0. In this caseζ(J1, . . . , Jk) can be written∑
n1>l1 ···nk−1>lk−1>lk
nk−1 − lk∏k−1i=1 n
ai,1i (ni − 1)ai,2 · · · (ni − li + 1)ai ,li
,
ME Hoffman Sequential Zeta Values
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SequentialZeta Values
ME Hoffman
Outline
Introduction
Proof of theSum Theoremfor H-series
IteratedIntegrals
SequentialZeta Values
SumConjecture forSZVs
ProductStructure
Convergence of SZVs cont’d
which is
ζ(J1, . . . , ak−1,1| · · · |ak−1,lk−1 − 1)− ζ(J1, . . . , Jk−1).
This is a sum of convergent SZVs if |Jk−1| > 1. Otherwise thefirst term has last sequence 0, and we can iterate. The onlycase that would produce a non-convergent SZV would be|J1| = 2, |J2| = · · · = |Jk−1| = 1, and |Jk | = 0, but this isexcluded by the hypothesis.We note that a leading 0 in J1 can simply be omitted withoutaffecting the value of the series; so henceforth we assume thatJ1 starts with a nonzero integer. If also |J1| > 1, Ji has notrailing 0 for i < k, |Ji | ≥ 1 for 1 < i < k , and|J1|+ · · ·+ |Jk | > k , we call (J1, . . . , Jk) admissible.
ME Hoffman Sequential Zeta Values
-
SequentialZeta Values
ME Hoffman
Outline
Introduction
Proof of theSum Theoremfor H-series
IteratedIntegrals
SequentialZeta Values
SumConjecture forSZVs
ProductStructure
Properties of SZVs
SZVs have some properties from partial fractions. Since
1
(n − p)(n − q)=
1
q − p
(1
n − p− 1
n − q
)it follows that
ζ(J1, . . . , ai ,1| · · · |ai ,q| · · · |ai ,p| · · · |ai ,li , . . . , Jk) =1
q − pζ(J1, . . . , ai ,1| · · · |ai ,p − 1| · · · |ai ,q| · · · |ai ,li , . . . , Jk)
− 1q − p
ζ(J1, . . . , ai ,1| · · · |ai ,p| · · · |ai ,q − 1| · · · |ai ,li , . . . , Jk)
whenever ai ,p, ai ,q > 0.
ME Hoffman Sequential Zeta Values
-
SequentialZeta Values
ME Hoffman
Outline
Introduction
Proof of theSum Theoremfor H-series
IteratedIntegrals
SequentialZeta Values
SumConjecture forSZVs
ProductStructure
Properties of SZVs cont’d
Recall that we assume J1 has no leading 0. A trailing 0 in Jk israther complicated to describe in general; one has, e.g.,
ζ(2, 1, 1|0) =∑
i>j>k≥2
1
i2jk=
∑i>j>k≥1
1
i2jk−∑i>j≥2
1
i2j=
ζ(2, 1, 1)−∑i>j≥1
1
i2j+∑i≥2
1
i2= ζ(2, 1, 1)− ζ(2, 1) + ζ(2)− 1.
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
-
SequentialZeta Values
ME Hoffman
Outline
Introduction
Proof of theSum Theoremfor H-series
IteratedIntegrals
SequentialZeta Values
SumConjecture forSZVs
ProductStructure
Strings of 1’s
By the preceding results we have for p > 1
ζ(1| · · · |1︸ ︷︷ ︸p
, J2, . . . , Jk) =1
p − 1ζ(1| · · · |1︸ ︷︷ ︸
p−1
|J2, J3, . . . , Jk).
Using this one can easily give a formula for ζ(J1, . . . , Jk) wheneach 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
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SequentialZeta Values
ME Hoffman
Outline
Introduction
Proof of theSum Theoremfor H-series
IteratedIntegrals
SequentialZeta Values
SumConjecture forSZVs
ProductStructure
Other Strings
A similar result is
ζ(1| 0| · · · |0︸ ︷︷ ︸p
|1, J2, . . . , Jk) =
1
p + 1
p∑j=0
ζ(1| 0| · · · |0︸ ︷︷ ︸j
|J2, J3, . . . , Jk).
For example,
ζ(1|0|0|1, 2, 1) = 13
[ζ(1|2, 1) + ζ(1|0|2, 1) + ζ(1|0|0|2, 1)] .
ME Hoffman Sequential Zeta Values
-
SequentialZeta Values
ME Hoffman
Outline
Introduction
Proof of theSum Theoremfor H-series
IteratedIntegrals
SequentialZeta Values
SumConjecture forSZVs
ProductStructure
Iterated Integrals Again
Provided that the last entry of Jk is nonzero, the SZVζ(J1, . . . , Jk) can be represented as an iterated integral. Forexample,
ζ(3|1, 2) =∑i ,j≥1
1
(i + j + 1)3(i + j)j2=
∫ 10ω20dtω1ω0ω1
ζ(2, 1|0|1) =∑i ,j≥1
1
(i + j + 2)2(i + 2)i=
∫ 10ω0ω1tdtω1
ME Hoffman Sequential Zeta Values
-
SequentialZeta Values
ME Hoffman
Outline
Introduction
Proof of theSum Theoremfor H-series
IteratedIntegrals
SequentialZeta Values
SumConjecture forSZVs
ProductStructure
Sum Conjecture
An MZV is a sequential zeta value of shape (1, 1, . . . , 1), whilean H-series is a sequential zeta value of shape (k). It turns outthat the sum theorems (2) and (4) can be put into a commonform for SZVs. Recall that a sequence (J1, . . . , Jk) isadmissible if J1 does not start with 0, |J1| > 1, Ji has notrailing 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), ∑p∈Am(S)
ζ(p) = ζ(m − `(S) + 1).
ME Hoffman Sequential Zeta Values
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SequentialZeta Values
ME Hoffman
Outline
Introduction
Proof of theSum Theoremfor H-series
IteratedIntegrals
SequentialZeta Values
SumConjecture forSZVs
ProductStructure
Sum Conjecture cont’d
It is simple to put the sum theorem (4) for H-series into thisform: in the identity∑
a1+···+ak=n, ai≥0ζ(a1| · · · |ak) = kζ(n),
the inadmissible terms on the right-hand side are those witha1 = 0. These add up to∑
a2+···+ak=n, ai≥0ζ(0|a2| · · · |ak) = (k − 1)ζ(n)
since ζ(0|a2| · · · |ak) = ζ(a2| · · · |ak); hence∑a1+···+ak=n, a1≥1, ai≥0
ζ(a1| . . . |ak) = ζ(n).
ME Hoffman Sequential Zeta Values
-
SequentialZeta Values
ME Hoffman
Outline
Introduction
Proof of theSum Theoremfor H-series
IteratedIntegrals
SequentialZeta Values
SumConjecture forSZVs
ProductStructure
Sum Conjecture cont’d
To put the MZV sum theorem (2) in this form is a little moreinvolved; to avoid notational complications we treat the specialcase depth 3, weight 5. The MZV sum theorem says
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), theresult 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
-
SequentialZeta Values
ME Hoffman
Outline
Introduction
Proof of theSum Theoremfor H-series
IteratedIntegrals
SequentialZeta Values
SumConjecture forSZVs
ProductStructure
Known Cases of Sum Conjecture
So the sum conjecture holds for shape (k) and for shape(1, . . . , 1). So far we have only been able to prove that the sumconjecture holds for a few other shapes. These include allshapes 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 ∑p∈Am(1,2)
ζ(p) = ζ(m − 1).
ME Hoffman Sequential Zeta Values
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SequentialZeta Values
ME Hoffman
Outline
Introduction
Proof of theSum Theoremfor H-series
IteratedIntegrals
SequentialZeta Values
SumConjecture forSZVs
ProductStructure
Known Cases of Sum Conjecture cont’d
The left-hand side is
ζ(m, 0|0) +∑
i+j=m, i≥2, j≥1[ζ(i , j |0) + ζ(i , 0|j)]
+m−2∑i=2
m−1−i∑j=1
ζ(i , j |m − i − j). (7)
Now
ζ(m, 0|0) = ζ(m − 1)− 2ζ(m)− 1ζ(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
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SequentialZeta Values
ME Hoffman
Outline
Introduction
Proof of theSum Theoremfor H-series
IteratedIntegrals
SequentialZeta Values
SumConjecture forSZVs
ProductStructure
Known Cases of Sum Conjecture cont’d
ζ(m−1)−2ζ(m)−1+m−1∑i=2
[2ζ(i ,m− i)−ζ(i |m− i)−ζ(i)+1]
= ζ(m − 1) + m − 2−m−2∑i=2
ζ(i)− ζ(1|m − 2)
using the sum theorem of MZVs. The second sum in (7) gives
m−2∑i=2
[ζ(i)− ζ(i |m − i − 1) + ζ(i)− 1] =
m−2∑i=2
ζ(i)− ζ(1|m − 3) + ζ(m − 2)− (m − 2)
and the conclusion follows.ME Hoffman Sequential Zeta Values
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SequentialZeta Values
ME Hoffman
Outline
Introduction
Proof of theSum Theoremfor H-series
IteratedIntegrals
SequentialZeta Values
SumConjecture forSZVs
ProductStructure
Product of SZVs
The product of two SZVs is a finite sum of SZVs, but theproduct structure (unlike that for MZVs) does not fall into theframework of quasi-shuffle products. As with the product ofMZVs, all shuffles appear in the product, together with termsin which sequences combine; a new feature is that a singlesequence in one factor can “paste together” two or moresequences 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
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SequentialZeta Values
ME Hoffman
Outline
Introduction
Proof of theSum Theoremfor H-series
IteratedIntegrals
SequentialZeta Values
SumConjecture forSZVs
ProductStructure
Product Structure cont’d
Here is another example.
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).
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:
ζ(2|1, 1|1) = ζ(1|1|1) + ζ(2|1)− ζ(2) + 1 + ζ(2|1|1)ζ(1|1, 2|1) = ζ(2|1)− ζ(1|1|1)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
IntroductionProof of the Sum Theorem for H-seriesIterated IntegralsSequential Zeta ValuesSum Conjecture for SZVsProduct Structure