Sum
Theorems,
Old and New
ME Hoffman
Outline
Introduction
Classical Sum,
Duality and
Derivation
Theorems
Multiple
Zeta-Star
Values
An Algebraic
Framework
Yamamoto’s
Product
New Sum
Theorems?
Sum Theorems for Multiple Zeta Values,
Old and New
Michael E. Hoffman
U. S. Naval Academy
Number Theory Talk
Max-Planck-Institut für Mathematik, Bonn
29 June 2016
ME Hoffman
Sum Theorems, Old and New
Outline
Sum
Theorems,
Old and New
ME Hoffman
Outline
Introduction
Classical Sum,
Duality and
Derivation
Theorems
1 Introduction
2 Classical Sum, Duality and Derivation Theorems
3 Multiple Zeta-Star Values
Multiple
Zeta-Star
Values
4 An Algebraic Framework
An Algebraic
Framework
5 Yamamoto’s Product
Yamamoto’s
Product
New Sum
Theorems?
6 New Sum Theorems?
ME Hoffman
Sum Theorems, Old and New
Introduction
Sum
Theorems,
Old and New
ME Hoffman
Outline
Introduction
Classical Sum,
Duality and
Derivation
Theorems
Multiple
Zeta-Star
Values
An Algebraic
Framework
Yamamoto’s
Product
New Sum
Theorems?
For positive integers a1 , . . . , ak with a1 > 1 we define the
corresponding multiple zeta value (MZV) by
X
1
.
ζ(a1 , a2 , . . . , ak ) =
a1 a2
n1 n2 · · · nkak
(1)
n1 >n2 >···>nk ≥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 Courtney Moen for
depth 3, and by Andrew Granville and Don Zagier for general
depth.
ME Hoffman
Sum Theorems, Old and New
Introduction cont’d
Sum
Theorems,
Old and New
ME Hoffman
Outline
Introduction
Classical Sum,
Duality and
Derivation
Theorems
Multiple
Zeta-Star
Values
An Algebraic
Framework
Yamamoto’s
Product
New Sum
Theorems?
In 1997 I introduced the algebraic notation which has since
become fairly standard. Let H be the underlying rational vector
space of the noncommutative polynomial algebra Qhx, y i,
graded by giving both x and y degree 1. Also, let
H1 = Q1 + Hy and H0 = Q1 + xHy ⊂ H1 . Then we can think
of MZVs as images under the Q-linear map ζ : H0 → R that
sends the empty word 1 ∈ H0 to 1 ∈ R, and the word
x a1 −1 y · · · x ak −1 y to ζ(a1 , . . . , ak ). We sometimes think of H1
as Qhz1 , z2 , . . .i, where zi = x i−1 y has degree i. Then H0 ⊂ H1
is the subspace generated by 1 and all words that don’t begin
with z1 .
ME Hoffman
Sum Theorems, Old and New
Introduction cont’d
Sum
Theorems,
Old and New
ME Hoffman
Outline
Introduction
The main point of my 1997 paper was that one can make H1 a
commutative algebra by giving it the product ∗ defined
inductively by w ∗ 1 = 1 ∗ w = w for all words w of H1 , and
zi u ∗ zj v = zi (u ∗ zj v ) + zj (zi u ∗ v ) + zi+j (u ∗ v )
Classical Sum,
Duality and
Derivation
Theorems
for all words u, v of H1 . Then one has the following results.
Multiple
Zeta-Star
Values
The graded algebra (H1 , ∗) is isomorphic to the algebra QSym
of quasi-symmetric functions, which is polynomial on Lyndon
words.
An Algebraic
Framework
Yamamoto’s
Product
New Sum
Theorems?
Theorem
Theorem
The Q-linear function ζ : H0 → R is a homomorphism of
algebras from (H0 , ∗) to R.
ME Hoffman
Sum Theorems, Old and New
Sum, Duality and Derivation Theorems
Sum
Theorems,
Old and New
ME Hoffman
Outline
Introduction
Classical Sum,
Duality and
Derivation
Theorems
Multiple
Zeta-Star
Values
An Algebraic
Framework
In the algebraic notation, the classical sum theorem (2) says
that ζ sends the sum of all words in H0 of degree n and
y -degree k < n to ζ(n), regardless of k. Two further striking
results for MZVs lend themselves to the algebraic notation. Let
τ be the antiautomorphism of Qhx, y i that exchanges x and y
(so, e.g., τ (xyx 2 y ) = xy 2 xy ), and let D be the derivation of
Qhx, y i that sends x to 0 and y to xy . Then we have the
following results.
Theorem
For all words w of H0 , ζ(τ (w )) = ζ(w ).
Yamamoto’s
Product
New Sum
Theorems?
Theorem
For all words w of H0 , ζ(D(w )) = ζ(τ Dτ (w )).
ME Hoffman
Sum Theorems, Old and New
Cyclic Sum Theorem
Sum
Theorems,
Old and New
ME Hoffman
Outline
There is another derivation theorem that is a bit more
mysterious. Regard H ⊗ H as a two-sided module over H via
a(b ⊗ c) = ab ⊗ c
and
(a ⊗ b)c = a ⊗ bc.
Introduction
Classical Sum,
Duality and
Derivation
Theorems
Multiple
Zeta-Star
Values
An Algebraic
Framework
Now let Ĉ : H → H ⊗ H be the derivation sending x to 0 and y
to y ⊗ x, and let µ̄(a ⊗ b) = ba. Then we define the “cyclic
derivation” C as µ̄Ĉ , so, e.g, C (z4 z2 ) = C (x 3 yxy ) is
µ̄Ĉ (x 3 yxy ) = µ̄(x 3 y ⊗ x 2 y + x 3 yxy ⊗ x) = x 2 yx 3 y + x 4 yxy
= z3 z4 + z5 z2 .
Yamamoto’s
Product
New Sum
Theorems?
Then the cyclic sum theorem, conjectured by me and proved by
Yasuo Ohno, reads as follows.
ME Hoffman
Sum Theorems, Old and New
Cyclic Sum Theorem cont’d
Sum
Theorems,
Old and New
Theorem
ME Hoffman
For any word of H1 not a power of y , ζ(C (w )) = ζ(τ C τ (w )).
Outline
Introduction
Classical Sum,
Duality and
Derivation
Theorems
Multiple
Zeta-Star
Values
An Algebraic
Framework
Yamamoto’s
Product
New Sum
Theorems?
For example, for w = z4 z2 it says that
ζ(5, 2) + ζ(3, 4) = ζ(4, 2, 1) + ζ(3, 2, 2) + ζ(2, 2, 3) + ζ(2, 4, 1)
while the derivation theorem applied to the same word is
ζ(5, 2) + ζ(4, 3) = ζ(4, 1, 2) + ζ(3, 2, 2) + ζ(2, 3, 2) + ζ(4, 2, 1).
Ohno’s proof (like mine for the derivation theorem) uses partial
fractions. Interestingly, the cyclic sum theorem implies the sum
theorem (which is not the case for the derivation theorem,
except for depth ≤ 3).
ME Hoffman
Sum Theorems, Old and New
Multiple Zeta-Star Values
Sum
Theorems,
Old and New
Euler actually worked in terms of what are now called multiple
zeta-star values (MZSVs), i.e.,
ME Hoffman
Outline
ζ ? (a1 , a2 , . . . , ak ) =
Introduction
Classical Sum,
Duality and
Derivation
Theorems
Multiple
Zeta-Star
Values
An Algebraic
Framework
Yamamoto’s
Product
New Sum
Theorems?
X
1
.
· · · nkak
n a1 n a2
n1 ≥n2 ≥···≥nk ≥1 1 2
(3)
The difference from eqn. (1) is that ≥ replaces >. Of course
ζ ? (n) = ζ(n). One has, e.g.,
ζ ? (3, 2, 1) = ζ(3, 2, 1) + ζ(5, 1) + ζ(3, 3) + ζ(6).
(4)
The MZSVs give the same rational vector space as the MZVs,
since equations like (4) can be inverted:
ζ(3, 2, 1) = ζ ? (3, 2, 1) − ζ ? (5, 1) − ζ ? (3, 3) + ζ ? (6).
ME Hoffman
Sum Theorems, Old and New
Multiple Zeta-Star Values cont’d
Sum
Theorems,
Old and New
ME Hoffman
Outline
Introduction
Classical Sum,
Duality and
Derivation
Theorems
Multiple
Zeta-Star
Values
An Algebraic
Framework
Yamamoto’s
Product
New Sum
Theorems?
In my 1992 paper I actually treated the MZVs and MZSVs on
a fairly equal footing (denoting ζ(a1 , . . . , ak ) by A(a1 , . . . , ak )
and ζ ? (a1 , . . . , ak ) by S(a1 , . . . , ak )). I noted that the sum
theorem (2) is equivalent to the statement that summing
ζ ? (a1 , . . . , ak ) over all strings with a1 > 1 and
a1 + · · · + ak = n gives
n−1
ζ(n).
k −1
But one loses the duality and derivation theorems. I didn’t
mention the zeta-star values in my 1997 paper, but I easily
could have, since the algebra of MZSVs parallels that for
MZVs.
ME Hoffman
Sum Theorems, Old and New
Algebra of MZSVs
Sum
Theorems,
Old and New
ME Hoffman
Outline
Instead of defining the product ∗ on H1 as above, one can
define an algebra ? on H1 inductively by w ? 1 = 1 ? w = w for
all words w of H1 , and
Introduction
Classical Sum,
Duality and
Derivation
Theorems
Multiple
Zeta-Star
Values
An Algebraic
Framework
Yamamoto’s
Product
New Sum
Theorems?
zi u ? zj v = zi (u ? zj v ) + zj (zi u ? v ) − zi+j (u ? v )
for all i, j ≥ 1 and words u, v of H1 . Then the Q-linear map
ζ ? : H0 → R sending za1 · · · zak , a1 > 1, to ζ ? (a1 , . . . , ak ) is a
homomorphism from (H0 , ?) to R. Also, (H1 , ?) is isomorphic
as a graded rational algebra to (H1 , ∗). From this point of view
MZVs and MZSVs look rather similar, but (despite the loss of
the duality theorem) there are some advantages to MZSVs.
ME Hoffman
Sum Theorems, Old and New
Cyclic Sum Theorem for MZSVs
Sum
Theorems,
Old and New
ME Hoffman
Outline
Introduction
Classical Sum,
Duality and
Derivation
Theorems
Multiple
Zeta-Star
Values
For example, despite the lack of an obvious analogue of the
derivation theorem for MZSVs, there is a cyclic sum theorem
for MZSVs that is actually simpler than the one for MZVs.
This result is due to Ohno and Noriko Wakabayashi and can be
stated as follows.
Theorem (Ohno-Wakabayashi)
An Algebraic
Framework
For all words w of H1 not a power of y ,
ζ ? (τ C τ (w )) = nζ(n + 1), where n = deg w .
Yamamoto’s
Product
Of course this implies the sum theorem.
New Sum
Theorems?
ME Hoffman
Sum Theorems, Old and New
Back to Algebra
Sum
Theorems,
Old and New
ME Hoffman
Outline
Introduction
Classical Sum,
Duality and
Derivation
Theorems
Multiple
Zeta-Star
Values
An Algebraic
Framework
Yamamoto’s
Product
New Sum
Theorems?
Let’s go back to the vector space H1 , which we now know has
two (isomorphic) commutative algebra structures (H1 , ∗) and
(H1 , ?). There’s actually some more structure here, which I
started to develop in my 2000 paper “Quasi-shuffle products”
and developed further in joint work here in 2012 with Kentaro
Ihara. Define a product ◦ on the zi by zi ◦ zj = zi+j . Suppose
w is a word of H1 , say w = a1 · · · ak , where each aj is a letter
zi . Given a composition I = (i1 , . . . , il ) of k (i.e., the ij are
positive integers whose sum is k), define I [w ] to be the word
(a1 ◦ · · · ◦ ai1 )(ai1 +1 ◦ · · · ◦ ai1 +i2 ) · · · (ak−il +1 ◦ · · · ◦ ak ).
For example,
(2, 1, 3)[z22 z1 z3 z12 ] = (z2 ◦ z2 )z1 (z3 ◦ z1 ◦ z1 ) = z4 z1 z5 .
ME Hoffman
Sum Theorems, Old and New
Formal Power Series
Sum
Theorems,
Old and New
ME Hoffman
Outline
Introduction
Classical Sum,
Duality and
Derivation
Theorems
Multiple
Zeta-Star
Values
An Algebraic
Framework
Yamamoto’s
Product
New Sum
Theorems?
Now let P be the set of formal power series
f = c1 t + c2 t 2 + c3 t 3 + · · ·
with c1 6= 0. Given f ∈ P, we can define a Q-linear function
Ψf : H1 → H1 by
X
Ψf (za1 · · · zak ) =
ci1 · · · cil I [za1 · · · zak ],
(i1 ,...,il )∈C(k)
where C(k) is the set of compositions of k. For example, if
f = e t − 1 (so that ci = (i!)−1 ) then
1
1
Ψf (z2 z1 z3 ) = z2 z1 z3 + (z32 + z2 z4 ) + z6 .
2
6
ME Hoffman
Sum Theorems, Old and New
Formal Power Series, cont’d
Sum
Theorems,
Old and New
Now if
ME Hoffman
Outline
Introduction
Classical Sum,
Duality and
Derivation
Theorems
f = c1 t + c2 t 2 + · · ·
and g = d1 t + d2 t 2 + · · ·
are two elements of P, they have a “functional composition”
f ◦ g = c1 (d1 t + d2 t 2 + · · · ) + c2 (d1 t + d2 t 2 + · · · )2 + · · ·
Multiple
Zeta-Star
Values
= c1 d1 t + (c1 d2 + c2 d12 )t 2 + · · ·
An Algebraic
Framework
in P, and the following result holds.
Yamamoto’s
Product
Theorem
New Sum
Theorems?
For f , g ∈ P, Ψf ◦g = Ψf Ψg .
ME Hoffman
Sum Theorems, Old and New
Functions Ψf
Sum
Theorems,
Old and New
ME Hoffman
Outline
Introduction
Classical Sum,
Duality and
Derivation
Theorems
Multiple
Zeta-Star
Values
An Algebraic
Framework
Yamamoto’s
Product
New Sum
Theorems?
The functions Ψf need not preserve the algebra structures
(H1 , ∗) or (H1 , ?), but we will shortly see some examples that
do. A key result from my work with Ihara is the following.
Theorem
If f = c1 t + c2 t 2 + · · · ∈ P, then
1
1
Ψf
=
.
1 − tzi
1 − f◦ (tzi )
Here f◦ (tzi ) means
tc1 zi + t 2 c2 zi ◦ zi + t 3 c3 zi ◦ zi ◦ zi + · · ·
= tc1 zi + t 2 c2 z2i + t 3 c3 z3i + · · ·
ME Hoffman
Sum Theorems, Old and New
Specific Examples
Sum
Theorems,
Old and New
ME Hoffman
Outline
Introduction
Classical Sum,
Duality and
Derivation
Theorems
Evidently Ψt = id. The function T = Ψ−t sends a word w to
(−1)deg w w . In fact we have the following result.
Proposition
T : (H1 , ?) → (H1 , ∗) is an algebra homomorphism, and so is
T : (H1 , ∗) → (H1 , ?).
Multiple
Zeta-Star
Values
Clearly T 2 = id, so T is an isomorphism. Here are two more
functions: exp = Ψe t −1 , with inverse log = Ψlog(1+t) . In my
2000 paper I proved the following.
An Algebraic
Framework
Theorem
Yamamoto’s
Product
New Sum
Theorems?
exp : (H1 , ) → (H1 , ∗) is an isomorphism of algebras, where
is the usual shuffle product on H1 = Qhz1 , z2 , . . .i.
This allows one to deduce the algebra structure of (H1 , ∗) from
known results about (H1 , ).
ME Hoffman
Sum Theorems, Old and New
Σ Takes MZVs to MZSVs
Sum
Theorems,
Old and New
ME Hoffman
Outline
But perhaps the most interesting function is Σ = Ψ t . Using
1−t
the composition theorem above it is easy to show that
T ΣT = Σ−1 and Σ = exp T log T . The key property of Σ is
the following.
Introduction
Classical Sum,
Duality and
Derivation
Theorems
Multiple
Zeta-Star
Values
An Algebraic
Framework
Yamamoto’s
Product
New Sum
Theorems?
Theorem
Σ : (H1 , ∗) → (H1 , ?) is an algebra isomorphism such that
ζ ? (w ) = ζ(Σ(w )) for all words w of H0 .
Having the isomorphism Σ turns out to be quite useful. One
can show that


X (−1)i−1 ζ(ik)
X
ti 
ζ(zkn )t n = exp 
i
n≥0
i≥1
ME Hoffman
Sum Theorems, Old and New
Σ takes MSVs to MZSVs cont’d
Sum
Theorems,
Old and New
ME Hoffman
Outline
Introduction
If we call this generating function Zk (t), then the theorems
above imply
X
1
.
ζ ? (zkn )t n =
Zk (−t)
n≥0
Thus from the well-known result
Classical Sum,
Duality and
Derivation
Theorems
Multiple
Zeta-Star
Values
ζ(z2n )
New Sum
Theorems?
i.e.,
we get
X
An Algebraic
Framework
Yamamoto’s
Product
π 2n
=
,
(2n + 1)!
ζ
?
n≥0
(z2n )t n
√
sinh π t
√
Z2 (t) =
π t
√
π t
√ ,
=
sin π t
and thus
ζ ? (z2n ) =
(−1)n 2(22n−1 − 1)B2n π 2n
.
(2n)!
ME Hoffman
Sum Theorems, Old and New
Σ takes MZVs to MZSVs cont’d
Sum
Theorems,
Old and New
But one can do much more. There is the following identity in
H1 [[t]].
ME Hoffman
Outline
Introduction
Classical Sum,
Duality and
Derivation
Theorems
Multiple
Zeta-Star
Values
An Algebraic
Framework
Yamamoto’s
Product
New Sum
Theorems?
Theorem (Ihara-Kajikawa-Ohno-Okuda, H-Ihara)
For any positive integers i, j,
1
1
1
Σ
=
∗Σ
.
1 − tzi zj
1 − tzi zj
1 − tzi+j
Applying ζ to the theorem with i = 2, j = 1 gives
X
X
X
ζ ? ((z2 z1 )n )t n =
ζ((z2 z1 )p )t p
ζ ? (z3q )t q ;
n≥0
p≥0
q≥0
using the duality relation ζ((z2 z1 )n ) = ζ(z3n ), this implies
ME Hoffman
Sum Theorems, Old and New
Σ takes MZVs to MZSVs cont’d
Sum
Theorems,
Old and New
X
ME Hoffman
Outline
Introduction
Classical Sum,
Duality and
Derivation
Theorems
Multiple
Zeta-Star
Values
An Algebraic
Framework
Yamamoto’s
Product
New Sum
Theorems?
ζ ? ((z2 z1 )n )t n =
n≥0
Z3 (t)
.
Z3 (−t)
Then (via a lemma from the theory of symmetric functions)
X
2i1 +i3 +i5 +··· ζ(3)i1 ζ(9)i3 ζ(15)i5 · · ·
ζ ? ((z2 z1 )n ) =
.
1i1 i1 !3i3 i3 !5i5 i5 ! · · ·
i1 +3i3 +5i5 +···=n
Similarly, from the Zagier-Broadhurst identity
t X
2π 4n
, i.e.,
ζ((z3 z1 )n )t n = Z4
ζ((z3 z1 )n ) =
(4n + 2)!
4
n≥0
we get
X
√
√
Z4 ( 4t )
cosh(π 4 t) − cos(π 4 t)
√
√
ζ ((z3 z1 ) )t =
=
.
Z4 (−t)
sinh(π 4 t) sin(π 4 t)
n≥0
?
n
n
ME Hoffman
Sum Theorems, Old and New
Yamamoto’s Product
Sum
Theorems,
Old and New
ME Hoffman
In fact, for any r ∈ Q we can define Σr as Ψ t . The
1−rt
composition theorem then gives us the following.
Outline
Proposition
Introduction
Classical Sum,
Duality and
Derivation
Theorems
Multiple
Zeta-Star
Values
An Algebraic
Framework
Yamamoto’s
Product
New Sum
Theorems?
For any r , s ∈ Q, Σr Σs = Σr +s .
Ihara and I got some interesting results involving fractional Σr
in our 2012 work, e.g.,
1
1
r
1−r
Σ
∗Σ
=1
1 − tzi
1 + tzi
for any rational r , but Shuji Yamamoto did something even
r
better in his 2013 paper; he managed to define a product ∗ on
r
H1 so that Σr (u ∗ v ) = Σr (u) ∗ Σr (v ).
ME Hoffman
Sum Theorems, Old and New
Yamamoto’s Product cont’d
Sum
Theorems,
Old and New
ME Hoffman
Outline
Introduction
Classical Sum,
Duality and
Derivation
Theorems
Multiple
Zeta-Star
Values
An Algebraic
Framework
Yamamoto’s
Product
New Sum
Theorems?
r
The definition of ∗ is similar to that of ∗, but with an
r
r
interesting twist. For words w of H1 , define w ∗ 1 = 1 ∗ w = w ;
r
for all i, j ≥ 1 define zi ∗ zj = zi zj + zj zi + (1 − 2r )zi+j ; and for
words u, v of H1 that are not both 1 and for i, j ≥ 1, define
r
r
r
r
zi u ∗ zj v = zi (u ∗ zj v ) + zj (zi u ∗ v ) + (1 − 2r )zi+j (u ∗ v )
r
+ (r 2 − r )zi+j ◦ (u ∗ v ).
The novel element is the fourth term on the right-hand side.
Thus, e.g.,
r
z2 ∗z3 z1 = z2 z3 z1 +z3 z1 z2 +z3 z2 z1 +(1−2r )(z32 +z5 z1 )+(r 2 −r )z6 .
r
Note that ∗ reduces to ∗ if r = 0 and to ? if r = 1.
ME Hoffman
Sum Theorems, Old and New
Yamamoto’s Product cont’d
Sum
Theorems,
Old and New
ME Hoffman
Outline
Introduction
Classical Sum,
Duality and
Derivation
Theorems
Multiple
Zeta-Star
Values
An Algebraic
Framework
Yamamoto’s
Product
New Sum
Theorems?
The additional term complicates computation of the product.
Even in the case of powers of z1 , things get interesting. Let
r
(r )
pn (w ) be the nth power of w using the ∗-product. For the
∗-product (i.e., r = 0) there is the well-known result
X n (0)
pn (z1 ) =
mλ ,
λ
λ`n
where the sum is over partitions λ of n, λn is the multinomial
coefficient, and mλ is the monomial symmetric function
corresponding to λ (in variables for which z1 , z2 , . . . are power
sums). For the ?-product (i.e., r = 1) the formula only adds an
alternation in sign:
X
(1)
n−`(λ) n
pn (z1 ) =
(−1)
mλ ,
λ
λ`n
ME Hoffman
Sum Theorems, Old and New
Yamamoto’s Product cont’d
Sum
Theorems,
Old and New
ME Hoffman
Outline
Introduction
Classical Sum,
Duality and
Derivation
Theorems
where `(λ) is the number of parts of λ. But for general r this
becomes
X n `(λ)
Y
(r )
pn (z1 ) =
aλi (r )mλ
λ
λ`n
for
Multiple
Zeta-Star
Values
An Algebraic
Framework
Yamamoto’s
Product
ap (r ) =
i=1
p X
p
j!(−r )j−1 ,
j
j=1
p
where j is the number of partitions of {1, 2, . . . , p} with j
blocks (Stirling number of the second kind).
New Sum
Theorems?
ME Hoffman
Sum Theorems, Old and New
Sum Theorems for Yamamoto’s Product
Sum
Theorems,
Old and New
ME Hoffman
Outline
Introduction
Classical Sum,
Duality and
Derivation
Theorems
Multiple
Zeta-Star
Values
An Algebraic
Framework
Yamamoto’s
Product
New Sum
Theorems?
Yamamoto proved a version of the sum theorem for his
product. The sum of ζ r (w ) = ζ(Σr (w )) over all words of
degree n and y -degree k < n in H0 is
k−1 X
n−1 i
r (1 − r )k−1−i ζ(n).
i
i=0
This follows from the following version of the cyclic sum
theorem for Yamamoto’s product.
Theorem (Yamamoto)
For words w of H1 not a power of y ,
ζ r (τ C τ (w )) = (1 − r )ζ r (C (w )) + r k nζ(n + 1),
where n = deg w and k is the y -degree of w .
ME Hoffman
Sum Theorems, Old and New
Reflections
Sum
Theorems,
Old and New
ME Hoffman
Outline
Introduction
Classical Sum,
Duality and
Derivation
Theorems
Multiple
Zeta-Star
Values
Note that the latter result interpolates nicely between the
(Hoffman-Ohno) cyclic sum theorem for MZVs (r = 0) and the
(Ohno-Wakabayashi) cyclic sum theorem for MZSVs (r = 1).
It seems that the cyclic sum theorem is the “right” result, in
the sense that it (unlike the derivation theorem) (1) implies the
sum theorem and (2) generalizes nicely to ζ r .
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Sum Theorems, Old and New
New Sum Theorems?
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Introduction
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The product ◦ leads to an interesting conjecture. Let
w1 , . . . , wk be words of H1 . For m ≤ k −1, define
Rk,m (w1 , . . . , wk ) as the sum of the k−1
terms of the form
m
w1 w2 · · · wk ,
where m of the boxes are ◦ and the others are empty. E.g.,
R3,0 (w1 , w2 , w3 ) = w1 w2 w3
R3,1 (w1 , w2 , w3 ) = w1 ◦ w2 w3 + w1 w2 ◦ w3
R3,2 (w1 , w2 , w3 ) = w1 ◦ w2 ◦ w3 .
Now let λ be a partition of n with k ≤ n parts. For m ≤ k − 1,
define
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New Sum Theorems?, cont’d
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Pk,m (λ) =
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X
ζ(xRk,m (w1 , . . . , wk )).
{|w1 |,...,|wk |}=λ
For example, if λ = (2, 1) then
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P2,1 (λ) = ζ(xy 3 +x 2 y 2 +xy 3 +xyxy ) = 2ζ(2, 1, 1)+ζ(3, 1)+ζ(2, 2).
Conjecture
For any partition λ of n with k parts,
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Pk,m (λ) =
(k − 1)!n
ζ(n + 1)
| Sym(λ)|
for all m ≤ k − 1, where Sym(λ) is the group that exchanges
identical parts of λ.
ME Hoffman
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Observations
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Some observations are in order. First, duality implies that
Pk,m (λ) = Pk,k−1−m (λ)
for any m ≤ k − 1. Of course the conjecture implies that
Pk,m (λ) doesn’t depend on m at all.
Second, the conjecture follows from the (classical) sum
theorem if k = n (i.e., if λ = (1, 1, 1, . . . , 1)) or if k = n − 1
(i.e., if λ = (2, 1, 1, . . . , 1)). Third, it holds in the case
λ = (3, 1, 1, . . . , 1), but here one must use the cyclic sum
theorem. The same is true of numerous special cases.
New Sum
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Sum Theorems, Old and New
Observations cont’d
Sum
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ME Hoffman
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Introduction
Classical Sum,
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An Algebraic
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Finally, the conjecture implies the following result. For positive
integers p, q, let
X
X
Sk,m (p, q) =
ζ(xRk,m (w1 , . . . , wk )).
i1 +···+ik =p |ws |=is +js +1
j1 +···+jk =q
1≤s≤k
Then Sk,m (p, q) is evidently symmetric in p and q. The
conjecture above implies that
p+q+k p+k −1 q+k −1
Sk,m (p, q) =
ζ(p+q+k+1).
k
k −1
k −1
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Addendum: Lemma on Symmetric Functions
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Here is the lemma used to derive the equation for ζ ? ((z2 z1 )n ).
Lemma
If E (t), H(t) are respectively the generating functions for the
elementary and complete symmetric functions, then the
coefficient of t n in E (t)H(t) is
X
i1 +3i3 +5i5 +···=n
2i1 +i3 +i5 +··· p1i1 p3i3 p5i5 · · ·
.
i1 !1i1 i3 !3i3 i5 !5i5 · · ·
This follows from two formulas that can be found in
Macdonald’s book Symmetric Functions and Hall Polynomials:
X
X
zλ−1 pλ t |λ|
E (t) =
λ zλ−1 pλ t |λ| , H(t) =
λ
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λ
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Addendum cont’d
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Outline
where in each case the sum is over all partitions λ, and for
λ = (λ1 , λ1 , . . . ),
Y
mi (λ)i mi (λ)
pλ = pλ1 pλ2 · · · , λ = (−1)|λ|−`(λ) , zλ =
i≥1
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(|λ| is the sum of the parts of λ and mi (λ) is the number of
parts of λ equal to i, so `(λ) = m1 (λ) + m2 (λ) + · · · and
|λ| = m1 (λ) + 2m2 (λ) + · · · ). From the formulas above,
X µ
X
X µ
E (t)H(t) =
pµ∪λ t |µ|+|λ| =
pν t |ν|
;
zµ zλ
zµ zλ
ν
µ,λ
µ∪λ=ν
so it suffices to show
X
µ∪λ=ν
µ
=
zµ zλ
(
2`(ν) zν−1 , if ν has all parts odd;
0,
otherwise.
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Sum Theorems, Old and New
Addendum cont’d
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This is equivalent to
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µ
µ∪λ=ν
Y mi (ν)
mi (µ)
i≥1
(
2`(ν) ,
=
0,
if ν has all parts odd;
otherwise.
But the latter
P is immediate if we note that
µ = (−1) i (i−1)mi (µ) and rewrite the left-hand side as
X Y
(i−1)mi (µ)
(−1)
µ∪λ=ν i≥1
mi (ν)
=
mi (µ)
mi (ν)
Yamamoto’s
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Y
New Sum
Theorems?
X
(−1)(i−1)j
mi (ν)6=0 j=0
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Sum Theorems, Old and New
mi (ν)
.
j