GSC ROOT Meeting 05.09.2020

1. Multiple Zeta Values Multiple Gamma Functions
Applications of Hoffman Algebra for the Multiple
Gamma Functions
Hanamichi Kawamura
Seifu High School
GSC-ROOT Zoom Meeting, 05.09.2020

2. Multiple Zeta Values Multiple Gamma Functions
Multiple Zeta Values
Let H = Q⟨x, y⟩ and its subalgebras h1, h0 be
h0 := Q + yhx ⊃ h1 := Q + yh ⊃ h.
We put zk = yxk−1
for k ∈ Z≥1, we define a Q-linear map Z : h0 → R
by
Z(1) = 1
Z(zk1
· · · zkr
) =
0<n1<···<nr
n−k1
1 · · · n−kr
r (r ∈ Z≥1).
For a positive integer r and an index (ki)i ∈ Zr−1
≥1 × Z≥2, we call the
series Z(zk1 · · · zkr ) the multiple zeta value.

3. Multiple Zeta Values Multiple Gamma Functions
If we define τ for the anti-automorphism interchanging variables, the
following is true.
Theorem 1.1 (Kontsevich, 1994)
(1 − τ)(h0) ∈ Ker Z.
Kontsevich proved this fact (duality) by the iterated integral expression,
but Seki-Yamamoto (2018) showed that one can prove the duality only
using deformations of series by the connector method.

4. Multiple Zeta Values Multiple Gamma Functions
The connector method can prove the q-Ohno relation (the q-analog of
the duality of the linear sum of some MZVs called the Ohno sum). This
is also result by Seki-Yamamoto.
While, the connector is applicable for the duality of certain general
iterated integrals found by Hirose-Iwaki-Sato-Tasaka (2017).
Subject 1
Obtaining the q-analog of the above HIST-duality.
Prof. Yamamoto (Keio Univ.) said
As this theorem includes the duality of T-values by Kaneko-Tsumura (a
kind of "shuffle analog" of Hoffman’s t-values), showing the q-analog of
that may require a transport relation obtained by deformations of the
q-hypergeometric series.
So far as I know, the duality of the Tz values has never included in any
papers.

5. Multiple Zeta Values Multiple Gamma Functions
We define two products x and ∗ in h1 by the following properties.
x , ∗ are both bilinear.
When • ∈ { x , ∗}, 1 • w = w • 1 = w for w ∈ h1.
For positive integers m, n and words w1, w2 ∈ h1,
w1zm ∗ w2zn = (w1 ∗ w2zn)zm + (w1zm ∗ w2)zn + (w1 ∗ w2)zmzn.
Fpr u1, u2 ∈ {x, y} and words w1, w2 ∈ h,
w1u1 x w2u2 = (w1 x w2u2)u1 + (w1u1 x w2)u2.
Then the following are true.
Theorem 1.2
Z(w1 ∗ w2 − w1 x w2) = 0 holds for w1, w2 ∈ h0.
The above theorem is called the finite double shuffle relation, (FDSR).

6. Multiple Zeta Values Multiple Gamma Functions
Some integral of elliptic multiple digamma functions gives the generating
functions of "2-parameter deformation" of MZVs.
The above result is obtained by Kato (2019) and the result which
generalizes that for the multiple L-values is obtained by Takeyama
(2020).
Subject 2
Letting the multiple elliptic gamma functions become the case of general
periods.
Subject 3
Extending the above theorem to the HIST-duality.

7. Multiple Zeta Values Multiple Gamma Functions
Multiple Gamma Functions
For r ∈ Z≥0 and w, ω1, · · · , ωr ∈ C having positive real parts, we put
ζr(s, w; ω) =
n∈Zr
≥0
(n · ω + w)−s
and call it the Barnes multiple zeta function. Here we assume that
ω = (ωi)1≥i≥r, n · ω is the ordinal inner-product and s ∈ C has the real
part which is greater than r.
This function can be continued holomorphically for s ∈ C {1, · · · , r}.
For k ∈ Z, we define
Γ(s)
Γ(s + k)
ζr(s, w; ω) =
∞
m=0
mPr,k(w; ω)
(s + k)m
m!
and call it the supermultiple gamma function of BM-type.

8. Multiple Zeta Values Multiple Gamma Functions
When we put
e−wt
r
i=1
(1 − e−ωit
) =
∞
c=−r
ar,c(w; ω)tc
,
the following is true.
Theorem 2.1 (Kawamura, 2019)
The approximation
mPr+l,k(w + a; ω, α) =
r+k+N
c=−l
al,c(a; α)mPr,k−c(w; ω)
+ O
(log w)m−1
wN+1
holds for N ∈ Z≥0 and |w| → ∞.
This is analogy of Stirling’s formula (n! has the approximation√
2πnnn
e−n
).

9. Multiple Zeta Values Multiple Gamma Functions
Theorem 2.2 (Kawamura, 2020)
The inductive series expression
mPr+1,k(w; ω0) = mPr,k(w; ω) + M1,r,k,0(ω0; w; ω0; ω)
+
∞
n=1
(mPr,k(w + nω0; ω) − M0,r,k,1(w; ω0; −; ω))
gives the analytic continuation of mPr,k(w; ω).
Subject 4
Giving the q-analog of the above theorem.
At the first, we need the q-analog of the function mPr,k. (Only known
case is m = 1 by Kawamura (2019).)

10. Multiple Zeta Values Multiple Gamma Functions
We define the q-analog of the Barnes multiple zeta functions as
ζq
r (s, w; ω) =
n∈Zr
≥0
n∈Z
(n · ω − nτ′
+ w)−s
for Im(τ′
) > 0 and q = exp(−2πi/τ′
) and the q-analog of
exp(1Pr,0(w; ω)) as
Γq
r(w; ω) = exp
∂
∂s
ζq
r (s, w; ω)
s=0
.
Theorem 2.3 (Tanaka, 2011)
Γq
r(w; ω) =
n∈Zr
≥0
(1 − qn·ω+w
)−1
.

11. Multiple Zeta Values Multiple Gamma Functions
Theorem 2.4 (Odawara, 2020)
Let Sn be the symmetric group of degree n and m(σ) be the major index
of each permutation σ ∈ Sn. Then
Γq
r(w; ω) =
∞
n=0 σ1···σr=id
r
i=1
qm(σi)ωi
n
j=1(1 − qωij)
qwn
holds.
Odawara used some method of combinatorics appeared in Garsia-Gessel
(1979).
Subject 5
Extending the above theorem to k ≥ 1.
Subject 6
Obtaining new q-analogs by substituting id on the right hand side by
general σ ∈ Sn.
Thank you for your attention.