A note on multiple zeta values in Tate algebras

0.3 MB·

by  F Pellarin

A NOTE ON MULTIPLE ZETA VALUES IN TATE ALGEBRAS
F.PELLARIN
6
1
0
2 Abstract. In this note, we shall discuss a generalization of Thakur’s multiple zeta values and
alliedobjects,intheframeworkoffunctionfieldsofpositivecharacteristic andmoreprecisely,of
n
periodsinTatealgebras.
a
J

7 1. Introduction
2
Let A=F [θ] be the ring of polynomials in an indeterminate θ with coefficients in F the finite
q q
] fieldwithq elementsandcharacteristicp,letK bethefractionfieldofAandK thecompletionof
T ∞
K at the infinity place ∞. For d≥0 an integer, we denote by A+(d) the set of monic polynomials
N
of A of degree d. Carlitz studied, in [9], the so-called Carlitz zeta values:
.
h
t ζC(n):= a−n ∈K∞, n≥1.
a
m aX∈A+
It is likely that the formal analogy of these objects with the classical zeta values
[

1 ζ(n)= i−n ∈R
v i≥1
X
8
with n integer (convergenceoccurs only if n≥2)was the main motivationfor his study. In a more
4
3 modernapproach,wecansaythatCarlitzsuggested,withhisfirstpioneeringpapers,todevelopan
7 arithmetic theory of periods over the ring F [θ] in parallel with the study of the arithmetic theory
q
0 of periods over Z.
1. Inallthe following,ifR is aring,R× denotesthe groupofthe multiplicative invertibleelements
0 of R. It was proved by Carlitz in [9] that, if n≡0 (mod q−1),
6
1 (1) ζC(n)∈K×πn,
:
v where π is the value in C =Kac (1) of a convergentinfinite product
i ∞ ∞ e
X ∞
r (2) e π :=d−(−θ)q−q1 (1−θ1−qi)−1 ∈(−θ)q−11K∞,
a
i=1
Y
uniquely defined up to thee multiplication by an element of F× = F \{0} (corresponding to the
q q
1
choice ofa root(−θ)q−1). It has been provedin a variety of ways(see [19] to see the most relevant
ones) that π is moreover transcendental over K.
Date:Janeuary, 2016.
2010 Mathematics Subject Classification. 11M38(primary).
Keywords and phrases. Multiplezetavalues,Carlitzmodule,A-harmonicsums.
1ThisisthecompletionofanalgebraicclosureofK∞.
1
2 F.PELLARIN
The element π is a fundamental period of the Carlitz exponential exp (Goss, [16, §3.2]), that
C
is, the unique surjective, entire, F -linear function
q
e exp :C →C
C ∞ ∞
of kernel πF [θ] such that its first derivative satisfies exp′ = 1 (note that, since we are in a
q C
characteristicp>0 environment,a function with constantderivative is not necessarilyC -linear).
∞
In his book [24, §5.10], Thakur also consider several variants of classical multiple zeta values
e
in the context of the Carlitzian arithmetic over the ring A. We mention here what we think is
the most relevant. For n ,...,n ∈ Z , Thakur defines, as one of the analogues of the classical
1 r ≥1
multiple zeta values in the Carlitzian setting:
1
(3) ζ (n ,...,n )= ∈K .
C 1 r an1···anr ∞
aiX∈A+ 1 r
|a1|>···>|ar|
Here, for x ∈ C×∞, we write |x| = q−v∞(x) where v∞ is the valuation of C∞ (so that v∞(θ) = −1)
and we define |0|:=0. If r =0 we further set the corresponding Thakur multiple zeta value ζ (∅)
C
to be equal to 1.
Classically, one of the reasons we could get interested in multiple zeta values is the need of
"enveloping" zeta values in the "simplest" Q-algebra possible. From Euler, it is well known that
the zeta values ζ(2),ζ(4),... all belong to the Q-algebra Q[ζ(2)]. In general, the other zeta values
arenotexpectedtobelongtothisalgebra. However,theybelongtotheQ-algebraZ ⊂Rgenerated
R
by the multiple zeta values. It is known that the product of two multiple zeta values is a Q-linear
combination of multiple zeta values, and this algebra also has a more natural structure.
WeexpectthatZR is isomorphictothe algebraQhf3,f5,...iX⊗QQ[ζ(2)], whereQhf3,f5,...iX
is the Q-algebra generated by the non-commutative words in the alphabet with letters f ,f ...
3 5
with,asaproduct,theshuffleproductX(seeBrown’s[8]). Afolkloreconjecturecomesinsupport
of this guess; the number π and the zeta values ζ(3),ζ(5),... are expected to be algebraically
independent over Q. Multiple zeta values are thus expected to provide a natural basis of this Q-
algebra. See also [17] for the definition of a Q-algebra of finite multi-zeta values which could offer
a nice realization of the algebra Qhf3,f5,...iX.
Similarly, in the Carlitzian setting we note that, after (1), the values ζ (n) with n>0 divisible
C
by q − 1 are all contained in the K-algebra K[ζ (q − 1)], which is isomorphic to K[X] for an
C
indeterminate X. However, the remaining Carlitz zeta values ζ (1),... do not belong to this
C
algebra (if q > 2). Indeed, Chang and Yu proved in [13] that π and the Carlitz zeta values ζ (n)
C
with n ≥ 1, q−1 ∤ n and p ∤ n with p the prime number dividing q are algebraically independent
(werecallthattheseauthors,inibid.,usethe powerfulalgebraicindependence methodsintroduced
e
by Papanikolas in [18]); see also [10, 11, 12].
Just as for the algebra Z , Thakur proved in [26] that the product of two multiple zeta values
R
as in (3) is a linear combination (this time with coefficients in F ) of such multiple zeta values.
p
Thakur also mentioned to the author of the present note that G. Todd’s numerical computations
have led to a good understanding of (conjectural) relations among Thakur’s multiple zeta values;
the relations are universal in a sense that is described in ibid. Compared to the classical setting,
the difficulty here is to handle the product of the so-called power sums (see later). We denote by
Z the K-sub-algebra of K generated by the multiple zeta values (3); even conjecturally, in
K∞ ∞
spite of the striking results of algebraic independence mentioned above, we know very little about
the structure of this algebra. In particular, we presently do not know what could be the analogue
structure which could play the role of the algebra Qhf3,f5,...iX in this setting.
MULTIPLE ZETA VALUES 3
Inthisnote,weshalldiscussofageneralizationofthe Thakurmultiplezetavalueswhich,sofar,
hasnocounterpartintheclassicalsetting. Forthispurpose,wenotethatAisanalgebraoverF (Z
q
isnotanalgebraoverafield). Therefore,aseriesofadvantagesoccursintheCarlitzianframework,
notably the possibility to use the tensor product over F . We consider variables t ,...,t over K
q 1 s
and we write t for the family of variables (t ,...,t ). We denote by F the field F (t ), so that
s 1 s s q s
F =F .
0 q
Inallthefollowing,ifRisaring,wedenotebyR∗theunderlyingmultiplicativemonoid(inclusive
oftheelement0). NotethatA+isamultiplicativesub-monoidofA∗. WedenotebyFacthealgebraic
q
closure of F in C .
q ∞
Definition 1. A monoid homomorphism σ : A+ →(Fac⊗ F )∗ is called a semi-character. The
q Fq s
trivial semi-character is the map 1 : A+ → {1}. Let σ be a semi-character. We say that it is of
Dirichlet type if there exist F -algebra homomorphisms
q
ρ :A→Fac⊗ F , i=1,...,s,
i q Fq s
such that σ(a) = ρ (a)···ρ (a) for all a ∈ A+. The integer s is called the length. By convention,
1 s
the semi-character 1 is the unique semi-character of Dirichlet type of length 0.
For example, setting t = t , the map χ : A+ →F [t]∗ ⊂(Fac⊗ F )∗ defined by χ (a) = a(t)
1 t q q Fq s t
(2) is a semi-character of Dirichlet type. Let ζ be an element of Fac. The map a ∈A+ 7→χ (a)=
q ζ
a(ζ) ∈ Fac ⊂ (Fac⊗ F )∗ is also a semi-character of Dirichlet type, and the same can be said if
q q Fq s
we now pickelements ζ ,...,ζ ∈Fac andconsiderthe map χ :a7→χ (a)···χ (a) (this is more
1 s q ζ ζ1 ζs
commonly called a “Dirichlet character"). The map a∈A+ 7→Fq[t]∗ which sends a to tdegθ(a) is a
semi-character,but it can be proved that it is not of Dirichlet type.
Definition 2. Let σ :A+ →(Fac⊗ F )∗ be a semi-character. The associatedtwisted power sum
q Fq s
of order k and degree d is the sum:
S (k;σ)= a−kσ(a)∈Fac⊗ K(t ).
d q Fq s
a∈XA+(d)
More generally, let σ ,...,σ be semi-characters, let n ,...,n be integers, and d a non-negative
1 r 1 r
integer. The associated multiple twisted power sum of degree d is the sum
σ σ ··· σ
S 1 2 r =S (n ;σ ) S (n ;σ )···S (n ;σ )∈Fac⊗ K(t ).
d n n ··· n d 1 1 i2 2 2 ir r r q Fq s
1 2 r
(cid:18) (cid:19) d>i2>X···>ir≥0
The integer n is called the weight and the integer r is called its depth.
i i
We can wPrite in both ways S (n;σ)=S σ . Observe also that
d d n
1 1 ··· (cid:0)1(cid:1)
S =S (n ,n ,...,n )∈K,
d n n ··· n d 1 2 r
1 2 r
(cid:18) (cid:19)
in the notations of Thakur in [26, §1.2]. We hope that all these slightly different notations will not
bother the reader.
2The"evaluationatθ=t",inother words,themapwhichsendsapolynomiala=a(θ)=a0+a1θ+···+arθr
withthecoefficients a0,...,ar ∈Fq tothepolynomiala(t)=a0+a1t+···+artr ∈Fq[θ].
4 F.PELLARIN
Definition 3. With n ,...,n ≥ 1 and semi-characters σ ,...,σ as above, we introduce the
1 r 1 r
associated multiple zeta value
σ σ ··· σ σ σ ··· σ
ζ 1 2 r := S 1 2 r ∈K ⊗ \Fac⊗ F .
C n n ··· n d n n ··· n ∞ Fq q Fq s
1 2 r 1 2 r
(cid:18) (cid:19) d≥0 (cid:18) (cid:19)
X
The sum converges in the completion of the field K ⊗ Fac ⊗ F with respect to the unique
∞ Fq q Fq s
valuationextendingthe∞-adicvaluationofK andinducingthetrivialvaluationoverFac⊗ F .
∞ q Fq s
We will say that this is the multiple zeta value associated to the matrix data
σ σ ··· σ
1 2 r .
n n ··· n
1 2 r
(cid:20) (cid:21)
The integer n is called the weight of the above multiple zeta value and the integer r is called
i i
its depth. If all the semi-characters σ ,...,σ are of Dirichlet type, then, for all 1 ≤ i ≤ r,
1 r
P
σ =ρ ···ρ forringhomomorphismsρ . Then,wesaythatthemultiplezetavalueassociated
i i,1 i,ni i,j
to the above matrix data is of Dirichlet type, and the cardinality of the set {ρ ;i,j} is called its
i,j
length.
Again note that if σ =···=σ =1, then, we can write
1 r
σ σ ··· σ
ζ 1 2 r =ζ (n ,...,n )∈K
C n n ··· n C 1 r ∞
1 2 r
(cid:18) (cid:19)
with ζ (n ,...,n ) as in (3) (of Dirichlet type, depth r and length 0). Further, let us assume that
C 1 r
r =1,n=n >0 and that σ =χ ···χ . Then, we have that
1 t1 ts
σ a(t )···a(t ) P(t )···P(t ) −1
ζ (n;σ)=ζ = 1 s = 1− 1 s ∈T×.
C C n an Pn s
(cid:18) (cid:19) aX∈A+ YP (cid:18) (cid:19)
Theseserieshavebeenintroducedin[20]andextensivelystudiedin[3,4,5]. Theproductrunsover
the irreducible polynomials of A+ and the convergence holds in the standard s-dimensional Tate
algebra, whichcanbeidentifiedwiththeC -algebraoftherigidanalyticfunctionsB(0,1)s →C ,
∞ ∞
where B(0,1) = {z ∈ C ;|z| ≤ 1}. In fact, these functions extend to entire functions Cs → C
∞ ∞ ∞
(see[3,Corollary8]). Moregenerally,ifσ ,...,σ aresemi-charactersofDirichlettypeconstructed
1 r
as monomials in the ring homomorphisms χ ,...,χ (including the trivial semi-character), the
t1 ts
multiple zeta value
σ σ ··· σ
(4) ζ 1 2 r
C n n ··· n
1 2 r
(cid:18) (cid:19)
belongs to T .
s
The following Proposition is easy to prove but the proof will appear elsewhere.
Proposition 4. With the above assumption over the semi-characters σ ,...,σ , the multiple zeta
1 r
value (4), hence of Dirichlet type and of length ≤s, extends to an entire function Cs →C .
∞ s
It is presently a work in progress of the author to show that the product of two multiple zeta
values as in Definition 3 is a linear combination, with coefficients in the field K⊗ Fac⊗ F , of
Fq q Fq s
such multiple zeta values (with the various matrices of associated data not including, necessarily,
the same semi-characters). We hope this will allow us to exhibit new multiple zeta values algebras
Z containingthealgebraZ andcollectingthealgebraicrelationsofZ infamilies
Faqc((θ\−1))⊗FqFs K∞ K∞
by specialization.
MULTIPLE ZETA VALUES 5
2. Content of the present note
Waiting for more general results, in this note we will accomplish a more modest objective, as
we will only give a few explicit examples of shuffle products of such multiple zeta values in the
following case: s=2, weight ≤2, and the semi-characters 1,σ,ψ and σψ of Dirichlet type, where
σ :a7→a(t )∈F [t ,t ], ψ :a7→a(t )∈F [t ,t ], (σψ)(a)=σ(a)ψ(a)=a(t )a(t ).
1 q 1 2 2 q 1 2 1 2
As an advantage of our explicit and restrictive viewpoint, we will see beautiful formulas dropping
outfromthisnewtheorythatwewillapplytosomenewpropertiesoftheso-called“Bernoulli-Goss"
polynomials.
The matrix data we are going to handle are:
Four in weight 1.
c1 cσ cψ cσψ
, , , .
1 1 1 1
(cid:20) (cid:21) (cid:20) (cid:21) (cid:20) (cid:21) (cid:20) (cid:21)
Four in weight 2 depth 1.
c1 cσ cψ cσψ
, , , .
2 2 2 2
(cid:20) (cid:21) (cid:20) (cid:21) (cid:20) (cid:21) (cid:20) (cid:21)
Nine in weight 2 depth 2.
1 1 σ 1 1 σ ψ 1 1 ψ
, , , , ,
1 1 1 1 1 1 1 1 1 1
(cid:20) (cid:21) (cid:20) (cid:21) (cid:20) (cid:21) (cid:20) (cid:21) (cid:20) (cid:21)
σψ 1 σ ψ ψ σ 1 σψ
, , , .
1 1 1 1 1 1 1 1
(cid:20) (cid:21) (cid:20) (cid:21) (cid:20) (cid:21) (cid:20) (cid:21)
We shall show the following Theorem, which provides, taking into account the above tables, a
completepictureofalltheproductsoftwoweightonemultiplezetavaluesintherestrictivecontext
we have prefixed (in two variables t ,t , and with the semi-characters 1,σ,ψ and σψ), unveiling
1 2
partly an extremely complex and mysterious algebra structure. From now on, we suppose that
q > 2. All the arguments presented below under this restriction can be also developed in the case
q =2 with appropriate modifications, but we refrain from giving full details here.
Theorem 5. The following formulas hold.
(1) ζ (1)2 =ζ (2)+2ζ (1,1),
C C C
σ 1
(2) ζ (1;σ)ζ (1)=ζ (2;σ)+ζ ,
C C C C 1 1
(cid:18) (cid:19)
ψ 1
(3) ζ (1;ψ)ζ (1)=ζ (2;ψ)+ζ ,
C C C C 1 1
(cid:18) (cid:19)
(4) ζ (1;σ)ζ (1;ψ)=ζ (2;σψ),
C C C
σ ψ ψ σ 1 σψ σψ 1
(5) ζ (1;σψ)ζ (1)=ζ (2;σψ)−ζ −ζ +ζ +ζ .
C C C C 1 1 C 1 1 C 1 1 C 1 1
(cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19)
We observe that the formula (1) can be found in Thakur’s [24, Theorem 5.10.13]. Further, due
to the symmetry of the roles of t and t , the formulas (2) and (3) are equivalent. Moreover, the
1 2
formula(4)isinfactwellknown(seePerkins’[22]formoregeneralformulasofthistype). However,
we will give a proof of this in the spirit of multiple zeta values. The formulas (2) is, on the other
side, new, as far as we can see.
6 F.PELLARIN
3. Twisted power sums
Weneedafewtoolsinordertoobtainourformulas;moreprecisely,wehavetoimproveourskill
in computing twisted powers sums. For this purpose, we are going to use the tools introduced in
the recent preprint of Perkins and the author [21]; we are going to use for a while the notations of
this reference. Let s be an integer ≥0. We set, for an integer d≥0:
a(t )···a(t )
1 s
S (n;s)= ∈K[t ].
d an s
a∈XA+(d)
We are thus considering a special case of Definition 2. We also set, for d≥0, F (n;s)=0 and
0
d−1
F (n;s)= S (n;s)∈K[t ],
d d s
i=0
X
so that
P(t )···P(t ) −1
lim F (n;s)=ζ (n;s):= 1− 1 s ∈T×,
d→∞ d C Pn s
P (cid:18) (cid:19)
Y
where the product runs over the irreducible polynomials of A+ (in general, all along this note,
empty sums are by conventionequal to zero). We are using the notation of [21]. In particular,if σ
is the semi-character χ ···χ (of Dirichlet type), then, the comparison between the notations of
t1 ts
ibid. and those of the present note are:
S (n;s)=S (n;σ), ζ (n;s)=ζ (n;σ).
d d C C
Itiseasytoshowthat,if0≤s′ <s,S (n;s′)isthecoefficientof(t ···t )d inF (n;s). We
d s′+1 s d+1
define inductively l =1 and l =(θ−θqi)l , and we set l =0 for n>0. We denote by b (Y)
0 i i−1 −n i
the product (Y −θ)···(Y −θqi−1)∈A[Y] (for an indeterminate Y) if i>0 and we set b (Y)=1.
0
We also write m = ⌊s−1⌋ (the brackets denote the integer part so that m is the biggest integer
q−1
≤ s−1). We set
q−1
b (t )···b (t )
d−m 1 d−m s
Π = ∈K[t ], d≥max{1,m}.
s,d l s
d−1
Now, we quote [21, Theorem 1]:
Theorem 6. For all integers s≥1, such that s ≡1 (mod q−1), there exists a non-zero rational
fraction H ∈K(Y,t ) such that, for all d≥m, the following identity holds:
s s
F (1;s)=Π H | .
d s,d s Y=θqd−m
If s=1, we have the explicit formula
1
H = .
1
t −θ
1
Further, if s=1+m(q−1) for an integer m>0, then the fraction H is a polynomial of A[Y,t ]
s s
with the following properties:
(1) For all i, deg (H )=m−1,
ti s
(2) deg (H )= qm−1 −m.
Y s q−1
The polynomial H is uniquely determined by these properties.
s
MULTIPLE ZETA VALUES 7
We apply this Theorem to compute explicitly the twisted power sums associatedto the data we
have chosen. For this, it suffices to choose s = q. In this case m = 1 and the Polynomial H of
q
Theorem 6 has degree 0 in Y as well as in t ,...,t . From [21, §2.6] we deduce that H = 1 (the
1 q q
same result is given as an example in Florent Demeslay’s thesis [14]). In particular, to compute
most of the twisted power sums associated to our data it suffices to analyze the polynomials
b (t )···b (t )
d 1 d q
(5) F (1;q)= .
d+1
l
d
The coefficients of (t ···t )d, (t ···t )d and (t ···t )d in F (1;q) are easily computed, and we
3 q 2 q 1 q d+1
get, for all d≥0 (note that these are well known formulas; see [22]):
1
(6) S (1;0) = ,
d
l
d
b (t )
d 1
(7) S (1;1) = ,
d
l
d
b (t )b (t )
d 1 d 2
(8) S (1;2) = .
d
l
d
To compute S (2;0),S (2,1),S (2,2) (3) we observe that, replacing θ with θq in (5):
d d d
b (t )···b (t )
d+1 1 d+1 q
F (q;q)= .
d+1 lq(t −θ)···(t −θ)
d 1 q
Note that the former is a polynomial in t , written as a reducible fraction. We get that
s
b (t )···b (t ) b (t )b (t )
d+1 1 d+1 q d+1 1 d+1 2
F (2;2)= = .
d+1 lq(t −θ)···(t −θ) l2(t −θ)(t −θ)
d 1 q (cid:12)t3=···=tq=θ d 1 2
(cid:12)
Calculatingthe coefficientsoftd2 and(t1t2)d, andsu(cid:12)(cid:12)btractingFd+1(2;2)−Fd(2;2),weeasilyobtain
the formulas, valid for d≥0 (the first one is well known):
1
S (2;0) = ,
d l2
d
b (t )
(9) S (2;1) = d 1 (t −θqd),
d (t −θ)l2 1
1 d
b (t )b (t )
S (2;2) = d 1 d 2 (t t −θqd(t +t )+2θ1+qd −θ2)
d (t −θ)(t −θ)l2 1 2 1 2
1 2 d
b (t )b (t )
(10) = d 1 d 2 [(t −θ)(t −θ)+(t −θ)(θ−θqd)+(t −θ)(θ−θqd)].
(t −θ)(t −θ)l2 1 2 1 2
1 2 d
WewillalsoneedthenextLemmawhichisalsowellknown,whereτ :A[t]→A[t]istheF [t]-linear
q
endomorphism such that τ(θ)=θq.
Lemma 7. We have the following formula, which holds in A[t].
d
b (t)
i
τ(b (t))=l , d≥0.
d d
l
i
i=0
X
3Inthenotations of[21];inournote, weshouldwriteSd(2;1),Sd(2;σ),Sd(2;σψ).
8 F.PELLARIN
Proof. We recallthe proofforconvenienceofthe reader. We proceedby inductionoverd. If d=0,
the formula is obvious. If d>0, it suffices to show that
d
b (t)=(t−θ)l l−1b (t).
d+1 d i i
i=0
X
Now, we compute easily, by using the induction hypothesis:
b (t) = (t−θ+θ−θqd)b
d+2 d+1
= (θ−θqd)b +(t−θ)b
d+1 d+1
d
= (t−θ)l (θ−θqd) l−1b (t)+l−1 b (t)
d i i d+1 d+1
!
i=0
X
d+1
= (t−θ)l l−1b (t).
d+1 i i
i=0
X
(cid:3)
4. Proof of Theorem 5
Proof of Theorem 5, (2). We compute, by using (6) and (7):
b (t )
S (1;σ)S (1;1)= d 1 .
d d l2
d
On the other hand, we have seen in (9) that
b (t )t −θqd τ(b (t ))
d 1 1 d 1
S (2;σ)= = ,
d l2 t −θ l2
d 1 d
where τ(b (t )) has the obvious meaning. We compute (the third identity follows from Lemma 7):
d 1
1 σ d−1
S = S (1;1) S (1;σ)
d 1 1 d i
(cid:18) (cid:19) i=0
X
d−1
1
= l−1b (t )
l i i 1
d
i=0
X
τ(b (t ))
d−1 1
=
l l
d d−1
b (t )θ−θqd
d 1
= .
l2 t −θ
d 1
Combining the above formulas, we see that
1 σ
(11) S (1;σ)S (1;1)=S (2;σ)−S .
d d d d 1 1
(cid:18) (cid:19)
MULTIPLE ZETA VALUES 9
With the obvious meaning of some new notations introduced below, we deduce:
d−1 d−1
F (1;σ)F (1;1) = S (1;σ) S (1;1)
d d i j
i=0 j=0
X X
1 σ σ 1 d−1
= F +F + S (1;σ)S (1;1)
d 1 1 d 1 1 d d
(cid:18) (cid:19) (cid:18) (cid:19) i=0
X
1 σ σ 1 d−1 1 σ
= F +F + S (2;σ)−S
d 1 1 d 1 1 i i 1 1
(cid:18) (cid:19) (cid:18) (cid:19) i=0(cid:18) (cid:18) (cid:19)(cid:19)
X
1 σ σ 1 1 σ
= F +F −F +F (2;σ)
d 1 1 d 1 1 d 1 1 d
(cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19)
σ 1
= F +F (2;σ),
d 1 1 d
(cid:18) (cid:19)
where we have used (11) in the third equality. We rewrite the resulting formula:
σ 1
(12) F (1;σ)F (1;1)=F +F (2;σ).
d d d 1 1 d
(cid:18) (cid:19)
Taking the limit d→∞ in (12) we obtain the required multiple zeta identity. (cid:3)
The above also implies the formula (3) of Theorem 5 by interchanging t and t . To continue,
1 2
we notice the next Lemma.
Lemma 8. We have that
bd(t1)bd(ts) ψ σ σ ψ
S (2;σψ)= +S +S .
d (t −θ)(t −θ)l2 d 1 1 d 1 1
1 2 d (cid:18) (cid:19) (cid:18) (cid:19)
Proof. We compute:
d−1
σ ψ
S = S (1;σ) S (1;ψ)
d 1 1 d i
(cid:18) (cid:19) i=0
X
d−1
b (t ) b (t )
d 1 i 2
=
l l
d i
i=0
X
b (t )b (t )
= d 1 d s [(t −θ)(θ−θqd)],
(t −θ)(t −θ)l2 1
1 2 d
in virtue of (7) and Lemma 7. Similarly, we have
S ψ σ = bd(t1)bd(ts) [(t −θ)(θ−θqd)].
d 1 1 (t −θ)(t −θ)l2 2
(cid:18) (cid:19) 1 2 d
The lemma follows from (10). (cid:3)
Proof of Theorem 5, (4). As we havealreadymentioned, this is a wellknownformulabut we want
to provide a new proof. We note, by (7), that
b (t )b (t )
d 1 d 2
S (1;σ)S (1;ψ)= [(t −θ)(t −θ)].
d d (t −θ)(t −θ)l2 1 2
1 2 d
10 F.PELLARIN
Hence, Lemma 8 implies the formula
ψ σ σ ψ
(13) S (1;σ)S (1;ψ)=S (2;σψ)−S −S .
d d d d 1 1 d 1 1
(cid:18) (cid:19) (cid:18) (cid:19)
We deduce:
d−1
ψ σ σ ψ
F (1;σ)F (1;ψ) = F +F + S (1;σ)S (1;ψ)
d d d 1 1 d 1 1 d d
(cid:18) (cid:19) (cid:18) (cid:19) i=0
X
d−1
ψ σ σ ψ ψ σ σ ψ
= F +F + S (2;σψ)−S −S
d 1 1 d 1 1 d d 1 1 d 1 1
(cid:18) (cid:19) (cid:18) (cid:19) i=0(cid:18) (cid:18) (cid:19) (cid:18) (cid:19)(cid:19)
X
ψ σ ψ σ σ ψ σ ψ
= F −F +F −F +F (2;σψ)
d 1 1 d 1 1 d 1 1 d 1 1 d
(cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19)
= F (2;σψ),
d
where we have applied the formula (13) in the third equality. The formula of the theorem follows
by letting d→∞. (cid:3)
Proof of Theorem 5, (5). The identities (6) and (8) imply that
b (t )b (t )
S (1;1)S (1;σψ)= d 1 d 2 [(t −θ)(t −θ)].
d d (t −θ)(t −θ)l2 1 2
1 2 d
Lemma 8 then implies that also:
ψ σ σ ψ
(14) S (1;1)S (1;σψ)=S (2;σψ)−S −S .
d d d d 1 1 d 1 1
(cid:18) (cid:19) (cid:18) (cid:19)
We deduce:
1 σψ σψ 1 d−1
F (1;1)F (1;σψ) = F +F + S (1;σ)S (1;ψ)
d d d 1 1 d 1 1 d d
(cid:18) (cid:19) (cid:18) (cid:19) i=0
X
1 σψ σψ 1 d−1 ψ σ σ ψ
= F +F + S (2;σψ)−S −S
d 1 1 d 1 1 d d 1 1 d 1 1
(cid:18) (cid:19) (cid:18) (cid:19) i=0(cid:18) (cid:18) (cid:19) (cid:18) (cid:19)(cid:19)
X
1 σψ σψ 1 ψ σ σ ψ
= F (2;σψ)+F +F −F −F ,
d d 1 1 d 1 1 d 1 1 d 1 1
(cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19)
so we have reached the formula
1 σψ σψ 1 ψ σ σ ψ
(15) F (1;1)F (1;σψ)=F (2;σψ)+F +F −F −F .
d d d d 1 1 d 1 1 d 1 1 d 1 1
(cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19)
Letting d tend to ∞ in (15) concludes the proof. (cid:3)
Remark 9. In fact, it is trivial that
1 σψ σψ 1 ψ σ σ ψ
ζ (1)ζ (1;σψ)−ζ (1;σ)ζ (1,ψ)=ζ +ζ −ζ −ζ ,
C C C C d 1 1 d 1 1 d 1 1 d 1 1
(cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19)
and that the corresponding identity for the sums F holds as well. Indeed, setting α =S (1;1)=
d i i
l−1, β =S (1;σ)=b (t )l−1, γ =S (1;ψ)=b (t )l−1 and δ =S (1;σψ)=b (t )b (t )l−1, we see
i i i i 1 i i i i 2 i i i i 1 i 2 i