article04. 383KB Jun 04 2011 12:06:51 AM
Journal de Th´eorie des Nombres
de Bordeaux 19 (2007), 595–640
Analytic and combinatoric aspects of Hurwitz
polyzˆ
etas
par Jean-Yves ENJALBERT et HOANG NGOC MINH
´sume
´. Dans ce travail, un codage symbolique des s´eries g´en´eraRe
trices de Dirichlet g´en´eralis´ees est obtenu par les techniques combinatoires des s´eries formelles en variables non-commutative. Il
permet d’expliciter les s´eries g´en´eratrices de Dirichlet g´en´eralis´ees
’p´eriodiques’ – donc notamment les polyzˆetas color´es – comme
combinaison lin´eaire de polyzˆetas de Hurwitz. De plus, la version non commutative du th´eor`eme de convolution nous fournit
une repr´esentation int´egrale des s´eries g´en´eratrices de Dirichlet
g´en´eralis´ees. Celle-ci nous permet de prolonger les polyzˆetas de
Hurwitz comme des fonctions m´eromorphes `a plusieurs variables.
Abstract. In this work, a symbolic encoding of generalized Dirichlet generating series is found thanks to combinatorial techniques of noncommutative rational power series. This enables to
explicit periodic generalized Dirichlet generating series – particularly the coloured polyzˆetas – as linear combinations of Hurwitz
polyzˆetas. Moreover, the noncommutative version of the convolution theorem gives easily rise to an integral representation of
Hurwitz polyzˆetas. This representation enables us to build the
analytic continuation of Hurwitz polyzˆetas as multivariate meromorphic functions.
Contents
1. Introduction
2. Background
2.1. The univariate zˆeta function
2.2. Symbolic computations on special functions
3. Generalized D.g.s. and their integral representation
3.1. Definitions and basic properties
3.2. Periodic D.g.s and Hurwitz polyzˆetas
3.3. Integral representation of Hurwitz polyzˆetas
4. Analytic continuation and structure of poles
4.1. Analytic continuation of Hurwitz polyzˆetas
Manuscrit re¸cu le 28 juillet 2006.
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Jean-Yves Enjalbert, Hoang Ngoc Minh
4.2. Structure of poles
5. A translation relation and prospects
5.1. The translation relation of Hurwitz polyzˆetas
5.2. Translation equality and analytic continuation
6. Conclusion
References
Appendix
623
632
632
633
634
635
636
1. Introduction
Technologies of generating series (g.s.) and of generating functions (g.f.)
on one variable are basic tools required for asymptotic, complexity and
probabilistic analysis of combinatorial and discrete structures. These technologies play a central role in several application areas like in arithmetics,
statistic physics, algorithmic information theory [8], analytic combinatorics
[10] and analysis of algorithms, . . . . This makes the natural link between
these structures and complex analysis and exploits intensively the association between ordinary generating functions (o.g.f.) and exponential generating functions (e.g.f.).
In this paper, we will study the association of o.g.f. and of a generalization of Dirichlet generating functions (D.g.f.) [22]. The value at 1 of a
D.g.f is a Dirichlet generating serie (D.g.s); for example Riemann polyzˆetas
[26], defined by
X
1
(1)
,
ζ(s1 , . . . , sr ) =
s1
n1 . . . nrsr
n >...>n
1
r
are D.g.s and Hurwitz polyzˆetas [27], defined by
X
1
(2) ζ(s1 , . . . , sr ; t1 , . . . , tr ) =
,
s
1
(n1 − t1 ) . . . (nr − tr )sr
n >...>n
1
r
are Parametrized D.g.s. Reciprocally, we show that ’periodic’ Parametrized
D.g.s. (and so periodic D.g.s.) can be written as a finite combination of
Hurwitz polyzˆetas (see Proposition 3.3). It is notably the case for the
coloured polyzˆetas as described by Equality (49). This equality enables us
to give the expression of any Hurwitz polyzˆeta as a finite combination of
coloured polyzˆetas (Proposition 3.4).
On the other hand, the moment sum of index l of basic sign algorithm
is given by [17]
X 1
2
1
(3)
.
ρ(l) = 1 + l +
ζ(2l)
2
cl dl
dnr >0
One thus has a encoding of the polyzˆeta ζ(s) in term of iterated integrals.
Analytic and combinatoric aspects of Hurwitz polyzˆetas
605
For 1 ≤ r ≤ N , let s = (s1 , . . . , sr ). The finite polyzˆetas (or multiple
harmonic sums) ζN (s) is defined as (see [30])
X
1
(33)
.
ζN (s) =
ns11 . . . nsrr
N ≥n1 >...>nr >0
and ζN (s) = 0 for 1 ≤ N < r. For r = 0, we put ζ0 (s) = 0 and ζN (s) = 1,
for any N ≥ 1. These can be obtained as the specialization in the quasimonomial functions (see [38])
X
(34)
Ms (t) =
tsn11 · · · tsnrr
n1 >...>nr >0
at ti = 1/i if 1 ≤ i ≤ k and ti = 0 if i > k. Let us extend linearly the
notation Ms when s is a linear combination of compositions. If r (resp. s)
is a composition of depth r and weight p (resp. of depth s and weight q),
Mr s is a quasi-monomial function of depth r + s and of weight p + q, and
one has
(35)
Ms
r
= M s Mr .
Therefore,
(36)
ζN (s
r) = ζN (s)ζN (r).
For s1 > 1, the limit when N → ∞ of ζN (s) is nothing but the polyzˆeta
ζ(s) [41], and thus by an Abel’s theorem,
X
1
(37)
= ζ(s).
lim ζN (s) = lim Lis (z) =
s1
z→1
N →∞
n1 . . . nsrr
n1 >...>nr >0
The asymptotic expansion of ζN (s), for N → ∞, was already treated by use
of Euler-Mac Laurin summation P
[12] or by use of a full singular expansion,
at z = 1, of its generating series N ≥0 ζN (s)z N [11].
Let us come back to Equality (37). On the one hand, the polyzˆetas ζ(s),
as limits of the polygarithms Lis (z), verify the shuffle relation. On the
other hand, the polyzˆetas ζ(s) can be obtained also as the specialization of
the quasi-monomial functions at ti = 1/i, for i ≥ 1. So, if two compositions
r, s correspond respectively to the two convergent words l1 , l2 , then we get
on the convergent polyzˆetas the three families of relations (see [23]) :
(38)
ζ(l1 ⊔⊔ l2 ) = ζ(l1 )ζ(l2 ),
(39)
ζ(l1
(40)
ζ(x1 ⊔⊔ l1 − x1
l2 ) = ζ(l1 )ζ(l2 ),
l1 ) = 0.
The first two families are called the double shuffle structure. The third
family involves the polynomials x1 ⊔⊔ l1 − x1 l1 that are convergent, even
when the two sums ζ(x1 ⊔⊔ l1 ) and ζ(x1 l1 ) are divergent. These divergent
terms can be regularized syntactically with respect to the associated shuffle
Jean-Yves Enjalbert, Hoang Ngoc Minh
606
products as explained in [27, 28]. Note that we only have to study that
point over a generator family : the set of Lyndon word [23].
3. Generalized D.g.s. and their integral representation
By now, r stands for a positive integer. Let T = {t1 , · · · , tr } and T =
{t1 , · · · , tr } two families of parameters connected by the change of variables
t = t1 − t2 ,
t 1 = t 1 + · · · + tr ,
1
t2 = t2 − t3 ,
t 2 = t 2 + · · · + tr ,
(41)
⇐⇒
..
..
.
.
t =t .
t =t .
r
r
r
r
For i = 1..m, let us consider the locally integrable function1 Fi and the
following associated differential 1-forms
(42)
ω0 (z) =
dz
dz
dz
.
, ωi (z) = Fi (z) , ωi,ti (z) = Fi (z)
z
z
z 1+ti
For any composition s = (s1 , . . . , sr ) and for any formal variables t1 , . . . , tr ,
we use the short notations t for (t1 , . . . , tr ), (s; t) for (s1 ; t1 ), . . . , (sr ; tr )
and F for (Fi1 , . . . , Fir ).
3.1. Definitions and basic properties. Here, we consider the case
X
(43)
Fi (z) =
fi,n z n , for i = 1..m.
n≥1
Definition. The D.g.s. associated to {Fi }i=1..m given in (43) is the sum
X
fi1 ,n1 −n2 . . . fir−1 ,nr−1 −nr fir ,nr
Di(F; s) =
.
ns11 . . . nsrr
n1 >...>nr >0
We get the following iterated integral interpretations :
Proposition 3.1 ([22]). Let i1 , . . . , ir = 1, .., m and let w = xs01 −1 xi1 . . .
. . . xs0r −1 xir associated to s. With the differential forms ω0 , ω1 , . . . , ωm of
(42), we get
X fi1 ,n1 −n2 . . . fir−1 ,nr−1 −nr fir ,nr
z n1 .
α0z (w) =
s1
sr
n
.
.
.
n
r
1
n >...>n
1
r
We call Generalized D.g.f. Dis (F|z) associated to {Fi }i=1..m given in (43)
this previous iterated integral. In particular, Di(F; s) = Dis (F|1) = α01 (w).
1Note that, here F is not necessary the function (1 − z)−1 except when m = 1 and it is in
1
the case of (29).
Analytic and combinatoric aspects of Hurwitz polyzˆetas
607
Example. Let z = (z1 , . . . , zr ). For the o.g.f.
zi z
Fi (z) =
with |zi | < 1, for i = 1..m,
1 − zi z
with the differential forms ω0 , ω1 , . . . , ωm , we have [22, 27]
α0z (w)
=
n
X
n1 >...>nr >0
−nr nr
zr n1
r−1
z1n1 −n2 . . . zr−1
ns11 . . . nsrr
z .
and Di(F1 , . . . , Fr ; s) is nothing but the multiple polylogarithm [4, 19]
X
z1n1 (z2 /z1 )n2 . . . (zr /zr−1 )nr
Lis (z) = Lis (z1 , . . . , zr ) =
.
ns11 . . . nrsr
n1 >...>nr >0
Definition. The Parametrized D.g.s. associated to {Fi }i=1..m given in (43)
is the sum
X
fi1 ,n1 −n2 . . . fir−1 ,nr−1 −nr fir ,nr
.
Di(Ft ; s) =
(n1 − t1 )s1 . . . (nr − tr )sr
n1 >...>nr >0
Proposition 3.2 ([22]). Let i1 , . . . , ir = 1, .., m and w = xs01 −1 xi1 . . .
. . . xs0r −1 xir associated to s. With the differential forms ω0 , ω1,t1 , . . . , ωm,tm
of (42), we get
X fi1 ,n1 −n2 . . . fir−1 ,nr−1 −nr fir ,nr
α0z (w) =
z n1 −t1 .
s1 . . . (n − t )sr
(n
−
t
)
1
1
r
r
n >...>n
1
r
We call Parametrized D.g.f. associated to {Fi }i=1..m given in (43) the
previous iterated integral. In particular, Di(Ft ; s) = α01 (w).
Example. Let z = (z1 , . . . , zr ). For the o.g.f.
zi z
with |zi | < 1, for i = 1..m,
Fi (z) =
1 − zi z
with the differential forms ω0 , ω1,t1 , . . . , ωm,tm , we have [22, 27]
α0z (w) =
X
n1 >...>nr
n
−n
r nr
r−1
z1n1 −n2 . . . zr−1
zr
z n1 −t1 ,
s
(n1 − t1 ) 1 . . . (nr − tr )sr
and Di(F1 , . . . , Fr ; s) is nothing but the multiple Lerch function
X
z1n1 (z2 /z1 )n2 . . . (zr /zr−1 )nr
α01 (w) =
.
(n1 − t1 )s1 . . . (nr − tr )sr
n1 >...>nr >0
Therefore, as iterated integral, the generalized and the parametrized
D.g.f. and the generalized Lerch function verify the shuffle relation (38)
over the alphabet {x0 , . . . , xm } [22]. In particular, their special values verify
this kind of relation.
Jean-Yves Enjalbert, Hoang Ngoc Minh
608
3.2. Periodic D.g.s and Hurwitz polyzˆ
etas.
Definition. The generalized Lerch function is defined as follows
Φs (z; t) =
X
n1 >···>nr >0
z n1
(n1 − t1 )s1 . . . (nr − tr )sr
for |z| < 1.
The associated Hurwitz polyzˆeta is defined by
ζ(s; t) =
X
n1 >···>nr >0
(n1 − t1
)s1
1
= Φs (1; t).
. . . (nr − tr )sr
With the notation of Example 3.1, z1 = . . . = zr and for t fixed, the
generalized Lerch function appear as an iterated integral and so verify the
shuffle product. Consequently, as z tends to 1, for any convergent compositions s and s′ ,
(44)
ζ(s ⊔⊔ s′ ; t) = ζ(s; t)ζ(s′ ; t).
Note that the Hurwitz polyzˆetas can be also encoded by noncommutative
rational series
(45)
xs01 −1 (t1 x0 )∗s1 x1 . . . xs0r −1 (tr x0 )∗sr x1
as given in [27] with two differential forms ω0 and ω1 .
These polyzˆetas contain divergent terms which are looked at via syntactic regularizations of divergent Hurwitz polyzˆetas with respect to the
associated shuffle products as in [27, 28]. Note also that if t1 = t2 = · · · = 0
then Φs (z; 0) = Lis (z) and ζ(s; 0) = ζ(s).
Proposition 3.3. If the {fi,n }n≥1 for i = 1..N , are periodic of the same
period K then with the differential forms ω0 , ω1,t1 , . . . , ωm,tm , we have
Di(Ft ; s) =
ζ[(s1 ;
1
K
Pr
t1 −
l=1 sl
Pr
fi1 ,b1 . . . fir ,br
b1 ,...,br =0
l=1 bl
K
K−1
X
), . . . , (sr ;
tr − br
)].
K
Proof. Let us sketch the proof in case of r = 2 :
Di(Ft ; s) =
X
n1 >n2 >0
=
X
m1 ,m2 >0
fi1 ,n1 −n2 fi2 ,n2
(n1 − t1 )s1 (n2 − t2 )s2
fi1 ,m1 fi2 ,m2
.
(m1 + m2 − t1 )s1 (m2 − t2 )s2
Analytic and combinatoric aspects of Hurwitz polyzˆetas
609
The assumption of periodicity gives fi,k+K = fi,k for all k ≥ 1 and i = 1, 2.
Therefore,
Di(Ft ; s)
=
K−1
X
X
b1 ,b2 =0 m1 ,m2 ≥1
=
K−1
X
fi1 ,b1 fi2 ,b2
X
n1 >n2 >0
b1 ,b2 =0
=
fi1 ,m1 K+b1 fi2 ,m2 K+b2
[(m1 + m2 )K + b1 + b2 − t1 ]s1 (m2 K + b2 − t2 )s2
1
K s1 +s2
K−1
X
b1 ,b2 =0
1
(n1 K + b1 + b2 − t1 )s1 (n2 K + b2 − t2 )s2
t2 − b 2
t1 − (b1 + b2 )
fi1 ,b1 fi2 ,b2 ζ s1 ;
, s2 ;
.
K
K
The proof is easily generalized for any positive integer r.
In other words, if {fi,n }n≥1 are periodic of the same period K then generalized D.g.s. is a linear combination of Hurwitz polyzˆetas. In particular,
if fi1 ,n = . . . = fir ,n = 1, for n ≥ 1, then one can generalize the well known
result for the Riemann and Hurwitz zˆeta functions.
Corollary 3.1. For K ∈ N+ , one has
ζ(s; t) =
1
K
Pr
l=1 sl
K−1
X
ζ[(s1 ;
t1 −
b1 ,...,br =0
Pr
l=1 bl
K
), . . . , (sr ;
tr − b r
)].
K
Now let q = e2iπ/m be a m-th primitive root of unity. For any integers i1 , . . . , ir , we will use the notation q i for q i1 , . . . , q ir . Let Q(q) be the
cyclotomic field generated by q.
Let us introduce also the o.g.f.
∞
X
Fi (z) =
(46)
q in z n , for i = 1..m.
n=1
The coloured polyzˆetas are a particular case of periodic D.g.s. Di(F; s) and
they are defined as follows (for 0 ≤ i1 < . . . < ir ≤ m − 1) [3, 4, 19]
X
s
q i1 n1 . . . q ir nr
ζ i =
(47)
.
ns11 . . . nsrr
q
n1 >...>nr >0
In other terms, these can be obtained as a special value at z = 1 of the
following D.g.s. associated to {Fi }i=1,..,m given in (46)
s
ζ i1 i2 −i1
(48)
= α01 (xs01 −1 xi1 . . . x0sr −1 xir ).
q ,q
. . . , q ir −ir−1
So these verify the first shuffle relation (38) over the alphabet {x0 , . . . , xm }.
By Proposition 3.3, the periodicity of {q n }n≥1 enables then to express the
610
Jean-Yves Enjalbert, Hoang Ngoc Minh
coloured polyzˆetas as linear combinations of Hurwitz polyzˆetas with coefficients in Q(q) [27] :
m−1
Pr
X
a
1
s
q l=1 il al ζ(s; − ).
(49) ζ i = Pr s
l
l=1
q
m
m
a −a ,...,a
−ar ,ar =0
1
2
r−1
Here, a/m stands
for (a1 /m, . . . , ar /m). Consequently, these coloured
s
polyzˆetas ζ qi (with 0 ≤ ik ≤ m − 1) are linear combinations, with coefficients in Q(q), of the rational parameters Hurwitz polyzˆetas ζ(s; −a/m)
with a1 − a2 , . . . , ar−1 − ar , ar = 1..m. They can also be viewed as the
evaluations, at the m-th primitive root of unit q of some commutative
polynomials on the Q-algebra of Hurwitz polyzˆetas. The reader can find
the shuffle structures of Hurwitz polyzˆetas in [27] inducing then the shuffle
structures of Riemann polyzˆetas. Conversely, by the distribution formula,
we have a partial result [30] :
Proposition 3.4. Let a = (a1 , . . . , ar ) be a composition. If the parameters
aj satisfy the conditions 1 ≤ a1 − a2 , . . . , ar−1 − ar , ar < m then
m−1
Pr
P
X
a
s
sl −r
− rl=1 il al
l=1
q
ζ(s; − ) = m
ζ i .
m
q
i1 ,...,ir =0
Proof. Let us set s and let
s1 , s2 , . . . , sr
,
f (i) = i1 ...+ir ζ i1 i2 −i1
q
q ,q
. . . , q ir −ir−1
g(c) = ζ (s1 , . . . , sr ; −a1 /m, . . . , −ar /m) ,
1
with aj = (cj + 1) + (cj+1 + 1) . . . + (cr + 1), for j = 1 . . . r. Let us consider
the lexicographical order on the index set
I = {i = (i1 , . . . , ir ) ∈ {0, . . . m − 1}r }
and on the index set
C = {c = (c1 , . . . , cr ) ∈ {0, . . . m − 1}r }.
Then f (i) i∈I (resp. g(c) c∈C ) can be viewed as the entries of a column
vector F (resp. G) of dimension
mr.
Let M (q) be the matrix q ij 0≤i,j≤m−1 of determinant
Y
(50)
(q i − q j ) 6= 0.
det M (q) =
0≤i 1, σ1 + σ2 > 1, . . . , σ1 + · · · + σr > 1. Either, if the power
series Fi (e−z ) has a suitable asymptotic expansion at 0 and at ∞ then the
Dirichlet generating functions associated to the sequences {fi,n }n≥1,i=1,..,N
are meromorphic functions in Cr and are given by
Proposition 3.5. The function Dis (F|z) can be represented as
Qj
Z
r
Y
l=1 ul ) duj
sj −1 1 Fij (z
Dis (F|z) =
log
( )
uj
Γ(sj )
uj
[0,1]r
j=1
Z
where ur+1 = 1.
Pj
r
Y
Fij (ze− l=1 ul ) duj
=
1−s .
Γ(sj )
Rr+
u j
j=1
j
Proof. The proof can be obtained by induction on r and by use of Lemma
3.1.
Therefore, by taking z = 1, one finally gets
Corollary 3.2. The generalized D.g.s. can be represented as
Qj
Z
r
Y
sj −1 1 Fij ( l=1 ul ) duj
Di(F; s) =
log
( )
uj
Γ(sj )
uj
[0,1]r
j=1
Z
where ur+1 = 1.
Pj
r
Y
Fij (e− l=1 ul ) duj
=
1−s .
Γ(sj )
Rr+
u j
j=1
j
For z = 1, the polylogarithmic transformation corresponds to the Mellin
transformation and onePobtains the D.g.s. Di(F/Γ(s); s) associated to
F (τ )/Γ(s) with F (τ ) = k≥1 fk q k and q = e−τ .
Example. Let s′ = (s2 , . . . , sr ) and F′ = (Fi2 , . . . , Fir ). By Lemma 3.1, the
Dirichlet function Dis (F|z) is a polylogarithmic transform of the function
Fi1 (z) Dis′ (F′ |z) :
Z 1
1 Fi (zt) Dis′ (F′ |zt) dt
Dis (F|z) =
logs1 −1 ( ) 1
t
Γ(s1 )
t
0
Z ∞
′
−t
−t
Fi1 (ze ) Dis′ (F |ze ) dt
=
.
Γ(s1 )
t1−s1
0
and the D.g.s. Di(F; s) is a Mellin transform of Fi1 (e−z ) Dis′ (F′ |e−z )/Γ(s1 ).
Therefore, by Corollary 3.2, Di(F; s) can be viewed as a multiple Mellin
Pj
Q
transform of the multivariate function rj=1 Fij (e− l=1 ul )/Γ(sj ).
Jean-Yves Enjalbert, Hoang Ngoc Minh
614
Corollary 3.3. Let f (z; t) = z 1−t /(1 − z). We have
Qj
Z
r
Y
sj −1 1 f ( l=1 ul ; tj ) duj
ζ(s; t) =
log
( )
uj
Γ(sj )
uj
[0,1]r
j=1
=
Z
Rr+
r
Y
f (e−
Pi
l=1
ul ; t
Γ(si )
i=1
i)
dui
.
i
u1−s
i
Therefore, in this way, the Hurwitz polyzˆeta ζ(s; t) can be viewed as a
multiple Mellin transform of the multivariate function
(54)
r
Y
f (e−
i=1
Pi
l=1
ul ; t
i)
Γ(si )
Pi
=
r
Y
i=1
Pi
1 e(1−ti ) l=1 ul
.
P
Γ(si ) 1 − e− il=1 ul
Its poles are l=1 ul , for i ∈ {1, . . . , r}. To isolate the poles, we use the
next proposition.
Proposition 3.6. Let s be a composition of depth r ≥ 2 and t ∈] − ∞, 1[r .
Then,
Pr
Z +∞ −(1−tr )xr
sj −1
xr j=1
e
dxr
ζ(s; t) =
1 − exr
Γ(sr )
0
P
i
Q
Z
r−1
Y e−(1−ti ) rj=i xj x j=1 sj −1 (1 − xi )si+1 −1
i
Qr
dxi .
Γ(si )
1 − e− j=i xj
[0,1]r−1
i=1
Qr
Q
Remark 3.1. Since 1 − e− j=i xj ∼ rj=i xj near 0, the equality of Proposition 3.6 gives an holomorphic expansion of the polyzˆeta ζ(s; t) over the
set of tuples (s, t) ∈ Cr ×] − ∞, 1[r such that ℜ(s1 ) > 1 and ℜ(si ) > 0,
P
ℜ( ij=1 sj ) > i for any i ∈ {2, . . . , r}.
Proof. We use the substitution
u1 = x1 . . . xr , u2 = (1 − x1 )x2 . . . xr , . . . , ur = (1 − xr−1 )xr .
P
Q
Note that ij=1 uj = rj=i xj for all i ∈ {1, . . . , r}. The Jacobian Jr =
Q
∂(u1 , . . . , ur )/∂(x1 , . . . , xr ) is equal to nk=2 xk−1
(see Lemma 6.1 in appenk
dix). The change of variable is so admissible for (u1 , . . . , ur ) in [0, +∞[r
i.e. when (x1 , . . . , xr ) is in Dr = [0, 1]r−1 × [0, +∞[.
ζ(s; t)
=
r
Y
Γ(si )
i=1
Qr
Z
r
e−(1−t1 ) j=1 xj Y s1 −1
Qr
xj
x
Dr 1 − e j=1 j j=1
Analytic and combinatoric aspects of Hurwitz polyzˆetas
Y
r
=
Z
i=2
+∞
e−(1−ti )
Qr
j=i
Qr
1−e
j=i
xj
xj
e−(1−tr )xr
si −1
(1 − xi−1 )
r
Y
xjsi −1
j=i
Pr
sj −1
Y
r
615
xi−1
i dxi
i=1
dxr
xr j=1
xr
1
−
e
0
Q
Z
r−1
Y e−(1−ti ) rj=i xj Pij=1 sj −1
Qr
xi
(1 − xi )si+1 −1 dxi .
xj
j=i
r−1
[0,1]
i=1 1 − e
4. Analytic continuation and structure of poles
In this section we assume that t ∈] − ∞, 1[r .
4.1. Analytic continuation of Hurwitz polyzˆ
etas.
4.1.1. Generality over the regularization near 0.
Notation. We denote by fxk1 ...xkr the partial differentiation
1
r
and by Z −1, then
Z ρ
xs f (x)dx.
Rρ [f ](s) =
0
Remark 4.1. Moreover, for any positive reals ρ1 , ρ2 and A in I,
Z ρ2
Rρ2 [f ](s) =
xs f (x)dx + Rρ1 [f ](s),
Z
A
ρ1
xs f (x)dx + Rρ1 [f ](s) =
ρ1
A
Z
ρ2
xs f (x)dx + Rρ2 [f ](s).
One can find this definition in some books about Distributions as [18].
Remark 4.2. Over any open set ℜ(s) > −ns , the expression of definition
4.1.1 show that s 7→ Rρ [f ](s) is holomophic over C \ Z −1, ℜ(b) > −1. To continue for any
(a, b) ∈ C2 , we decompose the integral between the singularities and use of
regularized : for ℜ(a) > −1 and ℜ(b) > −1 (where the integral is defined),
and with ρ1 and ρ2 in ]0, 1[,
Z ρ1
Z ρ2
Z 1
a
b
a
b
f (x)x (1 − x) dx =
x f (x)(1 − x) dx +
f (x)xa (1 − x)b dx
0
(55)
0
Z
ρ1
1
+ (1 − x)b f (x)xa dx
ρ2
Z ρ2
b
= Rρ1 [f (x)(1 − x) ](a) +
f (x)xa (1 − x)b dx
ρ1
(56)
+ R1−ρ2 [f (1 − x)(1 − x)a ](b).
Analytic and combinatoric aspects of Hurwitz polyzˆetas
619
Definition. Let J be an interval of R. Let f ∈ C 0 ([0, 1]) and indefinitely
differentiable at 0 and 1. The expression
Z ρ2
Rρ1 [f (x)(1 − x)b ](a) + f (x)xa (1 − x)b dx +R1−ρ2 [f (1 − x)(1 − x)a ](b)
ρ1
is defined for all (a, b) ∈ C \ Z −1 and ℜ(b) > −1, then
Z 1
1
(57)
f (x)xa (1 − x)b dx.
R0 [f ](a, b) =
0
Remark 4.3. ¿From Remark 4.2, we deduce that (a, b) 7→ R10 [f ](a, b) is
2
holomorphic over C \ Z i for all i ∈ {2, . . . , r},
Pr
Z +∞
sj −r−1
yr j=1
dyr
ζ(s; t) =
h(yr , tr )
Γ(sr )
0
Pi
sj −i−1
Z
r−1
r
Y Y
yi j=1
h( yi , ti )
(59)
(1 − yi )si+1 −1 dyi .
Γ(s
)
r−1
i
[0,1]
i=1
k=i
R +∞
R1
R +∞
By splitting the integral 0
onto 0 and 1 ,
1
(60)
Φ1 (s; t) + Φ2 (s; t)
ζ(s; t) = Qr
i=1 Γ(si )
where, with sr+1 = 1,
Z
r
r
Pi
Y
Y
sj −i−1
h( yi , ti )yi j=1
(61) Φ1 (s; t) =
(1 − yi )si+1 −1 dyi .
[0,1]r i=1
and,
Φ2 (s; t) =
Z
k=i
+∞
Pr
h(yr , tr )yr
j=1 sj −r−1
dyr
1
(62)
Z
r−1
Y
[0,1]r−1 i=1
h(
r
Y
Pi
yi , ti )yi
j=1 sj −i−1
(1 − yi )si+1 −1 dyi .
k=i
Lemma 4.5. The function Φ1 defined in (61) can be extended in a meroP
morphic function over Cr . Its set of poles is si ∈ −N, ij=1 sj ∈ i − N, for
i ∈ {1, . . . , r − 1}. These poles are all simple.
Analytic and combinatoric aspects of Hurwitz polyzˆetas
621
For i ∈ {1, . . . , r}, we put gi (x, y) = h(xy, ti ). Note that all gi are in
C ∞ (R2 ). In this way, Φ1 can be written, for ℜ(s1 ) > 1, ℜ(si ) > 0 and
P
ℜ( ij=1 si ) > i for all i ∈ {2, . . . , r},
Φ1 (s; t) = R10 [h(yr , tr )R10 [h(yr yr−1 , tr−1 )R10 [. . . R10 [h(yr . . . y2 , t2 )
R10 [h(yr . . . y1 , t1 )](s1 − 2, s2 − 1)](s1 + s2 − 3, s3 − 1) . . .
(63)
. . .](s1 + . . . + sr−1 − r, sr − 1)](s1 + . . . + sr − r − 1, 0)
= R10 [gr (yr , 1)R10 [gr−1 (yr−1 , yr )R10 [. . . R10 [g2 (y2 , yr . . . y3 )
R10 [g1 (y1 , yr . . . y2 )](s1 − 2, s2 − 1)](s1 + s2 − 3, s3 − 1) . . .
(64)
. . .](s1 + . . . + sr−1 − r, sr − 1)](s1 + . . . + sr − r − 1, 0)
Proof. Thanks to proposition 4.2, y 7→ R10 [g1 (y1 , y)](a1 , b1 ) exists and is in
2
C ∞ (R) for (a1 , b1 ) ∈ C \ Z
de Bordeaux 19 (2007), 595–640
Analytic and combinatoric aspects of Hurwitz
polyzˆ
etas
par Jean-Yves ENJALBERT et HOANG NGOC MINH
´sume
´. Dans ce travail, un codage symbolique des s´eries g´en´eraRe
trices de Dirichlet g´en´eralis´ees est obtenu par les techniques combinatoires des s´eries formelles en variables non-commutative. Il
permet d’expliciter les s´eries g´en´eratrices de Dirichlet g´en´eralis´ees
’p´eriodiques’ – donc notamment les polyzˆetas color´es – comme
combinaison lin´eaire de polyzˆetas de Hurwitz. De plus, la version non commutative du th´eor`eme de convolution nous fournit
une repr´esentation int´egrale des s´eries g´en´eratrices de Dirichlet
g´en´eralis´ees. Celle-ci nous permet de prolonger les polyzˆetas de
Hurwitz comme des fonctions m´eromorphes `a plusieurs variables.
Abstract. In this work, a symbolic encoding of generalized Dirichlet generating series is found thanks to combinatorial techniques of noncommutative rational power series. This enables to
explicit periodic generalized Dirichlet generating series – particularly the coloured polyzˆetas – as linear combinations of Hurwitz
polyzˆetas. Moreover, the noncommutative version of the convolution theorem gives easily rise to an integral representation of
Hurwitz polyzˆetas. This representation enables us to build the
analytic continuation of Hurwitz polyzˆetas as multivariate meromorphic functions.
Contents
1. Introduction
2. Background
2.1. The univariate zˆeta function
2.2. Symbolic computations on special functions
3. Generalized D.g.s. and their integral representation
3.1. Definitions and basic properties
3.2. Periodic D.g.s and Hurwitz polyzˆetas
3.3. Integral representation of Hurwitz polyzˆetas
4. Analytic continuation and structure of poles
4.1. Analytic continuation of Hurwitz polyzˆetas
Manuscrit re¸cu le 28 juillet 2006.
596
599
599
601
606
606
608
612
615
615
596
Jean-Yves Enjalbert, Hoang Ngoc Minh
4.2. Structure of poles
5. A translation relation and prospects
5.1. The translation relation of Hurwitz polyzˆetas
5.2. Translation equality and analytic continuation
6. Conclusion
References
Appendix
623
632
632
633
634
635
636
1. Introduction
Technologies of generating series (g.s.) and of generating functions (g.f.)
on one variable are basic tools required for asymptotic, complexity and
probabilistic analysis of combinatorial and discrete structures. These technologies play a central role in several application areas like in arithmetics,
statistic physics, algorithmic information theory [8], analytic combinatorics
[10] and analysis of algorithms, . . . . This makes the natural link between
these structures and complex analysis and exploits intensively the association between ordinary generating functions (o.g.f.) and exponential generating functions (e.g.f.).
In this paper, we will study the association of o.g.f. and of a generalization of Dirichlet generating functions (D.g.f.) [22]. The value at 1 of a
D.g.f is a Dirichlet generating serie (D.g.s); for example Riemann polyzˆetas
[26], defined by
X
1
(1)
,
ζ(s1 , . . . , sr ) =
s1
n1 . . . nrsr
n >...>n
1
r
are D.g.s and Hurwitz polyzˆetas [27], defined by
X
1
(2) ζ(s1 , . . . , sr ; t1 , . . . , tr ) =
,
s
1
(n1 − t1 ) . . . (nr − tr )sr
n >...>n
1
r
are Parametrized D.g.s. Reciprocally, we show that ’periodic’ Parametrized
D.g.s. (and so periodic D.g.s.) can be written as a finite combination of
Hurwitz polyzˆetas (see Proposition 3.3). It is notably the case for the
coloured polyzˆetas as described by Equality (49). This equality enables us
to give the expression of any Hurwitz polyzˆeta as a finite combination of
coloured polyzˆetas (Proposition 3.4).
On the other hand, the moment sum of index l of basic sign algorithm
is given by [17]
X 1
2
1
(3)
.
ρ(l) = 1 + l +
ζ(2l)
2
cl dl
dnr >0
One thus has a encoding of the polyzˆeta ζ(s) in term of iterated integrals.
Analytic and combinatoric aspects of Hurwitz polyzˆetas
605
For 1 ≤ r ≤ N , let s = (s1 , . . . , sr ). The finite polyzˆetas (or multiple
harmonic sums) ζN (s) is defined as (see [30])
X
1
(33)
.
ζN (s) =
ns11 . . . nsrr
N ≥n1 >...>nr >0
and ζN (s) = 0 for 1 ≤ N < r. For r = 0, we put ζ0 (s) = 0 and ζN (s) = 1,
for any N ≥ 1. These can be obtained as the specialization in the quasimonomial functions (see [38])
X
(34)
Ms (t) =
tsn11 · · · tsnrr
n1 >...>nr >0
at ti = 1/i if 1 ≤ i ≤ k and ti = 0 if i > k. Let us extend linearly the
notation Ms when s is a linear combination of compositions. If r (resp. s)
is a composition of depth r and weight p (resp. of depth s and weight q),
Mr s is a quasi-monomial function of depth r + s and of weight p + q, and
one has
(35)
Ms
r
= M s Mr .
Therefore,
(36)
ζN (s
r) = ζN (s)ζN (r).
For s1 > 1, the limit when N → ∞ of ζN (s) is nothing but the polyzˆeta
ζ(s) [41], and thus by an Abel’s theorem,
X
1
(37)
= ζ(s).
lim ζN (s) = lim Lis (z) =
s1
z→1
N →∞
n1 . . . nsrr
n1 >...>nr >0
The asymptotic expansion of ζN (s), for N → ∞, was already treated by use
of Euler-Mac Laurin summation P
[12] or by use of a full singular expansion,
at z = 1, of its generating series N ≥0 ζN (s)z N [11].
Let us come back to Equality (37). On the one hand, the polyzˆetas ζ(s),
as limits of the polygarithms Lis (z), verify the shuffle relation. On the
other hand, the polyzˆetas ζ(s) can be obtained also as the specialization of
the quasi-monomial functions at ti = 1/i, for i ≥ 1. So, if two compositions
r, s correspond respectively to the two convergent words l1 , l2 , then we get
on the convergent polyzˆetas the three families of relations (see [23]) :
(38)
ζ(l1 ⊔⊔ l2 ) = ζ(l1 )ζ(l2 ),
(39)
ζ(l1
(40)
ζ(x1 ⊔⊔ l1 − x1
l2 ) = ζ(l1 )ζ(l2 ),
l1 ) = 0.
The first two families are called the double shuffle structure. The third
family involves the polynomials x1 ⊔⊔ l1 − x1 l1 that are convergent, even
when the two sums ζ(x1 ⊔⊔ l1 ) and ζ(x1 l1 ) are divergent. These divergent
terms can be regularized syntactically with respect to the associated shuffle
Jean-Yves Enjalbert, Hoang Ngoc Minh
606
products as explained in [27, 28]. Note that we only have to study that
point over a generator family : the set of Lyndon word [23].
3. Generalized D.g.s. and their integral representation
By now, r stands for a positive integer. Let T = {t1 , · · · , tr } and T =
{t1 , · · · , tr } two families of parameters connected by the change of variables
t = t1 − t2 ,
t 1 = t 1 + · · · + tr ,
1
t2 = t2 − t3 ,
t 2 = t 2 + · · · + tr ,
(41)
⇐⇒
..
..
.
.
t =t .
t =t .
r
r
r
r
For i = 1..m, let us consider the locally integrable function1 Fi and the
following associated differential 1-forms
(42)
ω0 (z) =
dz
dz
dz
.
, ωi (z) = Fi (z) , ωi,ti (z) = Fi (z)
z
z
z 1+ti
For any composition s = (s1 , . . . , sr ) and for any formal variables t1 , . . . , tr ,
we use the short notations t for (t1 , . . . , tr ), (s; t) for (s1 ; t1 ), . . . , (sr ; tr )
and F for (Fi1 , . . . , Fir ).
3.1. Definitions and basic properties. Here, we consider the case
X
(43)
Fi (z) =
fi,n z n , for i = 1..m.
n≥1
Definition. The D.g.s. associated to {Fi }i=1..m given in (43) is the sum
X
fi1 ,n1 −n2 . . . fir−1 ,nr−1 −nr fir ,nr
Di(F; s) =
.
ns11 . . . nsrr
n1 >...>nr >0
We get the following iterated integral interpretations :
Proposition 3.1 ([22]). Let i1 , . . . , ir = 1, .., m and let w = xs01 −1 xi1 . . .
. . . xs0r −1 xir associated to s. With the differential forms ω0 , ω1 , . . . , ωm of
(42), we get
X fi1 ,n1 −n2 . . . fir−1 ,nr−1 −nr fir ,nr
z n1 .
α0z (w) =
s1
sr
n
.
.
.
n
r
1
n >...>n
1
r
We call Generalized D.g.f. Dis (F|z) associated to {Fi }i=1..m given in (43)
this previous iterated integral. In particular, Di(F; s) = Dis (F|1) = α01 (w).
1Note that, here F is not necessary the function (1 − z)−1 except when m = 1 and it is in
1
the case of (29).
Analytic and combinatoric aspects of Hurwitz polyzˆetas
607
Example. Let z = (z1 , . . . , zr ). For the o.g.f.
zi z
Fi (z) =
with |zi | < 1, for i = 1..m,
1 − zi z
with the differential forms ω0 , ω1 , . . . , ωm , we have [22, 27]
α0z (w)
=
n
X
n1 >...>nr >0
−nr nr
zr n1
r−1
z1n1 −n2 . . . zr−1
ns11 . . . nsrr
z .
and Di(F1 , . . . , Fr ; s) is nothing but the multiple polylogarithm [4, 19]
X
z1n1 (z2 /z1 )n2 . . . (zr /zr−1 )nr
Lis (z) = Lis (z1 , . . . , zr ) =
.
ns11 . . . nrsr
n1 >...>nr >0
Definition. The Parametrized D.g.s. associated to {Fi }i=1..m given in (43)
is the sum
X
fi1 ,n1 −n2 . . . fir−1 ,nr−1 −nr fir ,nr
.
Di(Ft ; s) =
(n1 − t1 )s1 . . . (nr − tr )sr
n1 >...>nr >0
Proposition 3.2 ([22]). Let i1 , . . . , ir = 1, .., m and w = xs01 −1 xi1 . . .
. . . xs0r −1 xir associated to s. With the differential forms ω0 , ω1,t1 , . . . , ωm,tm
of (42), we get
X fi1 ,n1 −n2 . . . fir−1 ,nr−1 −nr fir ,nr
α0z (w) =
z n1 −t1 .
s1 . . . (n − t )sr
(n
−
t
)
1
1
r
r
n >...>n
1
r
We call Parametrized D.g.f. associated to {Fi }i=1..m given in (43) the
previous iterated integral. In particular, Di(Ft ; s) = α01 (w).
Example. Let z = (z1 , . . . , zr ). For the o.g.f.
zi z
with |zi | < 1, for i = 1..m,
Fi (z) =
1 − zi z
with the differential forms ω0 , ω1,t1 , . . . , ωm,tm , we have [22, 27]
α0z (w) =
X
n1 >...>nr
n
−n
r nr
r−1
z1n1 −n2 . . . zr−1
zr
z n1 −t1 ,
s
(n1 − t1 ) 1 . . . (nr − tr )sr
and Di(F1 , . . . , Fr ; s) is nothing but the multiple Lerch function
X
z1n1 (z2 /z1 )n2 . . . (zr /zr−1 )nr
α01 (w) =
.
(n1 − t1 )s1 . . . (nr − tr )sr
n1 >...>nr >0
Therefore, as iterated integral, the generalized and the parametrized
D.g.f. and the generalized Lerch function verify the shuffle relation (38)
over the alphabet {x0 , . . . , xm } [22]. In particular, their special values verify
this kind of relation.
Jean-Yves Enjalbert, Hoang Ngoc Minh
608
3.2. Periodic D.g.s and Hurwitz polyzˆ
etas.
Definition. The generalized Lerch function is defined as follows
Φs (z; t) =
X
n1 >···>nr >0
z n1
(n1 − t1 )s1 . . . (nr − tr )sr
for |z| < 1.
The associated Hurwitz polyzˆeta is defined by
ζ(s; t) =
X
n1 >···>nr >0
(n1 − t1
)s1
1
= Φs (1; t).
. . . (nr − tr )sr
With the notation of Example 3.1, z1 = . . . = zr and for t fixed, the
generalized Lerch function appear as an iterated integral and so verify the
shuffle product. Consequently, as z tends to 1, for any convergent compositions s and s′ ,
(44)
ζ(s ⊔⊔ s′ ; t) = ζ(s; t)ζ(s′ ; t).
Note that the Hurwitz polyzˆetas can be also encoded by noncommutative
rational series
(45)
xs01 −1 (t1 x0 )∗s1 x1 . . . xs0r −1 (tr x0 )∗sr x1
as given in [27] with two differential forms ω0 and ω1 .
These polyzˆetas contain divergent terms which are looked at via syntactic regularizations of divergent Hurwitz polyzˆetas with respect to the
associated shuffle products as in [27, 28]. Note also that if t1 = t2 = · · · = 0
then Φs (z; 0) = Lis (z) and ζ(s; 0) = ζ(s).
Proposition 3.3. If the {fi,n }n≥1 for i = 1..N , are periodic of the same
period K then with the differential forms ω0 , ω1,t1 , . . . , ωm,tm , we have
Di(Ft ; s) =
ζ[(s1 ;
1
K
Pr
t1 −
l=1 sl
Pr
fi1 ,b1 . . . fir ,br
b1 ,...,br =0
l=1 bl
K
K−1
X
), . . . , (sr ;
tr − br
)].
K
Proof. Let us sketch the proof in case of r = 2 :
Di(Ft ; s) =
X
n1 >n2 >0
=
X
m1 ,m2 >0
fi1 ,n1 −n2 fi2 ,n2
(n1 − t1 )s1 (n2 − t2 )s2
fi1 ,m1 fi2 ,m2
.
(m1 + m2 − t1 )s1 (m2 − t2 )s2
Analytic and combinatoric aspects of Hurwitz polyzˆetas
609
The assumption of periodicity gives fi,k+K = fi,k for all k ≥ 1 and i = 1, 2.
Therefore,
Di(Ft ; s)
=
K−1
X
X
b1 ,b2 =0 m1 ,m2 ≥1
=
K−1
X
fi1 ,b1 fi2 ,b2
X
n1 >n2 >0
b1 ,b2 =0
=
fi1 ,m1 K+b1 fi2 ,m2 K+b2
[(m1 + m2 )K + b1 + b2 − t1 ]s1 (m2 K + b2 − t2 )s2
1
K s1 +s2
K−1
X
b1 ,b2 =0
1
(n1 K + b1 + b2 − t1 )s1 (n2 K + b2 − t2 )s2
t2 − b 2
t1 − (b1 + b2 )
fi1 ,b1 fi2 ,b2 ζ s1 ;
, s2 ;
.
K
K
The proof is easily generalized for any positive integer r.
In other words, if {fi,n }n≥1 are periodic of the same period K then generalized D.g.s. is a linear combination of Hurwitz polyzˆetas. In particular,
if fi1 ,n = . . . = fir ,n = 1, for n ≥ 1, then one can generalize the well known
result for the Riemann and Hurwitz zˆeta functions.
Corollary 3.1. For K ∈ N+ , one has
ζ(s; t) =
1
K
Pr
l=1 sl
K−1
X
ζ[(s1 ;
t1 −
b1 ,...,br =0
Pr
l=1 bl
K
), . . . , (sr ;
tr − b r
)].
K
Now let q = e2iπ/m be a m-th primitive root of unity. For any integers i1 , . . . , ir , we will use the notation q i for q i1 , . . . , q ir . Let Q(q) be the
cyclotomic field generated by q.
Let us introduce also the o.g.f.
∞
X
Fi (z) =
(46)
q in z n , for i = 1..m.
n=1
The coloured polyzˆetas are a particular case of periodic D.g.s. Di(F; s) and
they are defined as follows (for 0 ≤ i1 < . . . < ir ≤ m − 1) [3, 4, 19]
X
s
q i1 n1 . . . q ir nr
ζ i =
(47)
.
ns11 . . . nsrr
q
n1 >...>nr >0
In other terms, these can be obtained as a special value at z = 1 of the
following D.g.s. associated to {Fi }i=1,..,m given in (46)
s
ζ i1 i2 −i1
(48)
= α01 (xs01 −1 xi1 . . . x0sr −1 xir ).
q ,q
. . . , q ir −ir−1
So these verify the first shuffle relation (38) over the alphabet {x0 , . . . , xm }.
By Proposition 3.3, the periodicity of {q n }n≥1 enables then to express the
610
Jean-Yves Enjalbert, Hoang Ngoc Minh
coloured polyzˆetas as linear combinations of Hurwitz polyzˆetas with coefficients in Q(q) [27] :
m−1
Pr
X
a
1
s
q l=1 il al ζ(s; − ).
(49) ζ i = Pr s
l
l=1
q
m
m
a −a ,...,a
−ar ,ar =0
1
2
r−1
Here, a/m stands
for (a1 /m, . . . , ar /m). Consequently, these coloured
s
polyzˆetas ζ qi (with 0 ≤ ik ≤ m − 1) are linear combinations, with coefficients in Q(q), of the rational parameters Hurwitz polyzˆetas ζ(s; −a/m)
with a1 − a2 , . . . , ar−1 − ar , ar = 1..m. They can also be viewed as the
evaluations, at the m-th primitive root of unit q of some commutative
polynomials on the Q-algebra of Hurwitz polyzˆetas. The reader can find
the shuffle structures of Hurwitz polyzˆetas in [27] inducing then the shuffle
structures of Riemann polyzˆetas. Conversely, by the distribution formula,
we have a partial result [30] :
Proposition 3.4. Let a = (a1 , . . . , ar ) be a composition. If the parameters
aj satisfy the conditions 1 ≤ a1 − a2 , . . . , ar−1 − ar , ar < m then
m−1
Pr
P
X
a
s
sl −r
− rl=1 il al
l=1
q
ζ(s; − ) = m
ζ i .
m
q
i1 ,...,ir =0
Proof. Let us set s and let
s1 , s2 , . . . , sr
,
f (i) = i1 ...+ir ζ i1 i2 −i1
q
q ,q
. . . , q ir −ir−1
g(c) = ζ (s1 , . . . , sr ; −a1 /m, . . . , −ar /m) ,
1
with aj = (cj + 1) + (cj+1 + 1) . . . + (cr + 1), for j = 1 . . . r. Let us consider
the lexicographical order on the index set
I = {i = (i1 , . . . , ir ) ∈ {0, . . . m − 1}r }
and on the index set
C = {c = (c1 , . . . , cr ) ∈ {0, . . . m − 1}r }.
Then f (i) i∈I (resp. g(c) c∈C ) can be viewed as the entries of a column
vector F (resp. G) of dimension
mr.
Let M (q) be the matrix q ij 0≤i,j≤m−1 of determinant
Y
(50)
(q i − q j ) 6= 0.
det M (q) =
0≤i 1, σ1 + σ2 > 1, . . . , σ1 + · · · + σr > 1. Either, if the power
series Fi (e−z ) has a suitable asymptotic expansion at 0 and at ∞ then the
Dirichlet generating functions associated to the sequences {fi,n }n≥1,i=1,..,N
are meromorphic functions in Cr and are given by
Proposition 3.5. The function Dis (F|z) can be represented as
Qj
Z
r
Y
l=1 ul ) duj
sj −1 1 Fij (z
Dis (F|z) =
log
( )
uj
Γ(sj )
uj
[0,1]r
j=1
Z
where ur+1 = 1.
Pj
r
Y
Fij (ze− l=1 ul ) duj
=
1−s .
Γ(sj )
Rr+
u j
j=1
j
Proof. The proof can be obtained by induction on r and by use of Lemma
3.1.
Therefore, by taking z = 1, one finally gets
Corollary 3.2. The generalized D.g.s. can be represented as
Qj
Z
r
Y
sj −1 1 Fij ( l=1 ul ) duj
Di(F; s) =
log
( )
uj
Γ(sj )
uj
[0,1]r
j=1
Z
where ur+1 = 1.
Pj
r
Y
Fij (e− l=1 ul ) duj
=
1−s .
Γ(sj )
Rr+
u j
j=1
j
For z = 1, the polylogarithmic transformation corresponds to the Mellin
transformation and onePobtains the D.g.s. Di(F/Γ(s); s) associated to
F (τ )/Γ(s) with F (τ ) = k≥1 fk q k and q = e−τ .
Example. Let s′ = (s2 , . . . , sr ) and F′ = (Fi2 , . . . , Fir ). By Lemma 3.1, the
Dirichlet function Dis (F|z) is a polylogarithmic transform of the function
Fi1 (z) Dis′ (F′ |z) :
Z 1
1 Fi (zt) Dis′ (F′ |zt) dt
Dis (F|z) =
logs1 −1 ( ) 1
t
Γ(s1 )
t
0
Z ∞
′
−t
−t
Fi1 (ze ) Dis′ (F |ze ) dt
=
.
Γ(s1 )
t1−s1
0
and the D.g.s. Di(F; s) is a Mellin transform of Fi1 (e−z ) Dis′ (F′ |e−z )/Γ(s1 ).
Therefore, by Corollary 3.2, Di(F; s) can be viewed as a multiple Mellin
Pj
Q
transform of the multivariate function rj=1 Fij (e− l=1 ul )/Γ(sj ).
Jean-Yves Enjalbert, Hoang Ngoc Minh
614
Corollary 3.3. Let f (z; t) = z 1−t /(1 − z). We have
Qj
Z
r
Y
sj −1 1 f ( l=1 ul ; tj ) duj
ζ(s; t) =
log
( )
uj
Γ(sj )
uj
[0,1]r
j=1
=
Z
Rr+
r
Y
f (e−
Pi
l=1
ul ; t
Γ(si )
i=1
i)
dui
.
i
u1−s
i
Therefore, in this way, the Hurwitz polyzˆeta ζ(s; t) can be viewed as a
multiple Mellin transform of the multivariate function
(54)
r
Y
f (e−
i=1
Pi
l=1
ul ; t
i)
Γ(si )
Pi
=
r
Y
i=1
Pi
1 e(1−ti ) l=1 ul
.
P
Γ(si ) 1 − e− il=1 ul
Its poles are l=1 ul , for i ∈ {1, . . . , r}. To isolate the poles, we use the
next proposition.
Proposition 3.6. Let s be a composition of depth r ≥ 2 and t ∈] − ∞, 1[r .
Then,
Pr
Z +∞ −(1−tr )xr
sj −1
xr j=1
e
dxr
ζ(s; t) =
1 − exr
Γ(sr )
0
P
i
Q
Z
r−1
Y e−(1−ti ) rj=i xj x j=1 sj −1 (1 − xi )si+1 −1
i
Qr
dxi .
Γ(si )
1 − e− j=i xj
[0,1]r−1
i=1
Qr
Q
Remark 3.1. Since 1 − e− j=i xj ∼ rj=i xj near 0, the equality of Proposition 3.6 gives an holomorphic expansion of the polyzˆeta ζ(s; t) over the
set of tuples (s, t) ∈ Cr ×] − ∞, 1[r such that ℜ(s1 ) > 1 and ℜ(si ) > 0,
P
ℜ( ij=1 sj ) > i for any i ∈ {2, . . . , r}.
Proof. We use the substitution
u1 = x1 . . . xr , u2 = (1 − x1 )x2 . . . xr , . . . , ur = (1 − xr−1 )xr .
P
Q
Note that ij=1 uj = rj=i xj for all i ∈ {1, . . . , r}. The Jacobian Jr =
Q
∂(u1 , . . . , ur )/∂(x1 , . . . , xr ) is equal to nk=2 xk−1
(see Lemma 6.1 in appenk
dix). The change of variable is so admissible for (u1 , . . . , ur ) in [0, +∞[r
i.e. when (x1 , . . . , xr ) is in Dr = [0, 1]r−1 × [0, +∞[.
ζ(s; t)
=
r
Y
Γ(si )
i=1
Qr
Z
r
e−(1−t1 ) j=1 xj Y s1 −1
Qr
xj
x
Dr 1 − e j=1 j j=1
Analytic and combinatoric aspects of Hurwitz polyzˆetas
Y
r
=
Z
i=2
+∞
e−(1−ti )
Qr
j=i
Qr
1−e
j=i
xj
xj
e−(1−tr )xr
si −1
(1 − xi−1 )
r
Y
xjsi −1
j=i
Pr
sj −1
Y
r
615
xi−1
i dxi
i=1
dxr
xr j=1
xr
1
−
e
0
Q
Z
r−1
Y e−(1−ti ) rj=i xj Pij=1 sj −1
Qr
xi
(1 − xi )si+1 −1 dxi .
xj
j=i
r−1
[0,1]
i=1 1 − e
4. Analytic continuation and structure of poles
In this section we assume that t ∈] − ∞, 1[r .
4.1. Analytic continuation of Hurwitz polyzˆ
etas.
4.1.1. Generality over the regularization near 0.
Notation. We denote by fxk1 ...xkr the partial differentiation
1
r
and by Z −1, then
Z ρ
xs f (x)dx.
Rρ [f ](s) =
0
Remark 4.1. Moreover, for any positive reals ρ1 , ρ2 and A in I,
Z ρ2
Rρ2 [f ](s) =
xs f (x)dx + Rρ1 [f ](s),
Z
A
ρ1
xs f (x)dx + Rρ1 [f ](s) =
ρ1
A
Z
ρ2
xs f (x)dx + Rρ2 [f ](s).
One can find this definition in some books about Distributions as [18].
Remark 4.2. Over any open set ℜ(s) > −ns , the expression of definition
4.1.1 show that s 7→ Rρ [f ](s) is holomophic over C \ Z −1, ℜ(b) > −1. To continue for any
(a, b) ∈ C2 , we decompose the integral between the singularities and use of
regularized : for ℜ(a) > −1 and ℜ(b) > −1 (where the integral is defined),
and with ρ1 and ρ2 in ]0, 1[,
Z ρ1
Z ρ2
Z 1
a
b
a
b
f (x)x (1 − x) dx =
x f (x)(1 − x) dx +
f (x)xa (1 − x)b dx
0
(55)
0
Z
ρ1
1
+ (1 − x)b f (x)xa dx
ρ2
Z ρ2
b
= Rρ1 [f (x)(1 − x) ](a) +
f (x)xa (1 − x)b dx
ρ1
(56)
+ R1−ρ2 [f (1 − x)(1 − x)a ](b).
Analytic and combinatoric aspects of Hurwitz polyzˆetas
619
Definition. Let J be an interval of R. Let f ∈ C 0 ([0, 1]) and indefinitely
differentiable at 0 and 1. The expression
Z ρ2
Rρ1 [f (x)(1 − x)b ](a) + f (x)xa (1 − x)b dx +R1−ρ2 [f (1 − x)(1 − x)a ](b)
ρ1
is defined for all (a, b) ∈ C \ Z −1 and ℜ(b) > −1, then
Z 1
1
(57)
f (x)xa (1 − x)b dx.
R0 [f ](a, b) =
0
Remark 4.3. ¿From Remark 4.2, we deduce that (a, b) 7→ R10 [f ](a, b) is
2
holomorphic over C \ Z i for all i ∈ {2, . . . , r},
Pr
Z +∞
sj −r−1
yr j=1
dyr
ζ(s; t) =
h(yr , tr )
Γ(sr )
0
Pi
sj −i−1
Z
r−1
r
Y Y
yi j=1
h( yi , ti )
(59)
(1 − yi )si+1 −1 dyi .
Γ(s
)
r−1
i
[0,1]
i=1
k=i
R +∞
R1
R +∞
By splitting the integral 0
onto 0 and 1 ,
1
(60)
Φ1 (s; t) + Φ2 (s; t)
ζ(s; t) = Qr
i=1 Γ(si )
where, with sr+1 = 1,
Z
r
r
Pi
Y
Y
sj −i−1
h( yi , ti )yi j=1
(61) Φ1 (s; t) =
(1 − yi )si+1 −1 dyi .
[0,1]r i=1
and,
Φ2 (s; t) =
Z
k=i
+∞
Pr
h(yr , tr )yr
j=1 sj −r−1
dyr
1
(62)
Z
r−1
Y
[0,1]r−1 i=1
h(
r
Y
Pi
yi , ti )yi
j=1 sj −i−1
(1 − yi )si+1 −1 dyi .
k=i
Lemma 4.5. The function Φ1 defined in (61) can be extended in a meroP
morphic function over Cr . Its set of poles is si ∈ −N, ij=1 sj ∈ i − N, for
i ∈ {1, . . . , r − 1}. These poles are all simple.
Analytic and combinatoric aspects of Hurwitz polyzˆetas
621
For i ∈ {1, . . . , r}, we put gi (x, y) = h(xy, ti ). Note that all gi are in
C ∞ (R2 ). In this way, Φ1 can be written, for ℜ(s1 ) > 1, ℜ(si ) > 0 and
P
ℜ( ij=1 si ) > i for all i ∈ {2, . . . , r},
Φ1 (s; t) = R10 [h(yr , tr )R10 [h(yr yr−1 , tr−1 )R10 [. . . R10 [h(yr . . . y2 , t2 )
R10 [h(yr . . . y1 , t1 )](s1 − 2, s2 − 1)](s1 + s2 − 3, s3 − 1) . . .
(63)
. . .](s1 + . . . + sr−1 − r, sr − 1)](s1 + . . . + sr − r − 1, 0)
= R10 [gr (yr , 1)R10 [gr−1 (yr−1 , yr )R10 [. . . R10 [g2 (y2 , yr . . . y3 )
R10 [g1 (y1 , yr . . . y2 )](s1 − 2, s2 − 1)](s1 + s2 − 3, s3 − 1) . . .
(64)
. . .](s1 + . . . + sr−1 − r, sr − 1)](s1 + . . . + sr − r − 1, 0)
Proof. Thanks to proposition 4.2, y 7→ R10 [g1 (y1 , y)](a1 , b1 ) exists and is in
2
C ∞ (R) for (a1 , b1 ) ∈ C \ Z