2005 International Conference on Analysis of Algorithms
DMTCS proc. AD, 2005, 59–70
Algorithmic and combinatoric aspects of
multiple harmonic sums
Christian Costermans and Jean-Yves Enjalbert and Hoang Ngoc Minh
Université Lille II, 1, Place Déliot, 59024 Lille, France
Ordinary generating series of multiple harmonic sums admit a full singular expansion in the basis of functions
{(1 − z)α logβ (1 − z)}α∈Z,β∈N , near the singularity z = 1. A constructive proof of this result is given, and, by
combinatoric aspects, an explicit evaluation of Taylor coefficients of functions in some polylogarithmic algebra is
obtained. In particular, the asymptotic expansion of multiple harmonic sums is easily deduced.
Keywords: polylogarithms, polyzêtas, multiple harmonic sums, singular expansion, shuffle algebra, Lyndon words
1
Introduction
Hierarchical data structure occur in numerous domains, like computer graphics, image processing or biology (pattern matching). Among them, quadtrees, whose construction is based on a recursive definition
of space, constitute a classical data structure for storing and accessing collection of points in multidimensional space. Their characteristics (depth of a node, number of nodes in a given subtree, number of leaves)
are studied by Laforest [12], with probabilistic tools. In particular, she shows, for a quadtree of size N in
a d-dimension space, that the probability πN,k for the first subtree to have size k can be expressed as an
algebraic combination of j-th order harmonic numbers Hj (N ) and Hj (k), j ≥ 1, defined by
Hj (n) =
n
X
1
.
j
m
m=1
(1)
For instance, for d = 3, one has
πN,k =
[H1 (N ) − H1 (k)]2 + H2 (N ) − H2 (k)
.
2N
(2)
Flajolet et al. [2] give this general expression for the splitting probability
πN,k =
X
N ≥i1 ···≥id−1 >k
1
.
i1 · · · id−1
(3)
The probability πN,k appears as a particular case of the following sum As (N ) associated to the multi-index
s = (s1 , . . . , sr ), which is strongly related to multiple harmonic sums Hs (N ) :
As (N ) =
X
N ≥n1 ≥···≥nr >0
n1
s1
1
. . . n r sr
X
and Hs (N ) =
N ≥n1 >···>nr >0
1
.
n1 s1 . . . n r sr
(4)
Let us note that there exist explicit relations, given by Hoffman [10] between the As (N ) and Hs (N ).
Indeed, let Comp(n) be the set of compositions of n, i.e. sequences (i1 , . . . , ir ) of positive integers
summing to n. If I = (i1 , . . . , ir ) (resp. J = (j1 , . . . , jp )) is a composition of n (resp. of r) then
J ◦ I = (i1 + . . . + ij1 , ij1 +1 + . . . + ij1 +j2 , . . . , ik−jp +1 + . . . + ik ) is a composition of n. By Möbius
inversion, one has
X
X
As (N ) =
HJ◦s (N ) and Hs (N ) =
(−1)l(J)−r AJ◦s (N ),
(5)
J∈Comp(r)
J∈Comp(r)
where l(J) is the number of parts of J.
c 2005 Discrete Mathematics and Theoretical Computer Science (DMTCS), Nancy, France
1365–8050 60
Christian Costermans and Jean-Yves Enjalbert and Hoang Ngoc Minh
Example 1. For s = (1, 1, 1), since the set of compositions of 3 is {(1, 1, 1), (1, 2), (2, 1), (3)}, we get
A1,1,1 (N )
=
H1,1,1 (N ) + H1,2 (N ) + H2,1 (N ) + H3 (N ),
H1,1,1 (N )
=
A1,1,1 (N ) − A1,2 (N ) − A2,1 (N ) + A3 (N ).
Therefore, the As (N ) are Z-linear combinations on Hs (N ) (and vice versa). Thus, the remaining
problem is to know the asymptotic behaviour of πN,k , for N → ∞ [11]. For that, in this work, we
are interested in the combinatorial aspects of these sums by use of a symbolic encoding by words. This
enables then to transfer shuffle relations on words into algebraic relations between multiple harmonic
sums, or between polylogarithm functions, defined for a multi-index s = (s1 , . . . , sr ) by
X
z n1
, for |z| < 1.
(6)
Lis (z) =
s1
n . . . nsrr
n >...>n >0 1
1
r
These relations are recalled in SectionP
2. The reason to call upon polylogarithms is given in Section 3,
since the generating function Ps (z) = n≥0 Hs (n)z n of {Hs (n)}n≥0 , verifies
1
Lis (z).
(7)
1−z
So, we set the polylogarithmic algebra of {Ps }s , with coefficients in C = C[z, z −1 , (z − 1)−1 ], and
we then establish exact transfer results between a function g in this algebra and its Taylor coefficients
[z N ]g(z), in the C-algebra generated by {N k Hs (N )}s, k∈Z in both directions. The main result of this
paper is finally stated in Section 4, which gives a computation of the full singular expansion of g, in
the basis of functions {(1 − z)α logβ (1 − z)}α∈Z,β∈N , near the singularity z = 1. We deduce from
this a full asymptotic expansion of its Taylor coefficients. These results are based on the analysis of the
noncommutative generating series of functions of the form (7), in particular on its infinite factorization
indexed by Lyndon words.
Ps (z) =
2
2.1
Background
Combinatorics on words
To the multi-index s we can canonically associate the word u = x0s1 −1 x1 . . . xs0r −1 x1 over the finite
alphabet X = {x0 , x1 }. In the same way, s can be canonically associated to the word v = ys1 . . . ysr over
the infinite alphabet Y = {yi }i≥1 . Moreover, in both alphabets, the empty multi-index will correspond to
the empty word . We shall henceforth identify the multi-index s with its encoding by the word u (resp.
v). We denote by X ∗ (resp. Y ∗ ) the free monoid generated by X (resp. Y ), which is the set of words over
X (resp. Y ). Noting ChXi (resp. ChY i) the algebra of noncommutative polynomials with coefficients in
C, we obtain so a concatenation isomorphism from the C-algebra of multi-indexes into the algebra ChXi
(resp. ChY i). The coefficient of w ∈ X ∗ in a polynomial S ∈ ChXi is denoted by (S|w) or Sw . The
duality between polynomials is defined as follows
X
(S|p) =
Sw pw , p ∈ ChXi.
(8)
w∈X ∗
The set of Lie monomials is defined by induction: the letters in X are Lie monomials and the Lie
bracket [a, b] = ab − ba of two Lie monomials a and b is a Lie monomial. A Lie polynomial is a C-linear
combination of Lie monomials. The set of Lie polynomials is called the free Lie algebra.
2.2
Shuffle products
Let a, b ∈ X (resp. yi , yj ∈ Y ) and u, v ∈ X ∗ (resp. Y ∗ ). The shuffle (resp. stuffle) of u = au0 and
v = bv 0 (resp. u = yi u0 and v = yj v 0 ) is the polynomial recursively defined by
tt u = u tt = u
(resp. u = u = u
and u tt v = a(u0 tt v) + b(u tt v 0 ),
and u v = yi (u0 v) + yj (u v 0 ) + yi+j (u0
v 0 )).
(9)
(10)
Example 2.
x0 x1 tt x1
y2 y1
= x1 x0 x1 + 2x0 x21
= y1 y2 + y2 y1 + y 3 .
(11)
This product is extended to ChXi (resp. ChY i) by linearity. With this product, ChXi (resp. ChY i) is a
commutative and associative C-algebra.
Algorithmic and combinatoric aspects of multiple harmonic sums
61
l
x0
x1
x0 x1
x0 2 x1
x0 x1 2
x0 3 x1
..
.
Ql
x0
x1
[x0 , x1 ]
[x0 , [x0 , x1 ]]
[[x0 , x1 ], x1 ]
[x0 , [x0 , [x0 , x1 ]]]
..
.
Sl
x0
x1
x0 x1
x0 2 x1
x0 x1 2
x0 3 x1
..
.
x0 3 x1 3
x0 2 x1 x0 x1 2
x0 2 x1 2 x0 x1
x0 2 x1 4
x0 x1 x0 x1 3
x0 x1 5
[x0 , [x0 , [[[x0 , x1 ], x1 ], x1 ]]]
[x0 , [[x0 , x1 ], [[x0 , x1 ], x1 ]]]
[[x0 , [[x0 , x1 ], x1 ]], [x0 , x1 ]]
[x0 , [[[[x0 , x1 ], x1 ], x1 ], x1 ]]
[[x0 , x1 ], [[[x0 , x1 ], x1 ], x1 ]]
[[[[[x0 , x1 ], x1 ], x1 ], x1 ], x1 ]
x0 3 x1 3
3x0 3 x1 3 + x0 2 x1 x0 x1 2
6x0 3 x1 3 + 3x0 2 x1 x0 x1 2 + x0 2 x1 2 x0 x1
x0 2 x1 4
4x0 2 x1 4 + x0 x1 x0 x1 3
x0 x1 5
Tab. 1: Lyndon words, bracket forms and dual basis
2.3
Lyndon words and Radford’s theorem
By definition, a Lyndon word is a non empty word l ∈ X ∗ (resp. ∈ Y ∗ ) which is lower than any of its
proper right factors [14] (for the lexicographical ordering) i.e. for all u, v ∈ X ∗ \ {} (resp. ∈ Y ∗ \ {}),
l = uv ⇒ l < v. The set of Lyndon words of X (resp. Y ) is denoted by Lyn(X) (resp. Lyn(Y )).
Example 3. For X = {x0 , x1 } with the order x0 < x1 the Lyndon words of length ≤ 5 on X ∗ are (in
lexicographical increasing order)
{x0 , x40 x1 , x30 x1 , x30 x21 , x20 x1 , x20 x1 x0 x1 , x20 x21 , x20 x31 , x0 x1 , x0 x1 x0 x21 , x0 x21 , x0 x31 , x0 x41 , x1 }.
For Y = {yi , i ≥ 1}, with the order yi < yj when i > j, here are the corresponding Lyndon words over
Y
{y5 , y4 , y4 y1 , y3 , y3 y2 , y3 y1 , y3 y12 , y2 , y22 y1 , y2 y1 , y2 y12 , y2 y13 , y1 }.
Theorem 1 (Radford, [13, 14]). Let
C1 = C ⊕ (ChXi \ x0 ChXix1 ) and C2 = C ⊕ (ChY i \ y1 ChY i)
be the sets of convergent polynomials over X and Y respectively. Then,
(ChXi, tt ) ' (C[Lyn(X)], tt ) = (C1 [x0 , x1 ], tt ),
(ChY i, ) ' (C[Lyn(Y )], ) = (C2 [y1 ], ).
Example 4.
y 2 y4 y1 + y 2 y 1 y4 + y 1 y 2 y 4 + y 2 y 5 + y3 y4
2.4
= y 4 y2 y1 − y 4 y2
= y2 y4 y1 ∈ C2 [y1 ]
y1 − y6
y1 ∈ C[Lyn(Y )]
Bracket forms and the dual basis
The bracket form Ql of a Lyndon word l = uv, with l, u, v ∈ Lyn(X) and the word v being as long as
possible, is defined recursively by
Ql = [Qu , Qv ]
Qx = x for each letter x ∈ X,
It is classical that the set B1 = {Ql ; l ∈ Lyn(X)}, ordered lexicographically, is a basis for the free
Lie algebra. Moreover, each word w ∈ X ∗ can be expressed uniquely as a decreasing product of Lyndon
words:
w = l1α1 l2α2 . . . lkαk , l1 > l2 > · · · > lk , k ≥ 0.
(12)
62
Christian Costermans and Jean-Yves Enjalbert and Hoang Ngoc Minh
The Poincaré–Birkhoff–Witt basis B = {Qw ; w ∈ X ∗ } and its dual basis B ∗ = {Sw ; w ∈ X ∗ } are
obtained from (12) by setting [14]

αk
α2
1
Qw = Qα

l1 Ql2 . . . Qlk ,


α
α
Slt1t 1 tt . . . tt Sltkt k
,
S
=
w


α1 !α2 ! . . . αk !

Sl
= xSw , ∀l ∈ Lyn(X), where l = xw, x ∈ X, w ∈ X ∗ .
In [14], it is proved that B and B ∗ are dual bases of ChXi i.e. (Qu |Sv ) = δuv , for all words u, v ∈ X ∗
with δuv = 1 if u = v, otherwise 0.
Lemma 1. For all w ∈ x0 X ∗ x1 , one has Sw ∈ x0 ZhXix1 .
Proof. The Lyndon words involved in the decomposition (12) of a word w ∈ X ∗ x1 (resp. w ∈ x0 X ∗ x1 )
all belong to X ∗ x1 (resp. x0 X ∗ x1 ).
2.5
Polylogarithms
Let C = C[z, 1/z, 1/(z − 1)] and let ω0 and ω1 be the two following differential forms
ω0 (z) =
dz
z
and ω1 (z) =
dz
.
1−z
(13)
One verifies the polylogarithm Lis (z), defined by Formula (6), is also the following iterated integral with
respect to ω0 and ω1
Z
Lis (z) =
ω0s1 −1 ω1 · · · ω0sr −1 ω1 .
(14)
0
z
Thanks to the bijection from Y ∗ to X ∗ x1 previously explained, we can index the polylogarithms by the
words of X ∗ x1 , or indistinctly by the words of Y ∗ . We can extend (14) over X ∗ by putting
Z
Li (z) = 1, Lix0 (z) = log z, Lixi w (z) =
ωi (t) Liw (t), for xi ∈ X, w ∈ X ∗ .
(15)
0
z
Therefore, Liw verifies the following identity [4]
∀u, v ∈ X ∗ ,
Liu tt v = Liu Liv .
(16)
The extended definition enables to construct the noncommutative generating series [4]
X
L=
Liw w
(17)
w∈X ∗
as being the unique solution of the Drinfel’d equation, i.e. the differential equation [4]
dL = [x0 ω0 + x1 ω1 ]L,
(18)
satisfying the boundary condition
√
L(ε) = ex0 log ε + o( ε), when ε → 0+ .
(19)
Proposition 1 ([5]). Let σ be the monoid morphism defined over X ∗ by σ(x0 ) = −x1 and σ(x1 ) = −x0 .
Then,
&
Y
L(1 − z) = [σL(z)]
eζ(Sl )Ql .
l∈Lyn(X)\{x0 ,x1 }
Example 5 ([5]).
Lix0 x21 (1 − z)
Li2,1 (1 − z)
= − Lix20 x1 (z) + Lix0 (z) Lix0 x1 (z) −
= − Li3 (z) + log(z) Li2 (z) +
1 2
Li (z) Lix1 (z) + ζ(3),
2 x0
1
log2 (z) log(1 − z) + ζ(3).
2
i.e.
Algorithmic and combinatoric aspects of multiple harmonic sums
2.6
63
Harmonic sums
Definition 1. Let w = ys1 . . . ysr ∈ Y ∗ . For N ≥ r ≥ 1, the harmonic sum Hw (N ) is defined as
Hw (N )
X
=
N ≥n1 >...>nr >0
1
.
ns11 . . . nsrr
For 0 ≤ N < r, Hw (N ) = 0 and, for the empty word , we put H (N ) = 1, for any N ≥ 0.
Let w = ys1 . . . ysr ∈ Y ∗ . If s1 > 1 then, by an Abel’s theorem,
lim Hw (N ) = lim Liw (z) =
N →∞
z→1
X
n1 >...>nr
1
.
s1
.
. . nsrr
n
>0 1
That is nothing but the polyzêta (or MZV [16]) ζ(w) and the word w ∈ Y ∗ \y1 Y ∗ is said to be convergent.
A polynomial of ChY i is said to be convergent when it is a linear combination of convergent words. The
double shuffle algebra of polyzêtas is already pointed out and extensively studied in [3].
For w = ys w0 , we have
ζ(w)
=
X Hw0 (l − 1)
ls
l≥1
Hw (N + 1) − Hw (N )
=
−s
(N + 1)
,
Hw0 (N )
(20)
(21)
and, for any u, v ∈ Y ∗ [9]
Hu
3
v (N )
=
Hu (N )Hv (N ).
(22)
Generating series
3.1
Definition and first properties
Definition 2 ([8]). Let w ∈ Y ∗ and let Pw (z) be the ordinary generating series of {Hw (N )}N ≥0
X
Pw (z) =
Hw (N )z N .
N ≥0
Proposition 2 ([8]). Extended by linearity, the map P : u 7→ Pu is an isomorphism from (ChY i,
)
to the Hadamard algebra of ({Pw }w∈Y ∗ , ). Therefore, the map H : u 7→ Hu = {Hu (N )}N ≥0 is an
isomorphism from (ChY i, ) to the algebra of ({Hw }w∈Y ∗ , . ) .
P∞
P∞
P∞
n
Proof. The definition of the Hadamard product n=0 an z n n=0 bn z n =
n=0 an bn z , and the
formula (22) gives P as an algebra morphism. Since the functions {Liw }w∈X ∗ are linearly independent
over C [4], P is the expected isomorphism.
Proposition 3 ([8]). For every word w ∈ Y ∗ and for z ∈ C satisfying |z| < 1, one has Liw (z) =
(1 − z)Pw (z).
P
Proof. For w = ys w0 , since Pw (z) = N ≥0 Hw (N )z N and by using (21),
(1 − z)Pw (z) = Hw (0) +
X Hw0 (N − 1)
z N = Liw (z).
Ns
N ≥1
A direct consequence of this proposition and Identity (16) is
Corollary 1. For all u, v ∈ X ∗ , for all z ∈ C satisfying |z| < 1, Pu (z)Pv (z) = (1 − z)−1 Pu tt v (z).
Example 6. Since x1 tt x0 x1 = x1 x0 x1 + 2x0 x21 then we get
P1,2 (z) = (1 − z)P1 (z)P2 (z) − 2P2,1 (z).
64
Christian Costermans and Jean-Yves Enjalbert and Hoang Ngoc Minh
Proposition 3 allows to extend the definition of Pw over X ∗ as we have already extended the definition
of Liw over X ∗ . Moreover,
Definition 3 ([8]). Let P be the noncommutative generating series of {Pw }w∈X ∗ :
X
P=
Pw w.
w∈X ∗
Proposition 4 ([8]). Let σ be the monoid morphism defined over X ∗ by σ(x0 ) = −x1 and σ(x1 ) = −x0 .
Then
&
Y
1−z
P(1 − z) =
[σP(z)]
eζ(Sl )Ql .
z
l∈Lyn(X)\{x0 ,x1 }
Proof. It follows immediately from Proposition 1.
Example 7.
P2,1 (1 − z) =
ζ(3)
1−z
−P3 (z) + log(z)P2 (z) − log2 (z)P1 (z) +
z
1−z
Thus,
P2,1 (z) = −
z
z log(1 − z)
1 z log2 (1 − z)
ζ(3)
P3 (1 − z) +
P2 (1 − z) −
P1 (1 − z) +
.
1−z
1−z
2
1−z
1−z
By Formula (22) and Proposition 2, for w ∈ Y ∗ , there exist a finite set I and (ci )i∈I ∈ CI2 such that the
three following identities are equivalent
X
w =
ci y1 i ,
(23)
i∈I
Pw
=
X
Pci Pi
y1 ,
(24)
Hci Hiy1 .
(25)
i∈I
Hw
=
X
i∈I
In particular, for w = y1k , we have,
Lemma 2. Let M = mi,j 1≤i,j≤k be the matrix defined by mi,j = δi,j+1 (Kronecker symbol). Let ei,j
the matrix of size k × k, whose coefficients are all zero, except the one equal to 1 at line i and column j.
Let




Hy1
0
...
0
y1
0
... 0
Hy1
y1
 − Hy2
 − y22
... 0 
...
0 


2
2
2


.

..
..
A=
 and B = 
..
..
..
..
.
. 0
.
.

.
.
0 


(−1)k−2 yk−1
(−1)k−1 yk
(−1)k−2 Hyk−1
y1
(−1)k−1 Hyk
Hy1
... k
...
k
k
k
k
k
Then
 
 


 
Hy1
y1
1
k−1
`
k−1
`
Y
X
Y
X
 .. 
 
 . 
 
M ` A(t M )` +
eι,ι  ...  and  ..  = B
M ` B(t M )` +
eι,ι  ...  .
 . =A
ι=1
ι=1
`=1
`=1
Hy1k
1
y1k
P
k−1
Proof. The formula y1k = (−1)k−1 k −1 l=0 (−1)l y1l yk−l [6] can be written matricially as follows
 






0
...
0
y1
y12 
 y1 
0
 
y1
...
0 
 






..
..
 ..  = A  ..  = A  ..

 ..  .
.
.
 . 

.

.
0
. 
k−1
k−2
(−1)k−2 yk−1
k
y
1
y1
y1
y1
0
. . . k−1
k−1
Here all powers and products are carried out with the stuffle product. Successively, we get the expected
result.
Algorithmic and combinatoric aspects of multiple harmonic sums
65
The word y1k appears then as a computable stuffle product of words of length 1. Hence,
Proposition 5. Hy1k is a combination of {Hyr }1≤r<k which are algebraically independent.
Proof. The {Hyr }1≤r<k are algebraically independent according to Proposition 2, as image by the isomorphism H of the Lyndon words {yr }1≤r<k . By Lemma 2, we get the expected result.
Example 8. Since
y12 =
y1
y1 − y 2
2
and y13 =
2(y3 − y1
y2 ) + (y1
6
y 1 − y2 )
y1
then we have
Hy12 =
H2y1 − Hy2
2
and Hy13 =
2(Hy3 − Hy1 Hy2 ) + (H2y1 − Hy2 )Hy1
.
6
Identities (23-25) give rise to two interpretations : (24) enables to decompose Pw in a basis of singular
functions (1 − z)α logβ (1 − z) while (25) enables to compute an asymptotic expansion of its Taylor
coefficients in terms of N a logb N (or equivalently in terms of N a Hby1 (N )). Before stating a theorem
linking these two interpretations, we are interested in the action of C on Taylor coefficients; reciprocally,
we are interested in the effects of changing Taylor coefficients on a function in C[{Pw }w∈Y ∗ ].
3.2
Operations on the generating functions Pw
P
n
n
For f (z) =
n≥0 an z , we will henceforth denote [z ]f (z) = an its n-th Taylor coefficient. Since
n
multiplying or dividing by z acts very simply on [z ]f (z), we only have to study the effect of multiplying
or dividing by 1 − z.
[z n ](1 − z)Pw (z)
=
Pw (z)
1−z
=
[z n ]
Hw (n) − Hw (n − 1).
n
X
Hw (k)
(26)
(27)
k=0
(
(n + 1)Hw (n) − Hys−1 w0 (n) if w = ys w0 , with s 6= 1
=
Pn
(n + 1)Hw (n) − j=1 Hw0 (j − 1) if w = y1 w0 .
(28)
and, more generally,
Proposition 6.
n
k
[z ](1 − z) Pw (z) =
k X
k
j=0
3.3
j
(−1)j Hw (n − j) and [z n ]
Pw (z)
=
(1 − z)k
X
Hw (jk ).
n≥j1 ≥···≥jk ≥0
Operations on Taylor coefficients of Pw
We are now to find how multiplying or dividing Hw (N ) by N acts on Pw .
3.3.1
A particular case : w = The simple case w = , corresponding to H (N ) = 1, can be studied and treated by the following
Proposition 7. For any q ∈ Z, one has
 n
[z ](1 − z)P−q (z)


 n
[z ](1 − z)−1
nq =
1
z

n

Nq
 [z ]
1−z
1−z
if q < 0,
if q = 0,
if q > 0,
where Nq is defined by the following recurrence
N0 (X) = 1, and Nq (X) = X
X
q−1
j=0
(−1)q−1−j
q
Nj (X) .
j
66
Christian Costermans and Jean-Yves Enjalbert and Hoang Ngoc Minh
Example 9.
3.3.2
n
=
n2
=
z
1
1
n
=
[z
]
−
,
2
2
(1 − z) 1 − z
(1 − z)
z
2
3
1
2z
n
−
=
[z
]
−
+
.
[z n ]
(1 − z)3
(1 − z)2
(1 − z)3
(1 − z)2
1−z
[z n ]
How to divide by nk ?
Let w = ys1 · · · ysr and w0 = ys2 · · · ysr be the suffix of w, of length r − 1. The expression n−k Hw (n), k
positive integer, can be identified as follows
n−k Hw (n)
3.3.3
= n−k Hw (n − 1) + n−s1 −k Hw0 (n − 1)
= [z n ] Liyk w+ys1 +k w0 (z)
= [z n ][(1 − z)Pyk w+ys1 +k w0 (z)].
(29)
(30)
(31)
How to multiply by nk ?
In order to study the effect of multiplying by nk , k positive integer, we denote by θ = z∂/∂z the Euler
operator. Then for any integer k,
nk Hw (n) = [z n ]θk Pw (z).
(32)
So, we just have to compute θk Pw (z). As in [7], let us introduce
Definition 4. For any word w = xi1 · · · xik and for any composition r = (r1 , . . . , rk ), let τr (w) be
defined by τr (w) = τr1 (xi1 ) · · · τrk (xik ) with,
τ0 (x0 ) = x0 ,
and, for r ∈ N∗ ,
τr (x1 ) = x1 ,
τr (x0 ) = θr x0 = 0 and τr (x1 ) = θr
r!x1
zx1
=
.
1−z
(1 − z)r+1
We define the degree of r by deg(r) = k and its weight by wgt(r) = k + r1 + · · · + rk .
By applying successively the operator θ to L, we get
Lemma 3. θl L = Al L, where Al is defined by
X
Al (z) =
X
deg(r) Pi
Y
j=1 ri
wgt(r)=l w∈X deg(r) i=1
+j−1
ri
τr (w).
Proof. This is a consequence of the recurrence relation verified by Al , which is A0 (z) = 1, and, for all
l ∈ N, Al+1 (z) = [τ0 (x0 ) + τ0 (x1 )]Al (z) + θAl (z).
This lemma enables to extract the expression of θl Liw , for any word w ∈ X ∗ .
Example 10.
A0 (z)
A1 (z)
= 1,
= x0 +
z
x1 ,
1−z
z
z2
1
= x20 +
x0 tt x1 +
x21 +
x1 .
2
1−z
(1 − z)
(1 − z)2
A2 (z)
So, for w = x20 x1 ,
θ Lix20 x1
=
(x0 +
=
2
θ Lix20 x1
Lix x ,
0 1
=
(x20 +
z
x1 )L(z) x20 x1
1−z
=
Lix1 .
z
z2
1
2
2
t
t
x0 x1 +
x +
x1 )L(z) x0 x1
1−z
(1 − z)2 1 (1 − z)2
Algorithmic and combinatoric aspects of multiple harmonic sums
67
Lemma 4. Let ⊥ be the linear operator of Z[X] defined by ⊥X n = (n + 1)X n+1 + nX n and {Bl }l∈N ∈
Z[X] defined by B0 (X) = 1 and Bl+1 (X) = ⊥Bl (X). Then
θl (1 − z)−1 = (1 − z)−1 Bl (z(1 − z)−1 ).
Note that the head term of Bl , l ≥ 1, is l!X l and its trail term is X.
Example 11. B0 (X) = 1, B1 (X) = X, B2 (X) = 2X 2 + X, B3 (X) = 6X 3 + 6X 2 + X.
Proposition 8. With the notations of Lemma 4,
θk P(z) =
k
X
X
deg(r) Pi
Y
j=1 ri
X
+j−1 k
z
τr (w)Bj
P(z).
ri
j
1−z
j=1 wgt(r) w∈X deg(r) i=1
Using Leibniz formula, one has
k
θ Pw (z)
=
k X
k
j
j=0
=
k X
k
j
j=0
θk−j Liw (z)θj
Bj
z
1−z
1
1−z
(33)
1 k−j
θ
Liw (z).
1−z
(34)
Thanks to Lemma 3, we can extract the coefficient θl Liw of w in θl L : this can be written as C-linear
combination of Liv , with |v| ≤ |w| − l (where |u| denotes the length of a word u ∈ X ∗ ). We deduce so
the expression of θk Pw .
Example 12. For w = x20 x1 and k = 2,
θ2 Px20 x1 (z)
=
2 X
2
j=0
z
1−z
1 2−j
θ
Liw (z)
1−z
1
z
1
=
Lix1 (z) + 2
Lix0 x1 (z) + 2
1−z
1−z1−z
2z
z2 + z
= Px1 (z) +
Px0 x1 (z) +
P 2 (z).
1−z
1 − z x0 x1 2
2z
z +z
n2 H3 (n) = [z n ] P1 (z) +
P2 (z) +
P3 (z) .
1−z
1−z
So,
4
j
Bj
z
1−z
2
z
+
1−z
!
Lix20 x1 (z)
The main theorem
Throughout the section, we will write
fn ∼
∞
X
gi (n) for n → +∞,
i=0
for a scale of functions (gi )i∈N – i.e. verifying gi+1 (n) = O (gi (n)), for all i – to express that
fn =
I
X
gi (n) + O (gI+1 (n)) ,
for any I ≥ 0.
i=0
In the same way, given a scale of functions (hi )i∈N around z = 1 (i.e. verifying hi+1 (1 − z) =
O (hi (1 − z)), when z → 1) we will write
g(z) ∼
∞
X
hi (1 − z) for z → 1,
i=0
to mean
g(z) =
I
X
i=0
hi (1 − z) + O (hI+1 (1 − z)) for all I ≥ 0.
68
Christian Costermans and Jean-Yves Enjalbert and Hoang Ngoc Minh
For w = y1k , we know the expression of [z N ]Py1k (z) = Hy1k (N ) is given by Lemma 2. From the
second form of Euler-MacLaurin formula, involving the Bernoulli numbers {Bk }k≥0 , we get the following
asymptotic expansions
Hy1 (N ) ∼ log N + γ −
+∞
X
Bk 1
,
k Nk
k=1
Hyr (N ) ∼ ζ(r) −
+∞
X
1
Bk−r+1 k − 1 1
−
, for r > 1.
(r − 1)N r−1
k − r + 1 r − 1 Nk
k=r
Thus, we can deduce the asymptotic expansions of Hy1k (N ), for N → +∞, from the asymptotic expansions of {Hyr (N )}1≤r<k :
Example 13. From Example 8, we can deduce then
Hy12 (N )
=
Hy13 (N )
=
+
1
1
1
1 log(N ) + γ + 1
1
2
+O
,
(log(N ) + γ) − ζ(2) +
−
2
2
2
N
12N 2
N2
1
1
1
1
1
1
1 log2 (N )
log3 (N ) + γ log2 (N ) + (γ 2 − ζ(2)) log(N ) − ζ(2)γ + ζ(3) + γ 3 +
6
2
2
2
3
6
4 N
2
1
log(N ) 1
1
1
log
(N
)
1
γ
log(N
)
1
(γ + 1)
+
2γ + γ 2 − ζ(2)
−
−
+
+
O
.
2
N
4
N
24 N 2
8 12
N2
N2
Let us see in the general case how to reach the Taylor expansion of g ∈ C[(Pw )w∈Y ∗ ].
Theorem 2. Let g ∈ C[(Pw )w∈Y ∗ ]. There exist aj ∈ C, αj ∈ Z and βj ∈ N such that
g(z) ∼
+∞
X
aj (1 − z)αj logβj (1 − z), for z → 1.
j=0
Therefore, there exist bi ∈ C, ηi ∈ Z and κi ∈ N such that
[z n ]g(z) ∼
+∞
X
bi nηi logκi (n), for n → ∞.
i=0
Proof. Considering Corollary 1, we only have firstly to obtain the asymptotic expansion for the case
g(z) = Pw (z). Indeed, wePget then the expansions of f (z)g(z), for f ∈ C by remarking that z =
1 − (1 − z) and that z −1 = n≥0 (1 − z)n .
The first expansion can be derived from Proposition 4 which links the behaviour of Pw around z = 1
to the behaviour of some algebraic combination of functions {Pu }u∈X ∗ around z = 0. Moreover, by
Radford theorem 1, we can assume that each word u involved in this combination isP
a Lyndon word and
so belongs to x0 X ∗ x1 ∪ {x0 , x1 }. But, remind that, in this case, we have Pu (z) = n≥0 Hu (n)z n and
that Px0 (z) = (1 − z)−1 log(z). So, the expected first expansion follows.
From
(1 − z)α log(1 − z)β = (−1)β β!(1 − z)α+1 Pyβ (z),
1
(35)
we derive the second expansion by computing the Taylor coefficient [z n ](1 − z)α logβ (1 − z). Since we
have already explained how the multiplication by (1 − z)α acts on the Taylor coefficients, we just have
then to compute [z n ]Pyβ = Hyβ (n). For this, we use Lemma 2 which completes our proof.
1
1
Unfortunately, in the general case, knowing even the complete expansion of [z n ]g(z) only enables to
get an asymptotic expansion of g(z), as z → 1 up to order 0 (i.e. the singular part of the expansion).
Indeed, Taylor coefficients of all functions (1 − z)k , k ≥ 0 eventually vanish as in the following identity :
1
= [z n ] Li1 (z) = [z n ][Li1 (z) + (1 − z)2 ],
n
as soon as n > 2.
(36)
In fact, to obtain this singular part, it is sufficient to know the asymptotic expansion of [z n ]g(z) up to order
2 − , > 0 [15].
Algorithmic and combinatoric aspects of multiple harmonic sums
69
P
Remark 1. In the case of a finite sum i∈I bi nηi Hκ1 i (n), we are able to construct the unique function
f ∈ C[(Pw )w∈Y ∗ ] such that,
X
∀n ∈ N,
[z n ]f (z) =
(37)
bi nηi Hκ1 i (n),
i∈I
as illustrated in Examples 9 and 12.
Remark 2. Note that the proof of Theorem 2 gives an effective construction of the asymptotic expansion
of Taylor coefficients. In particular, applied to g(z) = Pw (z) directly, it enables to find an asymptotic
expansion of Hw (N ), as shown in the corollary below. Another algorithm, based on Euler Mac-Laurin
formula, is available in [1].
Corollary 2. Let Z be the Q-algebra generated by convergent polyzêtas and let Z 0 be the Q[γ]-algebra
generated by Z. Then there exist algorithmically computable coefficients bi ∈ Z 0 , κi ∈ N and ηi ∈ Z
such that, for any w ∈ Y ∗ ,
Hw (N ) ∼
+∞
X
bi N ηi logκi (N ), for N → +∞.
i=0
Example 14. From Example 7 we get, for z → 1
log2 (1 − z)
log2 (1 − z) log(1 − z)
ζ(3)
+ log(1 − z) − 1 −
+ (1 − z) −
+
+ O(|1 − z|).
P2,1 (z) =
1−z
2
4
4
But
[z N ]ζ(3)(1 − z)−1 = ζ(3),
[z N ] log(1 − z) = −N −1 ,
2!(1 − z)Py12 (z)
log2 (1 − z)
[z N ]
= [z N ]
2
2
= [z N ](1 − z)Py12 (z)
= Hy12 (N ) − Hy12 (N − 1),
..
.
We find finally, using Example 13 :
[z N ]P2,1 (z) = H2,1 (N ) = ζ(3) −
1 log(N )
1
log(N ) + 1 + γ
+
+
O
.
N
2 N2
N2
Otherwise, by Example 6,
P1,2 (z)
=
=
(1 − z)P1 (z)P2 (z) − 2P2,1
(z)
ζ(2)
− log(1 − z) z
−P2 (1 − z) + log(1 − z)P1 (1 − z) +
− 2P2,1 (z),
(1 − z)
1−z
1−z
z
calculated thanks to Proposition 4. So,
[z N ]P1,2 (z) = H1,2 (N ) = ζ(2)γ − 2ζ(3) + ζ(2) log(N ) +
1
ζ(2) + 2
+O
.
2N
N2
Corollary 3 ([8]). For any w ∈ Y ∗ , the N -free term in the asymptotic expansion of Hw (N ), when
N → +∞, is a polynomial qw in Z[γ]. This term is an element in Z, if and only if w is a convergent
word.
Example 15. qy1 y2 = ζ(2)γ − 2ζ(3) and qy2 y1 = ζ(3) = ζ(2, 1).
Question. For any convergent word w, are ζ(w) and γ algebraically independent ?
Now, let us go back to the As introduced in Section 1. We have seen that they are Z-linear combinations
on Hs , hence we get their asymptotic expansions with coefficients in Z 0 .
70
Christian Costermans and Jean-Yves Enjalbert and Hoang Ngoc Minh
Example 16. For s = (1, 1, 1),
A1,1,1 (N )
=
=
+
H1,1,1 (N ) + H1,2 (N ) + H2,1 (N ) + H3 (N ),
1
1
1
1
1
1
1 log2 (N )
log3 (N ) + γ log2 (N ) + [γ 2 + ζ(2)] log(N ) − ζ(2)γ + ζ(3) + γ 3 +
6
2
2
2
3
6
4 N
1
1
1
log(N ) 1 2
1 log2 (N )
log(N )
1
+ (9 − 2γ)
+O
.
(γ − 1)
+ [γ − 2γ + ζ(2)] −
2
N
4
N
24 N 2
24
N2
N2
Acknowledgements
We acknowledge the influence of Cartier’s lectures at the GdT Polylogarithmes et Polyzêtas. We greatly
appreciated fruitful discussions with Boutet de Monvel, Jacob, Petitot and Waldschmidt.
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