REPRESENTATION OF SOME SPECIAL FUNCTIONS ON TRANSCENDENCE BASIS

The special functions such asmultiple harmonic sums, polyzetas, or multiple polylogarithms are compatible with the structure of quasi‐shuffle algebras. We express non‐commutative generating series of these special functions on the transcendence bases of the algebras and then identify local coordinates to reduce their polynomial relations or asymptotic expansions indexed by these bases.


Introduction
A harmonic sum for the simple index, s ∈ N + , is defined by the sum H s (N ) := 1 + 1 2 s + . . . + 1 N s . We know that the limit lim Fortunately, the multiple harmonic sums are compatible with the algebra of the stuffle product; whereas, the multiple polylogarithms are compatible with the algebra of the shuffle product when they are observed in the forms of iterated Chen integrals. As a consequence of these results, the polyzetas are compatible with both of the structures. In this paper, we briefly review a general result about Hopf algebras (in Section 2), of the quasishuffle product and the concatenation product, constructed on a space of formal polynomials freely generated by some alphabet. They admit transcendence bases (see [1]) on which the special functions can be expressed as non-commutative generating series in respect of Hausdorff group 1 : Thanks to relations among the non-commutative 1 Hausdorff group is the group of group-like elements in a Hopf algebra. LynX, LynY denote the sets of Lyndon words generated by the alphabets Quasi-shuffle algebra with the deformation q Let's denote Y := {y k | k ∈ N + } an alphabet totally ordered by y 1 y 2 · · · . A word is a finite sequence of letters and Y * denotes the set of all words including the empty word, denoted by 1 Y * . This set is a free monoid 2 and 1 Y * is a neutral element. We call each linear combination, over the field Q, of words in Y * a (formal) polynomial and Q Y denotes the set of all polynomials. This set equipped with the concatenation product follows a free algebra with unit 1 Y * . A Lyndon word is a nonempty word that is smaller than all its nontrivial proper right factors and LynY denotes the set of all Lyndon words in Y * . For any q belonging to any field containing the field of rational number, the q−stuffle product, denoted by q , is defined by recurrent formula as follows:
This product is exactly the shuffle product (denoted by ) for q = 0 and the stuffle product (denoted by ) for q = 1. This product is commutative and associative hence, (A Y , q , 1 Y * ) is a commutative, associative algebra with unit, where A := Q[q] is the field extension of Q containing q. Here, we still use the notation q as a morphism We denote ∆ q and ∆ conc as the dual laws of the q−stuffle product and the concatenation product, respectively; this means that for all w in Y * , We proved (in paper [1]) that the coproduct ∆ q is compatible with the concatenation product. This means that ∆ q (uv) = ∆ q (u)∆ q (v), whereas ∆ conc is compatible with the q-stuffle product, that . An important point to note here is the weight of the word w = y s1 . . . y sr to be (and denoted by) (w) = s 1 +. . .+ s r . Due to these definitions we can see that ∆ q (w) is the polynomial of words in weight (w) and u q v is the polynomial of words in weight (u) + (v). Consequently, they all form the two algebraic structures in duality as follows:
This basis reduces a transcendence basis, . This permits us to express the diagonal series product on the left of the tensor and the concatenation product on the right.
where the last product takes Lyndon words in decreasing order.

Representation of multiple polylogarithms
We now consider the above algebra in the case of the alphabet X = {x 0 , x 1 }, totally ordered by x 0 ≺ x 1 , with the shuffle product (it means q = 0). At that time, the couple of bases in duality [2] is denoted by {P w } w∈X * , the PBW-basis, and {S w } w∈X * , Schützenberger basis. It follows from (11) that 5 We have seen at (3) that a multiple polylogarithms is determined for each multi-index s = (s 1 , . . . , s r ). In this section, we use encoding that each composition of positive integers s = (s 1 , . . . , s r ) associates with the word w = x s1−1 0 x 1 . . . x sr−1 0 x 1 . Thus, the multiple polylogarithms can be rewritten as: Using two differential forms ω 0 (z) := dz z and ω 1 (z) := dz 1−z with the conventions that Li 1 * X = 1 and Li x0 (z) = ∫ z 1 ω 0 (t) = log(z), one can express the multiple polylogarithms, thank to Frederich criterion, in the form of iterated integral [3,4], Following this representation, one proved that the multiple polylogarithms are compatible with the shuffle product, namely [2,5]: This permits us to extend Li as a morphism: The non-commutative generating series of multiple polylogarithms is defined as an image of the morphism 6 on the double series D X though Li • ⊗id X * : On the other hand, for any Lyndon word l ∈ (LynX) X, one has S l ∈ x 0 X * x 1 . Therefore, we can consider the non-commutative generating series, denoted by L reg , as well as its evaluation at z = 1 [3], we have Moreover, one studies about the monodromy of the multiple polylogarithms on close curves by Chen's series and the differential equation Drinfield [3,6] to state the following proposition: ii) In the special case of curve 1 − t, one has 7 We now use an automorphism of Q X of the concatenation product, denoted by σ, verified σ(x 0 ) = −x 1 , σ(x 1 ) = −x 0 . Note that, for all words w ∈ X * , P w and S w are homogeneous polynomial of weight 8 |w|, the length of w. Furthermore, Q X is a graded space admitting two graded bases {P w } w∈X * and {S w } w∈X * . We can see more precise by the following diagram illustrating a matrix representation of σ in a subspace of weight n, denoted by X n := span{u

Proposition 4. Let L(z) be the non-commutative generating series of multiple polylogarithms, we have
For this reason, we can rewrite relation (17) as follows: From this formula, by identifying local coordinates, we get relations among the multiple polylogarithms indexed by basis {S l } l∈LynX . The following example are computed by our program running under Maple.

Representation of multiple harmonic sums
We have seen at (1)  This allows us to prove, by induction, that multiple harmonic sums are compatible with the stuffle product [7]. It means that for all words w 1 , w 2 ∈ Y * , we have Proposition 5. The mapping w −→ H w is the isomorphism between (Q Y , , 1 Y * ) and the algebra of multiple harmonic sums with the standard product, denoted by (H R , ·, 1). 7 Here, we understand L(z) as L(x 0 , x 1 |z). 8 |w| denotes the length of the word w. Because the set of the Lyndon words freely generates the algebra of the quasi-shuffle product [8], it follows the isomorphism H R Q[H l , l ∈ LynY ]. Moreover, by using the expression of diagonal series D Y (see (11)), we can factorize the non-commutative generating series of multiple harmonic sums H := ∑ w∈Y * H w w as follows The original generating series of multiple harmonic sums forms a multiple polylogarithms deformed the factor 1 1−z , namely for all multiindices s = (s 1 , s 2 , . . . , s r ), Indeed, Here we accept that H s (n) = 0 for any n < r. In other words, H s (N ) is the coefficient of z N in the Taylor development of Lis(z) 1−z in the system {z N |N ∈ N}. By the way, according to the representations of multiple polylogarithms (in the above subsection) we obtain relations or asymptotic expansions of multiple harmonic sums.

3.2.1
Generating series of multiple harmonic sums on the alphabet X For any word w ∈ X * , we denote G X w (z) := Lw(z) 1−z and G X (z) := ∑ w∈X * G X w (z). By the way, using formula (20), we have the following expressions: Example 6. According to equality (27), we reduce the following relations by identifying local coordinates 9 : We use the notation From the representation of G X w (z), we can reduce asymptotic expansions of multiple harmonic sums in the scale of comparison {z i log j (n), i, j ∈ N}.

Generating series of multiple harmonic sums on the alphabet Y
We now use the linear projection π Y : Q X −→ Q Y which associates every word x 1 with the word y s1 . . . y sr and admits the convention π Y (wx 0 ) = 0 for any w ∈ X * . Then, for any word w = y s1 . . . y sr ∈ Y * , we set G Y w (z) := From this and formula (20), we have Moreover, by expanding exp((y 1 + 1) log 1 1−z ) in the form of the original generating series of y 1 , we get exp((y 1 + 1) log Consequently, Example 8. According to expression (28), we reduce the following relations by identifying local coordinates:

Representations of polyzetas
As we see the definition of polyzetas at (2), these convergent series are also compatible with the stuffle product like multiple harmonic sums. Using the expression in Proposition 6, we set On the other hand, we conclude from (16) that polyzetas are also obtained by letting z −→ 1 in multiple polylogarithms. Due to the isomorphism in the algebraic structures, we establish a bridge equation between the generating series Z and Z as follows.

Proposition 7 ([9]). We have a bridge equation between
the two spaces C X and C Y : where

B(y1)
Zγ πY Let Z n be the Q-vector space generated by polyzetas of weight n. Using this formula, we can rewrite the two sides on the same transcendence basis and then reduce the relations among polyzetas by identifying the local coordinates. On the one hand, by expressing the right hand side of (29) on the basis {Σ l } l∈LynY , we can identify coefficients on this basis [9,10]: Relations of polyzetas in terms of irreducible elements indexed by the basis {Σ l } l∈LynY : Weight 6: On the other hand, we use the inverse of π Y , denoted by π X , to express equality (29) on the basis {S l } l∈LynX . It means that π X is a morphism defined on the word by π X (y s1 . . . y sr ) = x s1−1 0 x 1 . . . x sr −1 0 x 1 , and applying to the two sides of (29) we have Weight 5: Weight 6: The above examples, one again, show us that each polyzetas only has a linear representation of plyzetas of the same weight. Hence, we can list the elements of linear bases of Z corresponding to the bases {Σ l } l∈LynY and {S l } l∈LynX that confirm the Zagier's dimension conjecture 10 (see [4]). Moreover, we can reduce algebraic bases (the normal product) from these representations. We show here two lists of irreducible elements up to weight 12 (see more [11]): 1. For the basis {Σ l } l∈LynY : ζ(Σ y 2 ), ζ(Σ y 3 ), ζ(Σ y 5 ), ζ(Σ y 7 ), ζ(Σ y 3 y 5 1 ), ζ(Σ y 9 ), ζ(Σ y 3 y 7 1 ), ζ(Σ y 11 ), ζ(Σ y 2 y 9 1 ), ζ(Σ y 2 2 y 8 1 ), ζ(Σ y 3 y 9 1 ).

Conclusion
We represented special functions (multiple harmonic sums, polyzetas, and multiple polylogarithms) in forms of non-commutative generating series indexed by transcendence bases of quasi-shuffle algebras. By identifying the local coordinates of the Hausdorff groups, in shuffle and stuffle Hopf algebras, we can reduce polynomial relations or asymptotic expansions of these special functions indexed by the bases.