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**Example text**

There is a discretised version P of the Rota-Baxter P(σ )(η) = |ξ |≤|η| σ (ξ ) dξ of Sect. 4: σ (k) ∀σ ∈ C S ∗,∗ (R), P(σ )(n) = (25) |k|≤|n| which has properties similar to those of P as the following lemma shows. Lemma 4. For any σ ∈ C S ∗,k (R), there is a symbol P(σ ) ∈ C S ∗,k+1 (R) with same order max(0, α + 1) (where α is the order of σ ) as P(σ ), which interpolates P(σ ). More precisely, P(σ )(n) = P(σ )(n) ∀n ∈ N and for any σ ∈ C S ∗,k (R), P(σ ) − P(σ ) lies in C S ∗,k (R). Remark 9.

M(σ )} (for some m(σ ) ≤ k + l) such that σ1 < · · · < σk and σk+1 < · · · < σk+l . The domain Pσ is defined by: Pσ = {(x1 , . . , xk+l ) / xσr > xσr +1 if σr = σr +1 and xr = xr +1 if σr = σr +1 }. The second shuffle relations are: ζk (z 1 , . . , z k ) ζl (z k+1 , . . , z k+l ) = ζm(σ ) (Z σ ), (30) σ ∈mix sh(k,l) where Z σ is the m(σ )-uple defined by: Z (σ ) j = zi . ,k+l}, σ (i)= j For k = l = 1 they read: ζ (z 1 )ζ (z 2 ) = ζ (z 1 , z 2 ) + ζ (z 2 , z 1 ) + ζ (z 1 + z 2 ). Using the identification −R σz (ξ )dξ = 2ζ (z) derived previously we can indeed compute: 2 4ζ (z 1 )ζ (z 2 ) = − D(σzi ) i=1 R = − D(σz 2 ) R +− |ξ1 |=|ξ2 | |ξ1 <|ξ2 | D(σz 1 ) + − D(σz 1 ) R |ξ2 |<|ξ1 | D(σz 2 ) D(σz 1 ) ⊗ D(σz 2 ) = 4ζ (z 1 , z 2 ) + 4ζ (z 2 , z 1 ) + 4ζ (z 1 + z 2 ).

Z k ) ζl (z k+1 , . . , z k+l ) = ζm(σ ) (Z σ ), (30) σ ∈mix sh(k,l) where Z σ is the m(σ )-uple defined by: Z (σ ) j = zi . ,k+l}, σ (i)= j For k = l = 1 they read: ζ (z 1 )ζ (z 2 ) = ζ (z 1 , z 2 ) + ζ (z 2 , z 1 ) + ζ (z 1 + z 2 ). Using the identification −R σz (ξ )dξ = 2ζ (z) derived previously we can indeed compute: 2 4ζ (z 1 )ζ (z 2 ) = − D(σzi ) i=1 R = − D(σz 2 ) R +− |ξ1 |=|ξ2 | |ξ1 <|ξ2 | D(σz 1 ) + − D(σz 1 ) R |ξ2 |<|ξ1 | D(σz 2 ) D(σz 1 ) ⊗ D(σz 2 ) = 4ζ (z 1 , z 2 ) + 4ζ (z 2 , z 1 ) + 4ζ (z 1 + z 2 ).