The functional τ_X on rational functions with simple integer poles

For a numerator A ∈ ℚ[x] and a finite set Pl ⊆ ℤ of poles, g = A / ∏_{r ∈ Pl} (x - r):

• polyPart A Pl = A /ₘ ∏ (x - r) (the polynomial part);
• resP A Pl r = A(r) / ∏_{s ≠ r} (r - s) (the residues);
• tauX A Pl = τ(polyPart) + ∑_r res_r (H⁽⁵⁾_{d(r)} - X) ∈ ℚ[X], with d(r) = r for r ≥ 0 and d(r) = -r - 1 for r < 0.

partial_fractions : A = P Π + ∑_r res_r ∏_{s ≠ r} (x - s).

def Zeta5Irrational.dd (r : ℤ) : ℕ

The harmonic index d(r).

Equations

• Zeta5Irrational.dd r = if 0 ≤ r then r.toNat else (-r - 1).toNat

noncomputable def Zeta5Irrational.tauX (A : Polynomial ℚ) (Pl : Finset ℤ) : Polynomial ℚ

The functional τ_X.

Equations

• One or more equations did not get rendered due to their size.