Identities arising from coproducts on multiple zeta values and multiple polylogarithms
The final version of my thesis is available in the Durham e-Theses repository. Alternatively a version of my thesis is hosted here (this version has the correct page numbering, and PDF bookmarks).
Supporting calculations
This page also contains Mathemaica worksheets with supporting calculations for the various identities and theorems in my thesis.
Most worksheets (those dealing with (multiple) polylogarithms) require Claude Duhr's PolylogTools package, which is not yet publicly available.
Each worksheet should function to perform one single stand-alone series of calculations, in order from start to finish, by evaluating the entire worksheet in one go. If there are problems with any worksheet, please let me know at charlton(at)math(dot)uni-tuebingen(dot)de.
Chapter 2: Cyclic insertion
These worksheets require the MZV functions defined in mathmzv2.m.
- Regularisation of divergent iterated integrals.
- Amenability of cyclic insertion to a proof by indution
- A different symmetrisation of \( \zeta_C(1, 3, 3, 3 \mid a_1, \ldots, a_5) \).
- Alternating block identities
Chapter 4: Relating weight 5 MPL's
- Memory and time problems.
- Symmetries of \( I_{4,1} \).
- Symmetries of \( I_{3,2} \).
- \( \Li_5 \) terms to express \( I_{3,2} \) in terms of \( I_{4,1} \) (as in Appendix B).
- Symmetries of \( I_{3,1,1} \) and the nullspace used.
- Symmetries of \( I_{2,2,1} \) and the nullspace used.
- Symmetries of \( I_{2,1,2} \) and the nullspace used.
- Symmetries of \( I_{1,2,2} \) and the nullspace used.
- \( \Li_5 \) terms for the \( I_{3,1,1} \) 4-term identities.
- \( \Li_5 \) terms for the \( I_{3,1,1} \) 5-term identities.
- Relating deph 2 iterated integrals.
- Relating the weight 5 depth 3 iterated integrals (as in Appendix B).
- Reduction of \( I_{2,1,1,1} \) to \( I_{3,1,1} \).
- Examples of \( I_{1,1,1,1,1} \) identities.
Chapter 5: Dan's reduction
- Dan's reduction applied to weight 4, and an automatically LaTeXed version of the result.
- Check Dan's reduction using the symbol.
- Dan's `cyclic' expression for \( I_{1,1,1,1} \), and the resulting \( I_{3,1} \) functional equation.
- Dan's procedure applied to weight 5.
- Symbol to check weight 5 reduction.
Chapter 6: \( I_{a,b} \) inversion identity
- \( I_{1,1} \) inversion identity.
- \( I_{2,1} \) inversion identity.
- \( I_{3,1} \) inversion identity.
- \( I_{4,1} \) inversion identity.
- \( I_{5,1} \) inversion identity.
- General \( I_{n,1} \) inversion identity (products).
- General \( I_{n,1} \) inversion identity (symbol).
- Numerically testable \( I_{n,1} \) inversion identity.
- Low weight \( I_{a,b} \) inversion identity tests.
- High weight \( I_{a,b} \) inversion identity experiments.
Chapter 7: Goncharov-motivated polylogarithm identities
\( I_{4,1} \) identities
- Identity for \( I_{4,1}^-(x, \text{algebraic \( \Li_2 \)}) \).
- Cancellation of terms in the proof of \( I_{4,1}^-(x, \text{algebraic \( \Li_2 \)}) \) identity in Appendix C.
- A \( \Li_4 \) functional equation needed for proof of \( I_{4,1}^-(x, \text{algebraic \( \Li_2 \)}) \) in Appendix C.
- Identity for \( I_{4,1}^+(\text{algebraic \( \Li_2 \)},y) \).
- Identity for \( I_{4,1}^-(\text{3-term \( \Li_3 \)},y) \).
- Identity for \( I_{4,1}^-(\text{algebraic \( \Li_3 \)},y) \).
- Data to verify \( I_{4,1}^-(\text{algebraic \( \Li_3 \)},y) \) identity.
- Weight 5 Nielsen identity, used when deriving \( \Li_5 \) functional equation.
- \( \Li_5 \) functional equation from \( I_{4,1}^-(\text{algebraic \( \Li_2 \)}, \text{3-term \( \Li_3 \)}) \).
- Rational \( a = 1, b = 2 \) case of \( \Li_5 \) functional equation from \( I_{4,1}^-(\text{algebraic \( \Li_2 \)}, \text{3-term \( \Li_3 \)}) \).
- \( \Li_5 \) functional equation from \( I_{4,1}^-(\text{algebraic \( \Li_2 \)}, \text{algebraic \( \Li_3 \)}) \).
- Rational \( a = 1, b = 2 \) case of \( \Li_5 \) functional equation from \( I_{4,1}^-(\text{algebraic \( \Li_2 \)}, \text{algebraic \( \Li_3 \)}) \).
\( I_{5,1} \) identities
- Identity for \( I_{5,1}^+(\text{algebraic \( \Li_3 \)},y) \).
- Data to verify \( I_{5,1}^+(\text{algebraic \( \Li_3 \)},y) \) identity.
- Identity for \( I_{5,1}^-(\text{algebraic \( \Li_4 \)},y) \).
- Data to verify \( I_{5,1}^-(\text{algebraic \( \Li_4 \)},y) \) identity.
- Identity for \( I_{5,1}^+(\text{3-term \( \Li_3 \)},y) \).
- Weight 6 Nielsen identity, used when deriving \( \Li_6 \) functional equation.
- \( \Li_6 \) functional equation from \( I_{5,1}^+(\text{3-term \( \Li_3 \)}, \text{3-term \( \Li_3 \)}) \).
- \( \Li_6 \) functional equation from \( I_{5,1}^+(\text{algebraic \( \Li_3 \)}, \text{3-term \( \Li_3 \)}) \).
- Rational \( a = 1, b = 2 \) case of \( \Li_6 \) functional equation from \( I_{5,1}^+(\text{algebraic \( \Li_3 \)}, \text{3-term \( \Li_3 \)}) \).
- \( \Li_6 \) functional equation from \( I_{5,1}^+(\text{algebraic \( \Li_3 \)}, \text{algebraic \( \Li_3 \)}) \).
- Rational \( a = 1, b = 2 \) case of \( \Li_6 \) functional equation from\( I_{5,1}^+(\text{algebraic \( \Li_3 \)}, \text{algebraic \( \Li_3 \)}) \).
- \( \Li_6 \) functional equation from \( I_{5,1}^-(\text{algebraic \( \Li_4\)}, \text{3-term \( \Li_4 \)}) \).
- Rational \( a = 1, b = 2 \) case of \( \Li_6 \) functional equation from\( I_{5,1}^-(\text{algebraic \( \Li_4\)}, \text{3-term \( \Li_4 \)}) \).
\( I_{6,1} \) identities
- Identity for \( \widehat{I_{6,1}^-}(\text{algebraic \( \Li_3 \)},y) \).
- Data to verify \( \widehat{I_{6,1}^-}(\text{algebraic \( \Li_3 \)},y) \) identity.
- Identity for \( I_{6,1}^+(\text{algebraic \( \Li_4 \)},y) \).
- Data to verify \( I_{6,1}^+(\text{algebraic \( \Li_4 \)},y) \) identity.
- Identity for \( \widehat{I_{6,1}^-}(\text{3-term \( \Li_3 \)},y) \).