Functional equations and ladders for polylogarithms

Abstract

We give a number of S 3 -symmetric functional equations for polylogarithms up to weight 7. This allows one to obtain the first proven ladder relations,à la Lewin, of weight 6 and 7.

On functional equations for Nielsen polylogarithms

“…The second identity, known to Landen, follows from duplication and the three-term relation (16). For the analytic functions we have…”

“…given in [26] for m = 2 and in [36], [20] for m ≤ 3. Inaugural results in weight 6 and 7 are treated in [15], [16].…”

On functional equations for Nielsen polylogarithms

“…given in [39] for m = 2 and in [50], [32] for m ≤ 3. Inaugural results in weight 6 and 7 are treated in [27], [28]. For further examples, and background, we refer also to these theses [15,26,47].…”

“…. With these (28) From this identity we extract a corresponding analytic identity for S 3,2 where we write β = √ 2 for convenience. From this, an analytic identity can again be extracted.…”

“…Applying Corollary 20 to the five-term combinations from(28) and simplifying via the two-term relations for S ¡ 3,2 from Propositions 11 and 12 gives us the desired S ¡ 3,2 evaluation. Because of its length, this initial reduction is only reproduced in Appendix B.Since the original reduction is rather long, we have applied a well-known lattice reduction algorithm ('LLL') to obtain the following shorter, but only numerically checked, identity instead…”