On functional equations for Nielsen polylogarithms

Abstract

We derive new functional equations for Nielsen polylogarithms. We show that, when viewed modulo Li 5 and products of lower weight functions, the weight 5 Nielsen polylogarithm S 3,2 satisfies the dilogarithm five-term relation. We also give some functional equations and evaluations for Nielsen polylogarithms in weights up to 8, and general families of identities in higher weight.

On Dirichlet eigenvalues of regular polygons

“…An important property of the space of multiple polylogarithms is that it is closed under mutliplication. More precisely, one has (9) Li w (z)Li w ′ (z) = Li w¡w ′ (z) .…”

“…Note that for w ∈ X × x 1 the function Li w (z) extends analytically to C [1, ∞), and for w ∈ x 0 X × x 1 it is moreover continuous on D. We will call the words w ∈ x 0 X × x 1 convergent and we will also call convergent any formal linear combination of convergent words in C X (in other words, all elements of x 0 C X x 1 are convergent). An important corollary of (9), is that for any w ∈ X × x 1 there exists a unique collection of convergent elements w 0 , w 1 , . .…”

“…The first fact that we will need is that the function F N (x) := 2 F 1 (2/N, 1/N, 1 + 1/N; x) can be expanded as a power series in 1/N (convergent for N ≥ 3) whose coefficients are multiple polylogarithms. Let us recall the definition of Nielsen polylogarithms (see [16], [9])…”

Explicit formulas for Grassmannian polylogarithms

“…x, 1) = 0 (mod Li 5 , products). Explicit expressions for these identities are given in Appendix B. Additionally, in [9] we have established the 5-term relation for I + 4,1 (x, 1) = I 4,1 (x, 1) in x (modulo Li 5 and products), or rather the Nielsen polylogarithm S 3,2 (x) ¡ = I 4,1 (x, 1) + 4 Li 5 (x), although we will not use this. Analogously to the weight 4 case it is convenient to introduce the following symmetrization of I + 4,1 .…”

On the Goncharov depth conjecture and a formula for volumes of orthoschemes

“…This result is sharp as it is easy to show using a coproduct (discussed in §2.1) that a general multiple polylogarithm of weight n can not be expressed via multiple polylogarithms of depth strictly less than tn{2u . The previous best-known result states that a polylogarithm of depth n can be expressed via polylogarithms of depth at most n ´2; see [Cha17], [CGR19b], and [CGR19a] for further results on the depth reduction problem for multiple polylogarithms.…”