# On functional equations for Nielsen polylogarithms

@article{Charlton2021OnFE, title={On functional equations for Nielsen polylogarithms}, author={Steven Charlton and Herbert Gangl and Danylo V. Radchenko}, journal={Communications in Number Theory and Physics}, year={2021} }

We derive new functional equations for Nielsen polylogarithms. We show that, when viewed modulo $\mathrm{Li}_5$ 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.

## 7 Citations

### Explicit formulas for Grassmannian polylogarithms

- Mathematics
- 2019

We give a new explicit formula for Grassmannian polylogarithms in terms of iterated integrals. We also explicitly reduce the Grassmannian polylogarithm in weight 4 and in weight 5 each to depth 2.…

### Clean Single-Valued Polylogarithms

- Mathematics, Computer ScienceSymmetry, Integrability and Geometry: Methods and Applications
- 2021

We define a variant of real-analytic polylogarithms that are single-valued and that satisfy ''clean'' functional relations that do not involve any products of lower weight functions. We discuss the…

### ITERATED INTEGRALS AND SPECIAL VALUES OF MULTIPLE POLYLOGARITHM AT ALGEBRAIC ARGUMENTS

- Mathematics
- 2022

We give systematic method that can convert many values of multiple polylogarithm at algebraic arguments into colored multiple zeta values (CMZV). Moreover, a new method to generate nonstandard…

### Iterated Integrals and Multiple Polylogarithm at Algebraic Arguments

- Mathematics
- 2022

. We give systematic method that can convert many values of multiple polylogarithm at algebraic arguments into colored multiple zeta values (CMZV). Moreover, a new method to generate nonstandard…

### Cluster Polylogarithms I: Quadrangular Polylogarithms

- Mathematics
- 2022

. We suggest a deﬁnition of cluster polylogarithms on an arbitrary cluster variety and classify them in type A . We ﬁnd functional equations for multiple polylogarithms which generalize equations…

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

- Mathematics
- 2020

We prove a conjecture of Goncharov, which says that any multiple polylogarithm can be expressed via polylogarithms of depth at most half of the weight. We give an explicit formula for this…

### Evaluation of one-dimensional polylogarithmic integral, with applications to infinite series

- Mathematics
- 2020

We give systematic method to evaluate a large class of one-dimensional integral relating to multiple zeta values (MZV) and colored MZV. We also apply the technique of iterated integrals and…

## References

SHOWING 1-10 OF 42 REFERENCES

### Functional equations for higher logarithms

- Mathematics
- 2002

AbstractFollowing earlier work by Abel and others,
Kummer gave in 1840 functional equations for the polylogarithm function
Lim
(z) up to
m = 5, but no example for larger
m was known until…

### On Nielsen's generalized polylogarithms and their numerical calculation

- Mathematics
- 1970

The generalized polylogarithms of Nielsen are studied, in particular their functional relations. New integral expressions are obtained, and relations for function values of particular arguments are…

### Functional equations and ladders for polylogarithms.

- Mathematics
- 2013

We give a number of S3-symmetric functional equations for polylogarithms up to weight 7. This allows one to obtain the first proven ladder relations, a la Lewin, of weight 6 and 7.

### Some New Transformation Properties of the Nielsen Generalized Polylogarithm

- MathematicsInt. J. Math. Math. Sci.
- 2014

By use of these transformation formulas presented, new fast algorithms for Nielsen generalized polylogarithm can be designed.

### Multiple polylogarithms in weight 4

- Mathematics
- 2016

We clarify the relationship between different multiple polylogarithms in weight~4 by writing suitable linear combinations of a given type of iterated integral I_{n_1,...,n_d}(z_1,...,z_d), in depth…

### Identities arising from coproducts on multiple zeta values and multiple polylogarithms

- Mathematics
- 2016

In this thesis we explore identities which can be proven on multiple zeta values using the derivation operators $ D_r $ from Brown's motivic MZV framework. We then explore identities which occur on…

### Multiple polylogarithms and mixed Tate motives

- Mathematics
- 2001

We develop the theory of multiple polylogarithms from analytic, Hodge and motivic point of view. Define the category of mixed Tate motives over a ring of integers in a number field. Describe…

### From polygons and symbols to polylogarithmic functions

- Mathematics
- 2011

A recipe is given for how to obtain the symbol of a multiple polylogarithm in terms of the combinatorial properties of an associated rooted decorated polygon, and it is indicated how that recipe relates to a similar explicit formula for it previously given by Goncharov.