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Draft:Multiple polylogarithm

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  • Comment: arXiv is an open-access so need to show papers were published in a reputable peer-reviewed journal. Also publications by the same author generally count as a single source. S0091 (talk) 17:58, 24 June 2024 (UTC)

In mathematics, the multiple polylogarithm is multivariable generalization of the polylogarithm. For special cases of it's arguments, the multiple polylogarithm reduces to the normal polylogarithm.

Definitions

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The multiple polylogarithms have numerous definitions.[1] Not all definitions are equivalent, but they are all related. Just like the polylogarithms, the multiple polylogarithm can be defined as either a recursive integral, or a convergant power series.

Recursive Integral

Define and for ,

.

Where denotes a path from to , and denotes travelling along that same path to a midway point . Often the subscript specifying the path is dropped.

Convergant Power Series

.[2]

We note that this power series definition allows us a natural generalization of the known identity between the classical polylogarithm and the Riemann zeta function, , by invoking the multiple zeta function:

.

Properties

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The recursive integral definition for integration beginning at a base-point can be broken up into sums and products of integrations beginning at .[3] For example:

Where is a path going , is from , is from , and the loop formed by traversing does not contain or .

References

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  1. ^ Duhr, Claude (2014-11-27). "Mathematical aspects of scattering amplitudes". p. 10. arXiv:1411.7538 [hep-ph].
  2. ^ Goncharov, A. B. (1998). "Multiple polylogarithms, cyclotomy and modular complexes". Math. Research Letters. 5 (4): 497–516. arXiv:1105.2076. doi:10.4310/MRL.1998.v5.n4.a7.
  3. ^ Duhr, Claude (2014-11-27). "Mathematical aspects of scattering amplitudes". p. 11. arXiv:1411.7538 [hep-ph].