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Erschienen in:
2002 | OriginalPaper | Buchkapitel
Multiple Polylogarithms: An Introduction
verfasst von : M. Waldschmidt
Erschienen in: Number Theory and Discrete Mathematics
Verlag: Hindustan Book Agency
Enthalten in: Professional Book Archive
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Multiple polylogarithms in a single variable are defined by <math display='block'> <mrow> <mi>L</mi><msub> <mi>i</mi> <mrow> <mrow><mo>(</mo> <mrow> <msub> <mi>s</mi> <mn>1</mn> </msub> <mo>,</mo><mo>⋯</mo><mo>,</mo><msub> <mi>s</mi> <mi>k</mi> </msub> </mrow> <mo>)</mo></mrow> </mrow> </msub> <mrow><mo>(</mo> <mi>z</mi> <mo>)</mo></mrow><mo>=</mo><mstyle displaystyle='true'> <munder> <mo>∑</mo> <mrow> <msub> <mi>n</mi> <mn>1</mn> </msub> <mo>></mo><msub> <mi>n</mi> <mn>2</mn> </msub> <mo>></mo><mo>⋯</mo><mo>></mo><msub> <mi>n</mi> <mi>k</mi> </msub> <mo>≥</mo><mn>1</mn> </mrow> </munder> <mrow> <mfrac> <mrow> <msup> <mi>z</mi> <mrow> <msub> <mi>n</mi> <mn>1</mn> </msub> </mrow> </msup> </mrow> <mrow> <msubsup> <mi>n</mi> <mn>1</mn> <mrow> <msub> <mi>s</mi> <mn>1</mn> </msub> </mrow> </msubsup> <mo>⋯</mo><msubsup> <mi>n</mi> <mi>k</mi> <mrow> <msub> <mi>s</mi> <mi>k</mi> </msub> </mrow> </msubsup> </mrow> </mfrac> </mrow> </mstyle> </mrow> </math>$$L{i_{\left( {{s_1}, \cdots ,{s_k}} \right)}}\left( z \right) = \sum\limits_{{n_1} > {n_2} > \cdots > {n_k} \geqslant 1} {\frac{{{z^{{n_1}}}}} {{n_1^{{s_1}} \cdots n_k^{{s_k}}}}}$$, when s1, … , s k are positive integers and z a complex number in the unit disk. For k = 1, this is the classical polylogarithm Li s (z). These multiple polylogarithms can be defined also in terms of iterated Chen integrals and satisfy shuffle relations. Multiple polylogarithms in several variables are defined for s i ≥ 1 and |z i | < 1(1 ≤ i ≤ k) by <math display='block'> <mrow> <mi>L</mi><msub> <mi>i</mi> <mrow> <mrow><mo>(</mo> <mrow> <msub> <mi>s</mi> <mn>1</mn> </msub> <mo>,</mo><mo>⋯</mo><mo>,</mo><msub> <mi>s</mi> <mi>k</mi> </msub> </mrow> <mo>)</mo></mrow> </mrow> </msub> <mrow><mo>(</mo> <mrow> <msub> <mi>z</mi> <mn>1</mn> </msub> <mo>,</mo><mo>⋯</mo><msub> <mi>z</mi> <mi>k</mi> </msub> </mrow> <mo>)</mo></mrow><mo>=</mo><mstyle displaystyle='true'> <munder> <mo>∑</mo> <mrow> <msub> <mi>n</mi> <mn>1</mn> </msub> <mo>></mo><msub> <mi>n</mi> <mn>2</mn> </msub> <mo>></mo><mo>⋯</mo><mo>></mo><msub> <mi>n</mi> <mi>k</mi> </msub> <mo>≥</mo><mn>1</mn> </mrow> </munder> <mrow> <mfrac> <mrow> <msubsup> <mi>z</mi> <mn>1</mn> <mrow> <msub> <mi>n</mi> <mn>1</mn> </msub> </mrow> </msubsup> <mo>⋯</mo><msubsup> <mi>z</mi> <mi>k</mi> <mrow> <msub> <mi>n</mi> <mi>k</mi> </msub> </mrow> </msubsup> </mrow> <mrow> <msubsup> <mi>n</mi> <mn>1</mn> <mrow> <msub> <mi>s</mi> <mn>1</mn> </msub> </mrow> </msubsup> <mo>⋯</mo><msubsup> <mi>n</mi> <mi>k</mi> <mrow> <msub> <mi>s</mi> <mi>k</mi> </msub> </mrow> </msubsup> </mrow> </mfrac> </mrow> </mstyle> </mrow> </math>$$L{i_{\left( {{s_1}, \cdots ,{s_k}} \right)}}\left( {{z_1}, \cdots {z_k}} \right) = \sum\limits_{{n_1} > {n_2} > \cdots > {n_k} \geqslant 1} {\frac{{z_1^{{n_1}} \cdots z_k^{{n_k}}}} {{n_1^{{s_1}} \cdots n_k^{{s_k}}}}}$$, and they satisfy not only shuffle relations, but also stuffle relations. When one specializes the stuffle relations in one variable at z = 1 and the stuffle relations in several variables at z1 = ⋯ = z k = 1, one gets linear or quadratic dependence relations between the Multiple Zeta Values <math display='block'> <mrow> <mi>ζ</mi><mrow><mo>(</mo> <mrow> <msub> <mi>s</mi> <mn>1</mn> </msub> <mo>,</mo><mo>⋯</mo><mo>,</mo><msub> <mi>s</mi> <mi>k</mi> </msub> </mrow> <mo>)</mo></mrow><mstyle displaystyle='true'> <munder> <mo>∑</mo> <mrow> <msub> <mi>n</mi> <mn>1</mn> </msub> <mo>></mo><msub> <mi>n</mi> <mn>2</mn> </msub> <mo>></mo><mo>⋯</mo><mo>></mo><msub> <mi>n</mi> <mi>k</mi> </msub> <mo>≥</mo><mn>1</mn> </mrow> </munder> <mrow> <mfrac> <mn>1</mn> <mrow> <msubsup> <mi>n</mi> <mn>1</mn> <mrow> <msub> <mi>s</mi> <mn>1</mn> </msub> </mrow> </msubsup> <mo>⋯</mo><msubsup> <mi>n</mi> <mi>k</mi> <mrow> <msub> <mi>s</mi> <mi>k</mi> </msub> </mrow> </msubsup> </mrow> </mfrac> </mrow> </mstyle> </mrow> </math>$$\zeta \left( {{s_1}, \cdots ,{s_k}} \right)\sum\limits_{{n_1} > {n_2} > \cdots > {n_k} \geqslant 1} {\frac{1} {{n_1^{{s_1}} \cdots n_k^{{s_k}}}}}$$ which are defined for k, s1, … ,s k positive integers with s1 ≥ 2. The Main Diophantine Conjecture states that one obtains in this way all algebraic relations between these MZV.