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MultiplePolyLog

MultiplePolyLog[{z₁,…,z_k},{s₁,…,s_k}]

gives the multiple polylogarithm $TemplateBox[{TemplateBox[{{{z, _, 1}, ,, ..., ,, {z, _, k}}}, ImplicitList], TemplateBox[{{{s, _, 1}, ,, ..., ,, {s, _, k}}}, ImplicitList]}, MultiplePolyLog]$ , defined by the nested sum .

Details

MultiplePolyLog is the depth- extension of PolyLog, formed by strictly ordered, nested series with terms combining argument powers and reciprocal index powers.
Series definition (primary): in its domain of absolute convergence,
$TemplateBox[{TemplateBox[{{{z, _, 1}, ,, ..., ,, {z, _, k}}}, ImplicitList], TemplateBox[{{{s, _, 1}, ,, ..., ,, {s, _, k}}}, ImplicitList]}, MultiplePolyLog]=sum_( n_1 > n_2 > ... > n_k >= 1)(s_1^(n_1)s_2^(n_2)...s_k^(n_k))/(z_1^(n_1)z_2^(n_2)...z_k^(n_k))$ .
This is a mathematical function that is suitable for both symbolic and numerical manipulation.
The function supports numerical evaluation of the following types:
machine precision N Yes

arbitrary precision N Yes

worst-case intervals CenteredInterval No

average-case intervals Around No
Convergence criteria for $TemplateBox[{TemplateBox[{{{z, _, 1}, ,, ..., ,, {z, _, k}}}, ImplicitList], TemplateBox[{{{s, _, 1}, ,, ..., ,, {s, _, k}}}, ImplicitList]}, MultiplePolyLog]$ with :

	Inside the open poly-disc: all p_i < 1	the series converges
	Outside the poly-disc: p_i > 1 for some j	the series diverges
	On the poly-disc boundary: p_j = 1 for some j	convergence depends on the values of

Outside the convergence region, MultiplePolyLog returns ComplexInfinity.
In $TemplateBox[{TemplateBox[{{{z, _, 1}, ,, ..., ,, {z, _, k}}}, ImplicitList], TemplateBox[{{{s, _, 1}, ,, ..., ,, {s, _, k}}}, ImplicitList]}, MultiplePolyLog]$ , the entries are the indices (also called weights), and are the decoration values (also called colors or numerator weights).
The iterated-integral realization of MultiplePolyLog is implemented as GeneralizedPolyLog. For positive integer indices and within the convergence region, they are related by ( denotes a sequence of zeros):
$TemplateBox[{TemplateBox[{{{z, _, 1}, ,, ..., ,, {z, _, k}}}, ImplicitList], TemplateBox[{{{s, _, 1}, ,, ..., ,, {s, _, k}}}, ImplicitList]}, MultiplePolyLog]=(-1)^kG(0^(z_1-1),1/(s_1),0^(z_2-1),1/(s_1s_2),...,0^(z_k-1),1/(s_1s_2...s_k);1)$ .
The following are a few relations satisfied by the function:
Relation to PolyLog

Relation to MultipleZeta $TemplateBox[{TemplateBox[{{{z, _, 1}, ,, , ..., ,, , {z, _, k}}}, ImplicitList], TemplateBox[{{1, ,, ..., ,, 1}}, ImplicitList]}, MultiplePolyLog]=TemplateBox[{TemplateBox[{{{z, _, 1}, ,, ..., ,, {z, _, k}}}, ImplicitList]}, MultipleZeta]$

Relation to MultipleHarmonicNumber (when the series converges) $TemplateBox[{TemplateBox[{{{z, _, 1}, ,, , ..., ,, , {z, _, k}}}, ImplicitList], TemplateBox[{{{s, _, {(, 1, )}}, ,, ..., ,, {s, _, {(, k, )}}}}, ImplicitList]}, MultiplePolyLog]=TemplateBox[{infty, TemplateBox[{{{z, _, 1}, ,, ..., ,, , {z, _, k}}}, ImplicitList], TemplateBox[{{{s, _, 1}, ,, ..., ,, {s, _, k}}}, ImplicitList]}, MultipleHarmonicNumber3]$
Sum, GeneratingFunction, ExponentialGeneratingFunction, ZTransform and DirichletTransform may return results in terms of MultiplePolyLog.

Examples

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Basic Examples (4)

Exact values are generated automatically for simple arguments:

Wolfram Language code: MultiplePolyLog[{1, 1}, {1 / 2, 1 / 3}]

The defining sum:

Wolfram Language code: Sum[((1 / 2)^n1(1 / 3)^n2/n1^2 n2), {n1, 1, Infinity}, {n2, 1, n1 - 1}]

Plot by varying the first index with decoration values fixed:

Wolfram Language code: Plot[MultiplePolyLog[{z, 1}, {1 / 2, 1 / 3}], {z, 1, 10}, PlotRange -> All]

Plot along a real slice of a decoration value:

Wolfram Language code: Plot[MultiplePolyLog[{1, 1}, {1 / 2, x}], {x, -1, 1}]

Scope (16)

Numerical Evaluation (3)

Evaluate numerically:

Wolfram Language code: N[MultiplePolyLog[{2, 4, 6}, {1 / 2, 1 / 3, 1 / 4}]]

Evaluate to high precision:

Wolfram Language code: N[MultiplePolyLog[{2, 4, 6}, {1 / 2, 1 / 3, 1 / 4}], 20]

Evaluate with complex/rational weights and decoration values:

Wolfram Language code: N[MultiplePolyLog[{2 + I, 4 / 3, 6}, {1 / 2 + I / 3, 1 / 3, 1 / 4}]]

Specific Values (6)

Evaluate for unity weights:

Wolfram Language code: MultiplePolyLog[{1, 1}, {1 / 2, 1 / 3}]

Evaluate at decoration values :

Wolfram Language code: MultiplePolyLog[{2, 1}, {1, -1}]

Evaluate at conjugate decoration values :

Wolfram Language code: MultiplePolyLog[{2, 1}, {-I, I}]

A MultiplePolyLog with negative integers and zeros as the indices evaluates to a rational number:

Wolfram Language code: MultiplePolyLog[{-2, -3}, {1 / 2, 1 / 3}]

Wolfram Language code: MultiplePolyLog[{-2, 0, 0}, {1 / 2, -1 / 2, 1 / 3}]

A zero in the decoration list makes the MultiplePolyLog vanish identically:

Wolfram Language code: MultiplePolyLog[{6, 2, 3, 4}, {-I, 0, 1 / 2, -1 / 2}]

A divergent MultiplePolyLog:

Wolfram Language code: MultiplePolyLog[{2, 3}, {6, 10}]

Visualization (7)

Plot of the diagonal depth-2 values across positive , highlighting rapid decay with increasing weight:

Wolfram Language code: Plot[MultiplePolyLog[{n, n}, {1 / 2, 1 / 3}], {n, 1, 10}]

Plot a MultiplePolyLog across positive and negative indices:

Wolfram Language code: Plot[Log10@MultiplePolyLog[{n, 1}, {1 / 2, 1 / 3}], {n, -10, 10}]

Side-by-side traces of and that emphasize order asymmetry under index swap:

Wolfram Language code: Plot[Evaluate@{MultiplePolyLog[{n, 2}, {1 / 2, 1 / 3}], MultiplePolyLog[{2, n}, {1 / 2, 1 / 3}]}, {n, 2, 20}, ...]

3D plot of Log10[MultiplePolyLog[{z1,z2},{1/2,1/3}]] over positive indices, showing how magnitude varies across the grid:

Wolfram Language code: Plot3D[Abs@Log10@MultiplePolyLog[{z1, z2}, {1 / 2, 1 / 3}], {z1, 1, 12}, {z2, 1, 12}, ...]

Plot along a real slice of a decoration value for a depth-3 MultiplePolyLog:

Wolfram Language code: Plot[MultiplePolyLog[{2, 3, 4}, {s, 1 / 3, 1 / 6}], {s, -1 / 2, 1 / 2}]

Visualize the dependence on two decoration values for a depth-2 MultiplePolyLog:

Wolfram Language code: Plot3D[MultiplePolyLog[{3, 2}, {s1, s2}], {s1, -1 / 2, 1 / 2}, {s2, -1 / 2, 1 / 2}, ...]

Render a complex surface over a rectangular region of the decoration -plane:

Wolfram Language code: ComplexPlot3D[MultiplePolyLog[{2, 3}, {s, 1 / 3}], {s, -1 / 2 - I / 2, 1 / 2 + I / 2}]

Applications (8)

Euler Sum (3)

Evaluate an Euler sum involving the three-argument HarmonicNumber using MultiplePolyLog:

Wolfram Language code: Underoverscript[∑, n = 1, ∞](HarmonicNumber[n, 2, (1/3)]^2/n^4)

Sum with complex exponents in the pre-factor results in MultiplePolyLog with complex weights:

Wolfram Language code: Underoverscript[∑, n = 1, ∞](HarmonicNumber[n, (2/3), (1/2)]^2/n^4 - I)

An Euler sum involving the MultipleHarmonicNumber:

Wolfram Language code: Underoverscript[∑, n = 1, ∞](MultipleHarmonicNumber[n, {2, 3}, {(1/4), (1/5)}]/n^6)

Generating Function (3)

Express the GeneratingFunction of the product of a HarmonicNumber and a HyperHarmonicNumber using MultiplePolyLog:

Wolfram Language code: GeneratingFunction[HarmonicNumber[n]HyperHarmonicNumber[3, n], n, x]

GeneratingFunction of a MultipleHarmonicNumber:

Wolfram Language code: GeneratingFunction[MultipleHarmonicNumber[n, {2, 3}, {1 / 4, 1 / 5}], n, x]

ExponentialGeneratingFunction of a product of StirlingS1 and HarmonicNumber, expressed using MultiplePolyLog:

Wolfram Language code: ExponentialGeneratingFunction[StirlingS1[n, 3]HarmonicNumber[n, 2, 1 / 3], n, x]

Discrete Transforms (2)

Compute the ZTransform of the square of a HarmonicNumber in terms of MultiplePolyLog:

Wolfram Language code: ZTransform[HarmonicNumber[n, 2, 1 / 3] ^ 2, n, z]

Compute the DirichletTransform of a MultipleHarmonicNumber in terms of MultiplePolyLog:

Wolfram Language code: DirichletTransform[MultipleHarmonicNumber[n, {2, 3}], n, s]

Properties & Relations (5)

Depth-1 MultiplePolyLog reduces to PolyLog:

Wolfram Language code: MultiplePolyLog[{10}, {1 / 3}]

When all the decoration values are , MultiplePolyLog reduces to MultipleZeta:

Wolfram Language code: MultiplePolyLog[{4, 2, 6, 3, 1}, {1, 1, 1, 1, 1}]

When all the decoration values are either or , MultiplePolyLog reduces to HarmonicPolyLog:

Wolfram Language code: MultiplePolyLog[{3, 6, 10, 2, 1}, {1, -1, -1, 1, -1}]

A convergent MultiplePolyLog is related to GeneralizedPolyLog by the relationship $TemplateBox[{TemplateBox[{{{z, _, 1}, ,, ..., ,, {z, _, k}}}, ImplicitList], TemplateBox[{{{s, _, 1}, ,, ..., ,, {s, _, k}}}, ImplicitList]}, MultiplePolyLog]=(-1)^kG(0^(z_1-1),1/(s_1),0^(z_2-1),1/(s_1s_2),...,0^(z_k-1),1/(s_1s_2...s_k);1)$ :

Wolfram Language code: MultiplePolyLog[{2, 3}, {1, (1/3)}] == GeneralizedPolyLog[{0, 1, 0, 0, 3}, 1]

When the defining series converges, MultiplePolyLog is same as the infinite version of the MultipleHarmonicNumber:

Wolfram Language code: MultipleHarmonicNumber[∞, {4, 2, 6, 3}, {1 / 2, 1 / 3, 1 / 5, 1 / 7}]

Possible Issues (1)

The two lists in the arguments of MultiplePolyLog must be of the same lengths:

Wolfram Language code: MultiplePolyLog[{2, 3, 4}, {1 / 2, 1 / 3}]

Neat Examples (1)

MultiplePolyLog serves as a bridge between Euler-type nested sums and iterated integrals:

Wolfram Language code:

Underoverscript[∑, n1 = 1, ∞]Underoverscript[∑, n2 = 1, n1 - 1](((1/4))^n1 ((1/5))^n2/n1^2 n2^3) == MultiplePolyLog[{2, 3}, {(1/4), (1/5)}] == Subsuperscript[∫, 0, 1]Subsuperscript[∫, 0, x1]Subsuperscript[∫, 0, x2]Subsuperscript[∫, 0, x3]Subsuperscript[∫, 0, x4](1/x1 (x2 - 4) x3 x4 (x5 - 20))ⅆx5ⅆx4ⅆx3ⅆx2ⅆx1

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MultiplePolyLog

Details

Examples

Basic Examples (4)