Function Repository Resource:

MultiplePolyLogStar

Compute the multiple polylogarithm star function

Contributed by: Jayanta Phadikar

ResourceFunction["MultiplePolyLogStar"][{r₁,r₂,…,r_k},{s₁,s₂,…,s_k}]

gives the multiple polylogarithm star function defined by the weakly ordered nested sum .

Details

ResourceFunction["MultiplePolyLogStar"] is the star, or weakly ordered, depth-k extension of PolyLog.

The first argument is the list of indices or weights; the second argument is the list of decoration values, also called colors or numerator weights. The two lists must have the same length.

In its domain of absolute convergence, ResourceFunction["MultiplePolyLogStar"] is defined by

For convergence, set p_m=|s₁⋯s_m|. The series converges absolutely when p_m<1 for every m, diverges when p_m>1 for some m, and, on the boundary p_m=1, convergence depends on the weights.

ResourceFunction["MultiplePolyLogStar"] is a mathematical function suitable for symbolic and numerical manipulation.

Divergent cases can evaluate to Infinity or ComplexInfinity.

For depth 1, ResourceFunction["MultiplePolyLogStar"][{r},{s}] agrees with MultiplePolyLog[{r},{s}] and PolyLog[r,s].

When all decoration values are 1, ResourceFunction["MultiplePolyLogStar"] gives the corresponding MultipleZetaStar value.

Examples

Basic Examples (3)

Exact values are generated automatically for simple arguments:

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Discretely sample across the first index with decoration values fixed:

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Plot along a real slice of a decoration value:

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Scope (15)

Numerical evaluation (3)

Evaluate numerically:

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Evaluate to high precision:

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Evaluate numerically with a complex decoration value:

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Specific Values (5)

Evaluate at decoration values {1, -1}:

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Evaluate at conjugate decoration values {-I,I}:

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

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

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A divergent MultiplePolyLogStar:

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Visualization (7)

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

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Discretely sample a MultiplePolyLogStar across integer (positive/zero/negative) indices:

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Side-by-side traces of MultiplePolyLogStar[{n,2},{1/2,1/3}] and MultiplePolyLogStar[{2,n},{1/2,1/3}] that emphasize order asymmetry under index swap:

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

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

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

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

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Properties & Relations (4)

Depth-1 convergent MultiplePolyLogStar is same as MultiplePolyLog and PolyLog:

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A convergent MultiplePolyLogStar can be written as a linear combination of MultiplePolyLog's:

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$Inactive[ ResourceFunction["MultiplePolyLogStar"]][{2, 3, 4}, {1/2, 1/3, 1/6}] == Sum[((1/2)^n1 (1/3)^n2 (1/6)^n3)/( n1^2 n2^3 n3^4), {n1, 1, Infinity}, {n2, 1, n1}, {n3, 1, n2}]$