Function Repository Resource:

MultipleHarmonicStarNumber

Source Notebook

Compute finite multiple harmonic star sums

Contributed by: Jayanta Phadikar

ResourceFunction["MultipleHarmonicStarNumber"][n]

gives the n^th harmonic number HarmonicNumber[n].

ResourceFunction["MultipleHarmonicStarNumber"][n,{r₁,…,r_k}]

gives the n^th multiple harmonic star number of orders (r₁,…,r_k).

ResourceFunction["MultipleHarmonicStarNumber"][n,{r₁,…,r_k},{s₁,…,s_k}]

gives the n^th decorated multiple harmonic star number of orders (r₁,…,r_k) and decoration values (s₁,…,s_k).

Details

ResourceFunction["MultipleHarmonicStarNumber"] represents finite harmonic sums that generalize HarmonicNumber to multiple, weakly ordered indices with optional decoration modifiers.

ResourceFunction["MultipleHarmonicStarNumber"] is the star analogue of the built-in MultipleHarmonicNumber: its defining nested sum uses weak inequalities, while MultipleHarmonicNumber uses strict inequalities.

ResourceFunction["MultipleHarmonicStarNumber"] of one argument is the same as HarmonicNumber.

ResourceFunction["MultipleHarmonicStarNumber"] is defined by the nested sums:

Two arguments

Three arguments

In ResourceFunction["MultipleHarmonicStarNumber"][n,{r₁,…,r_k},{s₁,…,s_k}], n is the truncation index; {r₁,…,r_k} are the weights, orders or indices; {s₁,…,s_k} are the decoration values, colors or numerator weights; and k is the depth.

In the two-argument form, all decoration values are 1: ResourceFunction["MultipleHarmonicStarNumber"][n,{r₁,…,r_k}]=ResourceFunction["MultipleHarmonicStarNumber"][n,{r₁,…,r_k},{1,…,1}].

The truncation index n must be a non-negative integer for finite sums. The weights and decoration values can be arbitrary complex numerical values; special all-zero and all-one order lists also simplify for symbolic n.

ResourceFunction["MultipleHarmonicStarNumber"] of depth 1 reproduces HarmonicNumber:

One argument

ResourceFunction["MultipleHarmonicStarNumber"][n]=HarmonicNumber[n]

Two arguments

ResourceFunction["MultipleHarmonicStarNumber"][n,{r}]=HarmonicNumber[n,r]

Three arguments

ResourceFunction["MultipleHarmonicStarNumber"][n,{r},{s}]=HarmonicNumber[n,r,s]

When the defining series converges, the infinity limit coincides with the corresponding multiple zeta star or multiple polylog star value.

Products of ResourceFunction["MultipleHarmonicStarNumber"] satisfy stuffle/quasi-shuffle relations from weakly nested sums.

Examples

Basic Examples (6)

Compute a depth-two star harmonic number exactly:

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Numerically evaluate a decorated depth-2 multiple harmonic star number:

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Plot over a subset of integer truncation indices:

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Plot by varying a weight parameter:

Plot by varying a decoration parameter:

Verify the weakly nested series definition:

Scope (9)

Specific Values (6)

Evaluate for integer weights:

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Evaluate for rational weights:

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Evaluate for rational decoration values:

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Depth-1 with decoration value -1 gives an alternating harmonic number:

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Some common star cases have closed forms for symbolic n:

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All-zero indices give a binomial coefficient:

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Numerical Evaluation (3)

Evaluate for complex weights:

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Evaluate for complex decoration values:

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Finite sums can use noninteger and complex indices together:

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Properties & Relations (6)

Depth-one star harmonic numbers agree with ordinary harmonic numbers:

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Star sums use weak inequalities; compare with the built-in strictly ordered MultipleHarmonicNumber:

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The defining weakly ordered double sum agrees with the recursive implementation:

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$ResourceFunction["MultipleHarmonicStarNumber"][4, {1, 1}] == \!$ \*UnderoverscriptBox[\(\[Sum]$, $i = 1$, $4$]$ \*UnderoverscriptBox[\(\[Sum]$, $j = 1$, $i$] \*FractionBox[$1$, $i\ j$]\)\)$

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Compare with built-in MultipleHarmonicNumber:

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$MultipleHarmonicNumber[4, {1, 1}] == \!$ \*UnderoverscriptBox[\(\[Sum]$, $i = 1$, $4$]$ \*UnderoverscriptBox[\(\[Sum]$, $j = 1$, $i - 1$] \*FractionBox[$1$, $i\ j$]\)\)$