Function Repository Resource:

BernoulliBarnesB

Source Notebook

Compute Bernoulli–Barnes numbers and polynomials

Contributed by: Jayanta Phadikar

ResourceFunction["BernoulliBarnesB"][n,{α₁,α₂,…}]

gives the Bernoulli-Barnes number of order n associated with the specified period parameters.

ResourceFunction["BernoulliBarnesB"][n,{α₁,α₂,…},x]

gives the Bernoulli-Barnes polynomial of order n in x associated with the specified period parameters.

Details

The order n must be a nonnegative integer.

The second argument must be a nonempty list of nonzero period parameters.

For r period parameters, BernoulliBarnesB uses the exponential generating function Exp[xt]Product[t/(Exp[α_jt]-1),{j,1,r}]==Sum[ResourceFunction["BernoulliBarnesB"][n,{α₁,α₂,…},x]t^n/n!,{n,0,Infinity}].

The periods may be exact or approximate, real or complex, and may remain symbolic.

ResourceFunction["BernoulliBarnesB"][n,periods] is defined as ResourceFunction["BernoulliBarnesB"][n,periods,0].

The periods {1} reproduce the built-in BernoulliB.

Examples

Basic Examples (4)

Compute a Bernoulli-Barnes number:

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Compute a Bernoulli-Barnes polynomial:

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Generate the first several numbers for fixed periods:

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Plot a family of Bernoulli-Barnes polynomials:

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Scope (6)

Use symbolic period parameters:

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The function supports any positive rank:

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Use algebraic or irrational period parameters:

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Obtain a high-precision numerical value:

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The polynomial variable can itself be an expression:

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Compare the defining exponential generating function with the reconstructed series:

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$Expand[Normal[ Series[Exp[x t] Times @@ (t/(Exp[#1 t] - 1) &) /@ {2, 3}, {t, 0, 5}]] - Total[ Table[(ResourceFunction["BernoulliBarnesB"][n, {2, 3}, x] t^n)/ n!, {n, 0, 5}]]]$

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Properties and Relations (8)

For one unit period, the function reduces to the built-in BernoulliB:

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For one general period alpha, use the scaled Bernoulli polynomial:

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$ResourceFunction["BernoulliBarnesB"][5, {alpha}, x] == Expand[alpha^4 BernoulliB[5, x/alpha]]$

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Bernoulli-Barnes polynomials form an Appell sequence in x:

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$Expand[\!$ \*SubscriptBox[\(\[PartialD]$, $x$]\(\* InterpretationBox[ TagBox[ FrameBox[ PaneBox[GridBox[{ { StyleBox[ StyleBox[ AdjustmentBox["\<\"[\[FilledSmallSquare]]\"\>", BoxBaselineShift->-0.25, BoxMargins->{{0, 0}, {-1, -1}}], "ResourceFunctionIcon", FontColor->RGBColor[ 0.8745098039215686, 0.2784313725490196, 0.03137254901960784]], ShowStringCharacters->False, FontFamily->"Source Sans Pro Black", FontSize->0.65 Inherited, FontWeight->"Heavy", PrivateFontOptions->{"OperatorSubstitution"->False}], StyleBox[ RowBox[{ StyleBox["\<\"BernoulliBarnesB\"\>", "ResourceFunctionLabel", FontFamily->"Source Sans Pro"], " "}], ShowAutoStyles->False, ShowStringCharacters->False, FontSize->0.9 Inherited, FontColor->GrayLevel[0.09999084492227013]]} }, GridBoxSpacings->{"Columns" -> {{0.25}}}], Alignment->Left, BaseStyle->{LineSpacing -> {0, 0}, LineBreakWithin -> False}, BaselinePosition->Baseline, FrameMargins->{{3, 0}, {0, 0}}], Background->RGBColor[ 0.968505860940661, 0.976440431253161, 0.984252931253161], BaselinePosition->Baseline, DefaultBaseStyle->{LightDarkAutoColorRules -> <|"MarkedColors" -> True|>}, FrameMargins->{{0, 0}, {1, 1}}, FrameStyle->RGBColor[ 0.831298829690661, 0.847045900003161, 0.850952150003161], RoundingRadius->4], {"FunctionResourceBox", RGBColor[0.8745098039215686, 0.2784313725490196, 0.03137254901960784], "BernoulliBarnesB"}, TagBoxNote->"FunctionResourceBox"], ResourceFunction["BernoulliBarnesB"], BoxID -> "BernoulliBarnesB", Selectable->False][6, {2, 3}, x]\)\) - 6 ResourceFunction["BernoulliBarnesB"][5, {2, 3}, x]]$

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Shifting x by one period removes that period and lowers the order:

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Simultaneously scaling x and all r periods gives a homogeneous scaling law of degree n-r:

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$FullSimplify[ ResourceFunction["BernoulliBarnesB"][5, {c alpha, c beta}, c x] == c^3 ResourceFunction["BernoulliBarnesB"][5, {alpha, beta}, x], Assumptions -> c alpha beta != 0]$

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Reflection about half the sum of the periods changes the sign according to the order:

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The result is invariant under permutations of periods:

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The Appell addition formula holds:

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$Expand[ResourceFunction["BernoulliBarnesB"][5, {2, 3}, x + y] - Total[Table[ Binomial[5, k] ResourceFunction["BernoulliBarnesB"][k, {2, 3}, x] y^(5 - k), {k, 0, 5}]]]$

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Possible Issues (2)

A zero period makes the defining generating function singular, so such input is intentionally left unevaluated:

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Negative or noninteger orders are outside the definition and remain unevaluated:

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Neat Examples (2)

Visualize the first eight polynomials for three unit periods:

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Display the real roots of a sequence of polynomials:

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Publisher

Jayanta Kumar Phadikar

Requirements

Wolfram Language 13.0 (December 2021) or above

Version History

1.0.0 – 29 July 2026

Source Metadata

Citation:
- Bayad, A. and Beck, M. "Relations for Bernoulli-Barnes Numbers and Barnes Zeta Functions." International Journal of Number Theory 10(5), 1321-1335, 2014.
- E. W. Barnes, "On the Theory of the Multiple Gamma Function", Transactions of the Cambridge Philosophical Society 19, 374-425, 1904.

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License Information

This work is licensed under a Creative Commons Attribution 4.0 International License