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NorlundB[n,a]

gives Nørlund polynomials TemplateBox[{n, a}, NorlundB] of degree n in a.

NorlundB[n,a,x]

gives generalized Bernoulli polynomials TemplateBox[{n, a, x}, NorlundB3].

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NorlundB

NorlundB[n,a]

gives Nørlund polynomials TemplateBox[{n, a}, NorlundB] of degree n in a.

NorlundB[n,a,x]

gives generalized Bernoulli polynomials TemplateBox[{n, a, x}, NorlundB3].

Details

  • Mathematical function, suitable for both symbolic and numerical manipulation.
  • The Nørlund polynomials satisfy the generating function relation (t/(e^t-1))^a=sum_(n=0)^(infty)TemplateBox[{n, a}, NorlundB](t^n/n!).
  • The Bernoulli numbers are given by TemplateBox[{n}, BernoulliB]=TemplateBox[{n, 1}, NorlundB]. Generalized Bernoulli numbers are given by higher integer values of a.
  • The generalized Bernoulli polynomials satisfy the generating function relation (t/(e^t-1))^ae^(xt)=sum_(n=0)^(infty)TemplateBox[{n, a, x}, NorlundB3](t^n/n!).
  • TemplateBox[{n, a}, NorlundB]=TemplateBox[{n, a, 0}, NorlundB3].
  • The Bernoulli polynomials are given by TemplateBox[{n, x}, BernoulliB2]=TemplateBox[{n, 1, x}, NorlundB3].
  • NorlundB can be evaluated to arbitrary numerical precision.
  • NorlundB automatically threads over lists.

Examples

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Basic Examples  (2)

First 5 Nørlund polynomials:

Wolfram Language code: Table[NorlundB[n, a], {n, 5}]

Generalized Bernoulli polynomials:

Wolfram Language code: Table[NorlundB[n, a, x], {n, 3}]

Scope  (3)

NorlundB threads element-wise over lists:

Wolfram Language code: NorlundB[{1, 2, 3, 4, 5}, a]

Plot Nørlund polynomials:

Wolfram Language code: Plot[Evaluate[Table[NorlundB[n, a], {n, 5}]], {a, -5, 5}]

TraditionalForm formatting:

Wolfram Language code: NorlundB[n, a]//TraditionalForm
Wolfram Language code: NorlundB[n, a, x]//TraditionalForm

Applications  (6)

Higher-order generalized Bernoulli numbers:

Wolfram Language code: Table[NorlundB[n, k], {k, 5}, {n, 10}]//Grid

First 10 Nørlund numbers:

Wolfram Language code: Table[NorlundB[n, n], {n, 10}]

Compare with their integral definition:

Wolfram Language code: Table[Integrate[FactorialPower[x - 1, n], {x, 0, 1}], {n, 10}]//FunctionExpand

Generate the Nørlund numbers from their exponential generating function:

Wolfram Language code: Rest[CoefficientList[Series[(z/(1 + z)Log[1 + z]), {z, 0, 10}], z]]Range[10]!

Define a function for computing the Gregory coefficients (also known as the Bernoulli numbers of the second kind):

Wolfram Language code: gregory[n_Integer ? Positive] := If[n == 1, 1 / 2, NorlundB[n, n - 1] / ((1 - n)n!)]

Compute the first 10 Gregory coefficients:

Wolfram Language code: Table[gregory[n], {n, 10}]

These coefficients appear in the series expansion of the function :

Wolfram Language code: Series[(z/Log[1 + z]), {z, 0, 10}]

Compare with their integral definition:

Wolfram Language code: Table[Integrate[Binomial[x, n], {x, 0, 1}], {n, 10}]//FunctionExpand

Express Stirling numbers of both kinds in terms of Nørlund polynomials:

Wolfram Language code: {Table[Binomial[n, k]NorlundB[k, n + 1], {n, 0, 5}, {k, 0, n}]//Grid, Table[StirlingS1[n + 1, n - k + 1], {n, 0, 5}, {k, 0, n}]//Grid}
Wolfram Language code: {Table[Binomial[n + k, k]NorlundB[k, -n], {n, 0, 5}, {k, 0, n}]//Grid, Table[StirlingS2[n + k, n], {n, 0, 5}, {k, 0, n}]//Grid}

Expand a ratio of Gamma functions at infinity using the Tricomi–Erdélyi formula:

Wolfram Language code: a = 1 / 2;b = 1;n = 10; Series[s^a - b Sum[((-1)^k/s^kk!)Pochhammer[b - a, k]NorlundB[k, a - b + 1, a], {k, 0, n + 1}], {s, ∞, n + 1}]

Compare with the direct expansion:

Wolfram Language code: % === Series[(Gamma[s + a]/Gamma[s + b]), {s, ∞, n + 1}]

An explicit expression for the k^(th)-order derivative of TemplateBox[{a, n}, Pochhammer]:

Wolfram Language code: n = 8;k = 3; (-1)^n - kFactorialPower[n, k]NorlundB[n - k, n + 1, 1 - a]//Expand

Compare with the result of D:

Wolfram Language code: D[Pochhammer[a, n], {a, k}]//Expand

Properties & Relations  (4)

Express NorlundB[n,a] in terms of the generalized Bell polynomial BellB:

Wolfram Language code: Table[BellY[Table[{(-1)^rPochhammer[a, r], (1/r + 1)}, {r, n}]]//Expand, {n, 9}]

Compare with NorlundB:

Wolfram Language code: Table[NorlundB[n, a], {n, 9}]

Demonstrate Gould's identity:

Wolfram Language code: Table[NorlundB[n, a] == Underoverscript[∑, k = 0, n](-1)^kBinomial[n + 1, k + 1]NorlundB[n, -a k]//Simplify, {n, 0, 9}]

Express NorlundB[n,a,x] in terms of NorlundB[n,a]:

Wolfram Language code: Table[NorlundB[n, a, x] == Underoverscript[∑, k = 0, n]Binomial[n, k]NorlundB[n - k, a]x^k, {n, 0, 9}]

Verify a recursive formula for NorlundB[n,a,x]:

Wolfram Language code: Table[NorlundB[n, a, x] == NorlundB[1, a, x]NorlundB[n - 1, a, x] - (a/n)Underoverscript[∑, k = 1, ⌊n / 2⌋]Binomial[n, 2k]BernoulliB[2k]NorlundB[n - 2k, a, x]//Simplify, {n, 9}]
Wolfram Research (2007), NorlundB, Wolfram Language function, https://reference.wolfram.com/language/ref/NorlundB.html.

Text

Wolfram Research (2007), NorlundB, Wolfram Language function, https://reference.wolfram.com/language/ref/NorlundB.html.

CMS

Wolfram Language. 2007. "NorlundB." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/NorlundB.html.

APA

Wolfram Language. (2007). NorlundB. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/NorlundB.html

BibTeX

@misc{reference.wolfram_2026_norlundb, author="Wolfram Research", title="{NorlundB}", year="2007", howpublished="\url{https://reference.wolfram.com/language/ref/NorlundB.html}", note=[Accessed: 12-June-2026]}

BibLaTeX

@online{reference.wolfram_2026_norlundb, organization={Wolfram Research}, title={NorlundB}, year={2007}, url={https://reference.wolfram.com/language/ref/NorlundB.html}, note=[Accessed: 12-June-2026]}