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:

Generalized Bernoulli polynomials:

Scope  (3)

NorlundB threads element-wise over lists:

Plot Nørlund polynomials:

TraditionalForm formatting:

Applications  (6)

Higher-order generalized Bernoulli numbers:

First 10 Nørlund numbers:

Compare with their integral definition:

Generate the Nørlund numbers from their exponential generating function:

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

Compute the first 10 Gregory coefficients:

These coefficients appear in the series expansion of the function :

Compare with their integral definition:

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

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

Compare with the direct expansion:

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

Compare with the result of D:

Properties & Relations  (4)

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

Compare with NorlundB:

Demonstrate Gould's identity:

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

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

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_2024_norlundb, author="Wolfram Research", title="{NorlundB}", year="2007", howpublished="\url{https://reference.wolfram.com/language/ref/NorlundB.html}", note=[Accessed: 22-December-2024 ]}

BibLaTeX

@online{reference.wolfram_2024_norlundb, organization={Wolfram Research}, title={NorlundB}, year={2007}, url={https://reference.wolfram.com/language/ref/NorlundB.html}, note=[Accessed: 22-December-2024 ]}