NorlundB

NorlundB[n,a]

gives Nørlund polynomials of degree n in a.

NorlundB[n,a,x]

gives generalized Bernoulli polynomials .

Details

  • Mathematical function, suitable for both symbolic and numerical manipulation.
  • The Nørlund polynomials satisfy the generating function relation .
  • The generalized Bernoulli polynomials satisfy the generating function relation .
  • The Bernoulli numbers are given by . Generalized Bernoulli numbers are given by higher integer values of a.
  • 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  (4)

Higher-order generalized Bernoulli numbers:

First 10 Nørlund numbers:

Compare with their integral definition:

Stirling numbers:

Expand ratio of Gamma functions at infinity:

Compare with direct expansion:

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_2023_norlundb, author="Wolfram Research", title="{NorlundB}", year="2007", howpublished="\url{https://reference.wolfram.com/language/ref/NorlundB.html}", note=[Accessed: 07-December-2023 ]}

BibLaTeX

@online{reference.wolfram_2023_norlundb, organization={Wolfram Research}, title={NorlundB}, year={2007}, url={https://reference.wolfram.com/language/ref/NorlundB.html}, note=[Accessed: 07-December-2023 ]}