# 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 Bernoulli numbers are given by . Generalized Bernoulli numbers are given by higher integer values of a.
• The generalized Bernoulli polynomials satisfy the generating function relation .
• .
• The Bernoulli polynomials are given by .
• 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)

Plot Nørlund polynomials:

## 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 TricomiErdélyi formula:

Compare with the direct expansion:

An explicit expression for the k-order derivative of :

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: 18-May-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: 18-May-2024 ]}