The Nørlund polynomial (note that the spelling Nörlund also appears in various publications) is a name given by Carlitz (1960) and Adelberg (1997) to the polynomial . These are implemented in the Wolfram Language as NorlundB[n, a], and are defined through the exponential generating function
(1)
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(Carlitz 1960).
Sums involving are given by
(2)
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(3)
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(Carlitz 1960, Gould 1960).
The Nørlund polynomials are related to the Stirling numbers by
(4)
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and
(5)
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(Carlitz 1960).
The Nørlund polynomials are a special case
(6)
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of the function sometimes known as the generalized Bernoulli polynomial, implemented in the Wolfram Language as NorlundB[n, a, z]. These polynomials are defined through the exponential generating function
(7)
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Values of for small positive integer and are given by
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(9)
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(10)
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(11)
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(12)
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(13)
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(14)
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(15)
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(16)
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The polynomial has derivative
(17)
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and Maclaurin series
(18)
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where are polynomials in .