The Nørlund polynomial (also spelled Nörlund polynomial) is a name given by Carlitz (1960) and Adelberg (1999) to the polynomial 
. These are implemented in the Wolfram
Language as NorlundB[n,
a], and are defined through the exponential
generating function
 |
(1)
|
(Carlitz 1960).
Sums involving 
are given by
 |  |  |
(2)
|
 |  |  |
(3)
|
(Carlitz 1960, Gould 1960).
The Nørlund polynomials are related to the Stirling numbers by
 |
(4)
|
and
 |
(5)
|
(Carlitz 1960).
The Nørlund polynomials are a special case
 |
(6)
|
of the generalized Bernoulli polynomial 
, defined by the exponential
generating function
 |
(7)
|
For 
this gives the ordinary Bernoulli polynomial 
(Roman 1984, p. 93). The generalized
Bernoulli polynomial is implemented in the
Wolfram Language as NorlundB[n,
a, x].
Values of 
for small positive integer 
and 
are given by
 |  |  |
(8)
|
 |  |  |
(9)
|
 |  |  |
(10)
|
 |  |  |
(11)
|
 |  |  |
(12)
|
 |  |  |
(13)
|
 |  |  |
(14)
|
 |  |  |
(15)
|
 |  |  |
(16)
|
The polynomial 
has derivative
 |
(17)
|
and Maclaurin series
 |
(18)
|
where 
are polynomials in 
.
."
Disc. Math. 204, 5-13, 1999. 
." Proc. Amer. Math. Soc. 11, 452-455,
1960.Gould, H. W. "Stirling Number Representation Problems."
Proc. Amer. Math. Soc. 11, 447-451, 1960.Nörlund,
N. E. Vorlesungen über Differenzenrechnung. Berlin, Germany: Springer-Verlag,
1924.Roman, S.