Nørlund Polynomial
The Nørlund polynomial (note that the spelling Nörlund also appears in various publications) is a name given by Carlitz (1960) and Adelberg (1999) to the polynomial B_n^((a)). These are implemented in the Wolfram Language as NorlundB [n, a], and are defined through the exponential generating function
(Carlitz 1960).
Sums involving B_n^((a)) are given by
(Carlitz 1960, Gould 1960).
The Nørlund polynomials are related to the Stirling numbers by
and
(Carlitz 1960).
The Nørlund polynomials are a special case
| B_n^((a))=B_n^((a))(0) |
(6)
|
of the function B_n^((a))(x) 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
Values of B_n^((a))(x) for small positive integer n and a are given by
The polynomial B_n^((a))(x) has derivative
and Maclaurin series
| B_n^((a))(x)=B_n^((a))+nB_(n-1)^((a))x+1/2n(n-1)B_(n-2)^((a))x^2+.... |
(18)
|
where B_n^((a)) are polynomials in a.
See also
Bernoulli PolynomialRelated Wolfram sites
https://reference.wolfram.com/language/ref/NorlundB.htmlExplore with Wolfram|Alpha
More things to try:
References
Adelberg, A. "Arithmetic Properties of the Nörlund Polynomial B_n^((x))." Disc. Math. 204, 5-13, 1999. https://doi.org/10.1016/S0012-365X(98)00363-X.Carlitz, L. "Note on Nörlund's Polynomial B_n^((z))." 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: Springer-Verlag, 1924.Referenced on Wolfram|Alpha
Nørlund PolynomialCite this as:
Weisstein, Eric W. "Nørlund Polynomial." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/NorlundPolynomial.html