Nørlund Polynomial

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)

(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 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)

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 .

REFERENCES:

Adelberg, A. "Arithmetic Properties of the Nörlund [sic] Polynomial ." Oct. 28, 1997. http://citeseer.ist.psu.edu/44033.html.

Carlitz, L. "Note on Nörlund's [sic] Polynomial ." 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. [sic]. Vorlesungen über Differenzenrechnung. Berlin: Springer-Verlag, 1924.