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Nørlund Polynomial

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NorlundPolynomial

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 polynomialB_n^((a)). These are implemented in theWolfram Language asNorlundB[n,a], and are defined through theexponential generating function

(t/(e^t-1))^a=sum_(n=0)^inftyB_n^((a))(t^n)/(n!)
(1)

(Carlitz 1960).

Sums involvingB_n^((a)) are given by

B_k^((a))=sum_(j=0)^(k)(-1)^j(k+1; j+1)B_k^((-ja))
(2)
(-1)^k(z; k)B_k^((k-z))=sum_(k=0)^(k)(j+k-1; k)(k-z; j+k)(k+z; k-j)B_k^((j+k))
(3)

(Carlitz 1960, Gould 1960).

The Nørlund polynomials are related to the Stirling numbers by

s(n,n-k)=(n-1; k)B_k^((n))
(4)

and

S(k+n,n)=(k+n; k)B_k^((-n))
(5)

(Carlitz 1960).

The Nørlund polynomials are a special case

B_n^((a))=B_n^((a))(0)
(6)

of the functionB_n^((a))(x) sometimes known as the generalized Bernoulli polynomial, implemented in theWolfram Language asNorlundB[n,a,z]. These polynomials are defined through theexponential generating function

(t/(e^t-1))^ae^(zt)=sum_(n=0)^inftyB_n^((a))(z)(t^n)/(n!).
(7)

Values ofB_n^((a))(x) for small positive integern anda are given by

B_1^((1))(x)=x-1/2
(8)
B_1^((2))(x)=x-1
(9)
B_1^((3))(x)=x-3/2
(10)
B_2^((1))(x)=x^2-x+1/6
(11)
B_2^((2))(x)=x^2-2x+5/6
(12)
B_2^((3))(x)=(1-x)(2-x)
(13)
B_3^((1))(x)=x^3-3/2x^2+1/2x
(14)
B_3^((2))(x)=x^3-3x^2+5/2x-1/2
(15)
B_3^((3))(x)=x^3-9/2x^2+6x-9/4.
(16)

The polynomialB_n^((a))(x) hasderivative

(dB_n^((a))(x))/(dx)=nB_(n-1)^((a))(x)
(17)

andMaclaurin series

B_n^((a))(x)=B_n^((a))+nB_(n-1)^((a))x+1/2n(n-1)B_(n-2)^((a))x^2+....
(18)

whereB_n^((a)) are polynomials ina.

See also

Bernoulli Polynomial

Related Wolfram sites

http://functions.wolfram.com/Polynomials/NorlundB/

Explore with Wolfram|Alpha

References

Adelberg, A. "Arithmetic Properties of the Nörlund [sic] PolynomialB_n(x)." Oct. 28, 1997.http://citeseer.ist.psu.edu/44033.html.Carlitz, L. "Note on Nörlund's [sic] PolynomialB_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. [sic].Vorlesungen über Differenzenrechnung. Berlin: Springer-Verlag, 1924.

Referenced on Wolfram|Alpha

Nørlund Polynomial

Cite this as:

Weisstein, Eric W. "Nørlund Polynomial."FromMathWorld--A Wolfram Resource.https://mathworld.wolfram.com/NorlundPolynomial.html

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Created, developed and nurtured by Eric Weisstein at Wolfram Research
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