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Bernoulli Polynomial of the Second Kind

The Bernoulli polynomials of the second kind b_n(x) form a Sheffer sequence with

g(t)=t/(e^t-1)
(1)
f(t)=e^t-1,
(2)

giving exponential generating function

sum_(k=0)^infty(b_k(x))/(k!)t^k=(t(t+1)^x)/(ln(1+t)).
(3)

Roman (1984) defines Bernoulli numbers of the second kind as b_n=b_n(0). They are related to the Stirling numbers of the first kind s(n,m) by

b_n(x)=b_n(0)+sum_(k=1)^nn/ks(n-1,k-1)x^k
(4)

(Roman 1984, p. 115), and obey the reflection formula

b_n(1/2n-1-x)=(-1)^nb_n(1/2n-1+x)
(5)

(Roman 1984, p. 119).

The first few Bernoulli polynomials of the second kind are

b_0(x)=1
(6)
b_1(x)=1/2(2x+1)
(7)
b_2(x)=1/6(6x^2-1)
(8)
b_3(x)=1/4(4x^3-6x^2+1)
(9)
b_4(x)=1/(30)(30x^4-120x^3+120x^2-19).
(10)

See also

Bernoulli Number of the Second Kind, Bernoulli Polynomial, Sheffer Sequence, Stirling Number of the First Kind

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References

Roman, S. "The Bernoulli Polynomials of the Second Kind." In The Umbral Calculus. New York: Academic Press, pp. 113-119, 1984.

Referenced on Wolfram|Alpha

Bernoulli Polynomial of the Second Kind

Cite this as:

Weisstein, Eric W. "Bernoulli Polynomial of the Second Kind." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/BernoulliPolynomialoftheSecondKind.html

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