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Stirling Polynomial

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Polynomials S_k(x) which form the Sheffer sequence for

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

where f^(-1)(t) is the inverse function of f(t), and have generating function

sum_(k=0)^infty(S_k(x))/(k!)t^k=(t/(1-e^(-t)))^(x+1).
(3)

The first few polynomials are

S_0(x)=1
(4)
S_1(x)=1/2(x+1)
(5)
S_2(x)=1/(12)(3x+2)(x+1)
(6)
S_3(x)=1/8x(x+1)^2.
(7)

The Stirling polynomials are related to the Stirling numbers of the first kind s(n,m) by

S_n(m)=((-1)^n)/((m; n))s(m+1,m-n+1),
(8)

where (m; n) is a binomial coefficient and m is an integer with m>=n (Roman 1984, p. 129).

See also

Stirling Number of the First Kind, Stirling Number of the Second Kind

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References

Erdélyi, A.; Magnus, W.; Oberhettinger, F.; and Tricomi, F. G. Higher Transcendental Functions, Vol. 3. New York: Krieger, p. 257, 1981.Roman, S. "The Stirling Polynomials." §4.8 in The Umbral Calculus. New York: Academic Press, pp. 128-129, 1984.

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Stirling Polynomial

Cite this as:

Weisstein, Eric W. "Stirling Polynomial." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/StirlingPolynomial.html

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