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Mahler Polynomial

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MahlerPolynomials

Polynomials s_n(x) which form the Sheffer sequence for

f^(-1)(t)=1+t-e^t,
(1)

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=e^(x(1+t-e^t)).
(2)

The first few are

s_0(x)=1
(3)
s_1(x)=0
(4)
s_2(x)=-x
(5)
s_3(x)=-x
(6)
s_4(x)=3x^2-x
(7)
s_5(x)=10x^2-x.
(8)

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References

Erdélyi, A.; Magnus, W.; Oberhettinger, F.; and Tricomi, F. G. Higher Transcendental Functions, Vol. 3. New York: Krieger, p. 254, 1981.Roman, S. The Umbral Calculus. New York: Academic Press, 1984.

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Mahler Polynomial

Cite this as:

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

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