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Power Polynomial

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The power polynomials x^n are an associated Sheffer sequence with

f(t)=t,
(1)

giving generating function

sum_(k=0)^inftyx^kt^k=1/(1-tx)
(2)

and exponential generating function

sum_(k=0)^infty(x^k)/(k!)t^k=e^(xt)
(3)

and binomial identity

(x+y)^n=sum_(k=0)^n(n; k)x^ky^(n-k).
(4)

See also

Sheffer Sequence

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References

Roman, S. "The Sequence x^n." §4.1.1 in The Umbral Calculus. New York: Academic Press, p. 55, 1984.

Referenced on Wolfram|Alpha

Power Polynomial

Cite this as:

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

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