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Narumi Polynomial


Polynomials s_k(x;a) which form the Sheffer sequence for

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

which have generating function

 sum_(k=0)^infty(s_k(x))/(k!)t^k=[t/(ln(1+t))]^a(1+t)^x.
(3)

The first few are

s_0(x;a)=1
(4)
s_1(x;a)=1/2(2x+a)
(5)
s_2(x;a)=1/(12)[12x^2+12(a-1)x+a(3a-5)].
(6)

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References

Boas, R. P. and Buck, R. C. Polynomial Expansions of Analytic Functions, 2nd print., corr. New York: Academic Press, p. 37, 1964.Erdélyi, A.; Magnus, W.; Oberhettinger, F.; and Tricomi, F. G. Higher Transcendental Functions, Vol. 3. New York: Krieger, p. 258, 1981.Roman, S. The Umbral Calculus. New York: Academic Press, 1984.

Referenced on Wolfram|Alpha

Narumi Polynomial

Cite this as:

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

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