Narumi Polynomial
Polynomials s_k(x;a) which form the Sheffer sequence for
g(t) = [画像:((e^t-1)/t)^(-a)]
(1)
f(t) = e^t-1
(2)
which have generating function
![画像: sum_(k=0)^infty(s_k(x))/(k!)t^k=[t/(ln(1+t))]^a(1+t)^x.](/index.cgi/contrast/https://mathworld.wolfram.com/images/equations/NarumiPolynomial/NumberedEquation1.svg){kind=link}
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 PolynomialCite this as:
Weisstein, Eric W. "Narumi Polynomial." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/NarumiPolynomial.html