Central Factorial
The central factorials x^([k]) form an associated Sheffer sequence with
f(t) = e^(t/2)-e^(-t/2)
(1)
= 2sinh(1/2t),
(2)
giving the generating function
![画像: sum_(k=0)^infty(x^([k]))/(k!)t^k=e^(2xsinh^(-1)(t/2)).](/index.cgi/contrast/https://mathworld.wolfram.com/images/equations/CentralFactorial/NumberedEquation1.svg){kind=link}
The first central factorials are
x^([0]) = 1
(4)
x^([1]) = x
(5)
x^([2]) = x^2
(6)
x^([3]) = 1/4(4x^3-x)
(7)
= -1/4(1-2x)x(1+2x)
(8)
x^([4]) = x^4-x^2
(9)
= -(1-x)x^2(1+x)
(10)
x^([5]) = 1/(16)(16x^5-40x^3+9x)
(11)
= 1/(16)(1-2x)(3-2x)x(1+2x)(3+2x).
(12)
See also
Factorial, Falling Factorial, Gould Polynomial, Rising FactorialExplore with Wolfram|Alpha
WolframAlpha
References
Roman, S. The Umbral Calculus. New York: Academic Press, pp. 133-134, 1984.Referenced on Wolfram|Alpha
Central FactorialCite this as:
Weisstein, Eric W. "Central Factorial." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/CentralFactorial.html