TOPICS
Search

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

sum_(k=0)^infty(x^([k]))/(k!)t^k=e^(2xsinh^(-1)(t/2)).
(3)

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 Factorial

Explore with Wolfram|Alpha

References

Roman, S. The Umbral Calculus. New York: Academic Press, pp. 133-134, 1984.

Referenced on Wolfram|Alpha

Central Factorial

Cite this as:

Weisstein, Eric W. "Central Factorial." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/CentralFactorial.html

Subject classifications