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Falling Factorial


FallingFactorial

The falling factorial (x)_n, sometimes also denoted x^(n__) (Graham et al. 1994, p. 48), is defined by

 (x)_n=x(x-1)...(x-(n-1))
(1)

for n>=0. Is also known as the binomial polynomial, lower factorial, falling factorial power (Graham et al. 1994, p. 48), or factorial power.

The falling factorial is related to the rising factorial x^((n)) (a.k.a. Pochhammer symbol) by

 (x)_n=(-1)^n(-x)^((n)),
(2)

The falling factorial is implemented in the Wolfram Language as FactorialPower[x, n].

A generalized version of the falling factorial can defined by

 (x)_n^((h))(x)=x(x-h)...(x-(n-1)h)
(3)

and is implemented in the Wolfram Language as FactorialPower[x, n, h].

The usual factorial is related to the falling factorial by

 n!=(n)_n
(4)

(Graham et al. 1994, p. 48).

In combinatorial usage, the falling factorial is commonly denoted (x)_n and the rising factorial is denoted (x)^((n)) (Comtet 1974, p. 6; Roman 1984, p. 5; Hardy 1999, p. 101), whereas in the calculus of finite differences and the theory of special functions, the falling factorial is denoted x^((n)) and the rising factorial is denoted (x)_n (Roman 1984, p. 5; Abramowitz and Stegun 1972, p. 256; Spanier 1987). Extreme caution is therefore needed in interpreting the meanings of the notations (x)_n and x^((n)). In this work, the notation (x)_n is used for the falling factorial, potentially causing confusion with the Pochhammer symbol.

The first few falling factorials are

(x)_0=1
(5)
(x)_1=x
(6)
(x)_2=x(x-1)
(7)
=x^2-x
(8)
(x)_3=x(x-1)(x-2)
(9)
=x^3-3x^2+2x
(10)
(x)_4=x(x-1)(x-2)(x-3)
(11)
=x^4-6x^3+11x^2-6x
(12)

(OEIS A054654).

The derivative is given by

 d/(dz)(z)_n=(H_z-H_(z-n))(z)_n,
(13)

where H_z is a harmonic number.

A sum formula connecting the falling factorial (x)_n and rising factorial x^((n)),

 (x)_n=sum_(k=0)^nc_(nk)x^((k)),
(14)

is given using the Sheffer formalism with

g(t)=1
(15)
f(t)=e^t-1
(16)
h(t)=1
(17)
l(t)=1-e^(-t),
(18)

which gives the generating function

 sum_(n=0)^infty(t_n(x))/(n!)t^n=sum_(n=0)^infty1/(n!)sum_(k=0)^nc_(nk)x^kt^k 
=e^(tx/(1+t)) 
=1+xt+1/2(x^2-2x)t^2+1/6(x^3-6x^2+6x)t^3+1/(24)(x^4-12x^3+36x^2-24x)t^4+...,
(19)

where

 t_n(x)=sum_(k=0)^nc_(nk)x^k.
(20)

Reading the coefficients off gives

 c_(00)=1 
c_(11)=1    c_(10)=0 
c_(22)=1    c_(21)=-2    c_(20)=0 
c_(33)=1    c_(32)=-6    c_(31)=6    c_(30)=0,
(21)

so,

(x)_0=x^((0))
(22)
(x)_1=x^((1))
(23)
(x)_2=x^((2))-2x^((1))
(24)
(x)_3=x^((3))-6x^((2))+6x^((1)),
(25)

etc. (and the formula given by Roman 1984, p. 133, is incorrect).

The falling factorial is an associated Sheffer sequence with

 f(t)=e^t-1
(26)

(Roman 1984, p. 29), and has generating function

sum_(k=0)^(infty)((x)_k)/(k!)t^k=e^(xln(1+t))
(27)
=(1+t)^x,
(28)

which is equivalent to the binomial theorem

 sum_(k=0)^infty(x; k)t^k=(1+t)^x.
(29)

The binomial identity of the Sheffer sequence is

 (x+y)_n=sum_(k=0)^n(n; k)(x)_k(y)_(n-k),
(30)

where (n; k) is a binomial coefficient, which can be rewritten as

 (x+y; n)=sum_(k=0)^n(x; k)(y; n-k),
(31)

known as the Chu-Vandermonde identity. The falling factorials obey the recurrence relation

 x(x)_n=(x)_(n+1)+n(x)_n
(32)

(Roman 1984, p. 61).


See also

Binomial Theorem, Central Factorial, Chu-Vandermonde Identity, Pochhammer Symbol, Rising Factorial, Sheffer Sequence, Sigma Polynomial

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References

Abramowitz, M. and Stegun, I. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, 1972.Comtet, L. Advanced Combinatorics: The Art of Finite and Infinite Expansions, rev. enl. ed. Dordrecht, Netherlands: Reidel, 1974.Graham, R. L.; Knuth, D. E.; and Patashnik, O. Concrete Mathematics: A Foundation for Computer Science, 2nd ed. Reading, MA: Addison-Wesley, 1994.Hardy, G. H. Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work, 3rd ed. New York: Chelsea, p. 101, 1999.Roman, S. "The Lower Factorial Polynomial." §1.2 in The Umbral Calculus. New York: Academic Press, pp. 5, 28-29, and 56-63, 1984.Sloane, N. J. A. Sequences A054654 in "The On-Line Encyclopedia of Integer Sequences."Spanier, J. and Oldham, K. B. "The Pochhammer Polynomials (x)_n." Ch. 18 in An Atlas of Functions. Washington, DC: Hemisphere, pp. 149-165, 1987.

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Falling Factorial

Cite this as:

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

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