The falling factorial 
,
sometimes also denoted 
(Graham et al. 1994, p. 48), is defined
by
 |
(1)
|
for a nonnegative integer 
, with 
. It 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 
(a.k.a. Pochhammer
symbol) by
 |
(2)
|
The falling factorial is implemented in the Wolfram Language as FactorialPower[x, n].
A generalized version of the falling factorial can be defined by
 |
(3)
|
and is implemented in the Wolfram Language as FactorialPower[x, n, h].
The usual factorial is related to the falling factorial by
 |
(4)
|
(Graham et al. 1994, p. 48).
In combinatorial usage, the falling factorial is commonly denoted 
and the rising factorial
is denoted 
(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 
and the rising factorial is denoted 
(Abramowitz and Stegun 1972, p. 256; Roman 1984,
p. 5; Spanier and Oldham 1987). Extreme caution is therefore needed in interpreting
the meanings of the notations 
and 
. In this work, the notation 
is used for the falling factorial, potentially
causing confusion with the Pochhammer symbol.
The first few falling factorials are
 |  |  |
(5)
|
 |  |  |
(6)
|
 |  |  |
(7)
|
 |  |  |
(8)
|
 |  |  |
(9)
|
 |  |  |
(10)
|
 |  |  |
(11)
|
 |  |  |
(12)
|
(OEIS A054654).
The derivative is given by
 |
(13)
|
where 
is a harmonic number and removable
singularities on the right side are interpreted by limits.
A sum formula connecting the falling factorial 
and rising factorial 
,
 |
(14)
|
is given using the Sheffer sequence formalism with
 |  |  |
(15)
|
 |  |  |
(16)
|
 |  |  |
(17)
|
 |  |  |
(18)
|
which gives the generating function
 |
(19)
|
where
 |
(20)
|
Reading the coefficients off gives
 |  |  |
(21)
|
 |  |  |
(22)
|
 |  |  |
(23)
|
 |  |  |
(24)
|
 |  |  |
(25)
|
 |  |  |
(26)
|
 |  |  |
(27)
|
 |  |  |
(28)
|
 |  |  |
(29)
|
 |  |  |
(30)
|
so,
 |  |  |
(31)
|
 |  |  |
(32)
|
 |  |  |
(33)
|
 |  |  |
(34)
|
etc. (and the formula given by Roman 1984, p. 133, is incorrect).
The falling factorial is an associated Sheffer sequence with
 |
(35)
|
(Roman 1984, p. 29), and, for 
or as a formal power
series, has generating function
 |  |  |
(36)
|
 |  |  |
(37)
|
which is equivalent to the binomial theorem
 |
(38)
|
The binomial identity of the Sheffer sequence is
 |
(39)
|
where 
is a binomial coefficient, which can be rewritten
as
 |
(40)
|
known as the Chu-Vandermonde identity. The falling factorials obey the recurrence relation
 |
(41)
|
(Roman 1984, p. 61).
." Ch. 18 in