Falling Factorial

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

for . 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

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

A generalized version of the falling factorial can defined by

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

The usual factorial is related to the falling factorial by

(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 (Roman 1984, p. 5; Abramowitz and Stegun 1972, p. 256; Spanier 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

(OEIS A054654).

The derivative is given by

where is a harmonic number.

A sum formula connecting the falling factorial and rising factorial ,

is given using the Sheffer formalism with

which gives the generating function

where

Reading the coefficients off gives

so,

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

The falling factorial is an associated Sheffer sequence with

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

which is equivalent to the binomial theorem

The binomial identity of the Sheffer sequence is

where is a binomial coefficient, which can be rewritten as

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

(Roman 1984, p. 61).