falling factorial

falling factorial

For

n

∈

ℕ

, the rising and

falling factorials

are

n

th

degree

polynomial

described, respectively, by

x

n

¯

=

x

⁢

(

x

+

1

)

⁢

…

⁢

(

x

+

n

-

1

)

x

n

¯

=

x

⁢

(

x

-

1

)

⁢

…

⁢

(

x

-

n

+

1

)

The two types of polynomials are related by:

x

n

¯

=

(

-

1

)

n

⁢

(

-

x

)

n

¯

.

The

rising factorial

is often written as

(

x

)

n

, and referred to as
the

Pochhammer symbol

(see

hypergeometric series

). Unfortunately, the
falling factorial is also often denoted by

(

x

)

n

, so great care must
be taken when encountering this notation.

Notes.

Unfortunately, the notational conventions for the rising and falling
factorials lack a common standard, and are plagued with a fundamental
inconsistency. An examination of reference works and textbooks reveals
two fundamental sources of notation: works in

combinatorics

and works
dealing with

hypergeometric functions

.

Works of combinatorics [1,2,3] give greater focus to the falling
factorial because of its role in defining the

Stirling numbers

.
The symbol

(

x

)

n

almost always denotes the falling factorial. The
notation for the rising factorial varies widely; we find

⟨

x

⟩

n

in [1] and

(

x

)

(

n

)

in [3].

Works focusing on

special functions

[4,5] universally use

(

x

)

n

to
denote the rising factorial and use this symbol in the description of
the various flavours of hypergeometric series. Watson [5] credits
this notation to Pochhammer [6], and indeed the special functions
literature eschews “falling factorial” in favour of “Pochhammer
symbol”. Curiously, according to Knuth [7], Pochhammer himself used

(

x

)

n

to denote the

binomial coefficient

(Note: I haven’t verified
this.)

The notation featured in this entry is due to D. Knuth [7,8]. Given
the fundamental inconsistency in the existing notations, it seems
sensible to break with both traditions, and to adopt new and
graphically suggestive notation for these two concepts. The
traditional notation, especially in the hypergeometric camp, is so
deeply entrenched that, realistically, one needs to be familiar with
the traditional modes and to take care when encountering the symbol

(

x

)

n

.

References

1.

Comtet,

Advanced combinatorics.

2.

Jordan,

Calculus of finite differences.

3.

Riordan,

Introduction to combinatorial analysis.

4.

Erdélyi, et. al.,

Bateman manuscript project.

5.

Watson,

A treatise on the theory of

Bessel functions

.

6.

Pochhammer, “Ueber hypergeometrische Functionen

n

ter

Ordnung,”

Journal für die reine und angewandte Mathematik

71

(1870), 316–352.

7.

Knuth, “Two notes on notation”

http://www-cs-faculty.stanford.edu/ knuth/papers/tnn.tex.gz

download

8.

Greene, Knuth,

Mathematics for the analysis of algorithms.

Title

falling factorial

Canonical

name

FallingFactorial

Date of creation

2013-03-22 12:23:58

Last modified on

2013-03-22 12:23:58

Owner

rmilson (146)

Last modified by

rmilson (146)

Numerical id

11

Author

rmilson (146)

Entry type

Definition

Classification

msc 05A10

Defines

rising factorial

Defines

Pochhammer symbol

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