falling factorial

For $n\\in\\mathbb{N}$ , the rising and falling factorials are $n^{\\text{th}}$ degree polynomial described, respectively, by

	$\\displaystyle x^{\\overline{n}}$	$\\displaystyle=x(x+1)\\ldots(x+n-1)$
	$\\displaystyle x^{\\underline{n}}$	$\\displaystyle=x(x-1)\\ldots(x-n+1)$

The two types of polynomials are related by:

x^{\\overline{n}}=(-1)^{n}(-x)^{\\underline{n}}.

The rising factorial is often written as $(x)_{n}$ , and referred to asthe Pochhammer symbol (see hypergeometric series). Unfortunately, thefalling factorial is also often denoted by $(x)_{n}$ , so great care mustbe taken when encountering this notation.

Notes.

Unfortunately, the notational conventions for the rising and fallingfactorials lack a common standard, and are plagued with a fundamentalinconsistency. An examination of reference works and textbooks revealstwo fundamental sources of notation: works in combinatorics and worksdealing with hypergeometric functions.

Works of combinatorics [1,2,3] give greater focus to the fallingfactorial because of its role in defining the Stirling numbers.The symbol $(x)_{n}$ almost always denotes the falling factorial. Thenotation for the rising factorial varies widely; we find $\\left<x\\right>_{n}$ in [1] and $(x)^{(n)}$ in [3].

Works focusing on special functions [4,5] universally use $(x)_{n}$ todenote the rising factorial and use this symbol in the description ofthe various flavours of hypergeometric series. Watson [5] creditsthis notation to Pochhammer [6], and indeed the special functionsliterature eschews “falling factorial” in favour of “Pochhammersymbol”. Curiously, according to Knuth [7], Pochhammer himself used $(x)_{n}$ to denote the binomial coefficient (Note: I haven’t verifiedthis.)

The notation featured in this entry is due to D. Knuth [7,8]. Giventhe fundamental inconsistency in the existing notations, it seemssensible to break with both traditions, and to adopt new andgraphically suggestive notation for these two concepts. Thetraditional notation, especially in the hypergeometric camp, is sodeeply entrenched that, realistically, one needs to be familiar withthe 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 $n^{\\rm ter}$ Ordnung,” Journal für die reine und angewandte Mathematik71 (1870), 316–352.
7.
Knuth, “Two notes on notation” http://www-cs-faculty.stanford.edu/ knuth/papers/tnn.tex.gzdownload
8.
Greene, Knuth, Mathematics for the analysis of algorithms.