Falling and rising factorials information

In mathematics, the falling factorial (sometimes called the descending factorial,^[1] falling sequential product, or lower factorial) is defined as the polynomial

{\begin{aligned}(x)_{n}=x^{\underline {n}}&=\overbrace {x(x-1)(x-2)\cdots (x-n+1)} ^{n{\text{ factors}}}\\&=\prod _{k=1}^{n}(x-k+1)=\prod _{k=0}^{n-1}(x-k).\end{aligned}}

The rising factorial (sometimes called the Pochhammer function, Pochhammer polynomial, ascending factorial,^[1] rising sequential product, or upper factorial) is defined as

{\begin{aligned}x^{(n)}=x^{\overline {n}}&=\overbrace {x(x+1)(x+2)\cdots (x+n-1)} ^{n{\text{ factors}}}\\&=\prod _{k=1}^{n}(x+k-1)=\prod _{k=0}^{n-1}(x+k).\end{aligned}}

The value of each is taken to be 1 (an empty product) when $n = 0$ . These symbols are collectively called factorial powers.^[2]

The Pochhammer symbol, introduced by Leo August Pochhammer, is the notation $(x) n$ , where $n$ is a non-negative integer. It may represent either the rising or the falling factorial, with different articles and authors using different conventions. Pochhammer himself actually used $(x) n$ with yet another meaning, namely to denote the binomial coefficient ${\tbinom {x}{n}}.$ ^[3]

In this article, the symbol $(x) n$ is used to represent the falling factorial, and the symbol $x (n)$ is used for the rising factorial. These conventions are used in combinatorics,^[4] although Knuth's underline and overline notations $x^{\underline {n}}$ and $x^{\overline {n}}$ are increasingly popular.^[2]^[5] In the theory of special functions (in particular the hypergeometric function) and in the standard reference work Abramowitz and Stegun, the Pochhammer symbol $(x) n$ is used to represent the rising factorial.^[6]^[7]

When $x$ is a positive integer, $(x) n$ gives the number of $n$ -permutations (sequences of distinct elements) from an $x$ -element set, or equivalently the number of injective functions from a set of size $n$ to a set of size $x$ . The rising factorial $x (n)$ gives the number of partitions of an $n$ -element set into $x$ ordered sequences (possibly empty).^[a]

^ ^a ^b Steffensen, J.F. (17 March 2006). Interpolation (2nd ed.). Dover Publications. p. 8. ISBN 0-486-45009-0. — A reprint of the 1950 edition by Chelsea Publishing.
^ ^a ^b Knuth, D.E. The Art of Computer Programming. Vol. 1 (3rd ed.). p. 50.
^ Knuth, D.E. (1992). "Two notes on notation". American Mathematical Monthly. 99 (5): 403–422. arXiv:math/9205211. doi:10.2307/2325085. JSTOR 2325085. S2CID 119584305. The remark about the Pochhammer symbol is on page 414.
^ Olver, P.J. (1999). Classical Invariant Theory. Cambridge University Press. p. 101. ISBN 0-521-55821-2. MR 1694364.
^ Harris; Hirst; Mossinghoff (2008). Combinatorics and Graph Theory. Springer. ch. 2. ISBN 978-0-387-79710-6.
^ Abramowitz, Milton; Stegun, Irene A., eds. (December 1972) [June 1964]. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. National Bureau of Standards Applied Mathematics Series. Vol. 55. Washington, DC: United States Department of Commerce. p. 256 eqn. 6.1.22. LCCN 64-60036.
^ Slater, Lucy J. (1966). Generalized Hypergeometric Functions. Cambridge University Press. Appendix I. MR 0201688. — Gives a useful list of formulas for manipulating the rising factorial in $(x) n$ notation.

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[Steffensen-1] Steffensen, J.F. (17 March 2006). Interpolation (2nd ed.). Dover Publications. p. 8. ISBN 0-486-45009-0. — A reprint of the 1950 edition by Chelsea Publishing.

[The_Art_of_Computer_Programming-2] Knuth, D.E. The Art of Computer Programming. Vol. 1 (3rd ed.). p. 50.

[Knuth-3] Knuth, D.E. (1992). "Two notes on notation". American Mathematical Monthly. 99 (5): 403–422. arXiv:math/9205211. doi:10.2307/2325085. JSTOR 2325085. S2CID 119584305. The remark about the Pochhammer symbol is on page 414.

[4] Olver, P.J. (1999). Classical Invariant Theory. Cambridge University Press. p. 101. ISBN 0-521-55821-2. MR 1694364.

[5] Harris; Hirst; Mossinghoff (2008). Combinatorics and Graph Theory. Springer. ch. 2. ISBN 978-0-387-79710-6.

[6] Abramowitz, Milton; Stegun, Irene A., eds. (December 1972) [June 1964]. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. National Bureau of Standards Applied Mathematics Series. Vol. 55. Washington, DC: United States Department of Commerce. p. 256 eqn. 6.1.22. LCCN 64-60036.

[7] Slater, Lucy J. (1966). Generalized Hypergeometric Functions. Cambridge University Press. Appendix I. MR 0201688. — Gives a useful list of formulas for manipulating the rising factorial in $(x) n$ notation.

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