Falling and rising factorials

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In mathematics, the falling factorial (sometimes called the descending factorial,[1] falling sequential product, lower factorial) is defined:

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


The Pochhammer symbol introduced by Leo August Pochhammer is the notation (x)n, where n is a non-negative integer. Depending on the context the Pochhammer symbol may represent either the rising factorial or the falling factorial as defined above. Care needs to be taken to check which interpretation is being used in any particular article. Pochhammer himself actually used (x)n with yet another meaning, namely to denote the binomial coefficient .[2]

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.[3] In the theory of special functions (in particular the hypergeometric function) the Pochhammer symbol (x)n is used to represent the rising factorial.[4] A useful list of formulas for manipulating the rising factorial in this last notation is given in (Slater 1966, Appendix I). Knuth uses the term factorial powers to comprise rising and falling factorials.[5]

When x is a non-negative integer, then (x)n gives the number of n-permutations of an x-element set, or equivalently the number of injective functions from a set of size n to a set of size x. However, for these meanings other notations like xPn and P(x,n) are commonly used. The Pochhammer symbol serves mostly for more algebraic uses, for instance when x is an indeterminate, in which case (x)n designates a particular polynomial of degree n in x.


Rising Factorial:

Falling Factorial:




A rising factorial can be expressed as a falling factorial that starts from the other end,

or as a falling factorial with opposite argument,


The rising factorial can be extended to real values of n using the Gamma function provided x and x + n are complex numbers that are not negative integers:

and so can the falling factorial:

If D denotes differentiation with respect to x, one has



Relation to umbral calculus[edit]

The falling factorial occurs in a formula which represents polynomials using the forward difference operator Δ and which is formally similar to Taylor's theorem of calculus. In this formula and in many other places, the falling factorial (x)k in the calculus of finite differences plays the role of xk in differential calculus. Note for instance the similarity of


A similar result holds for the rising factorial.





Connection coefficients[edit]

Since the falling factorials are a basis for the polynomial ring, we can re-express the product of two of them as a linear combination of falling factorials:


Alternate notations[edit]

An alternate notation for the rising factorial

and for the falling factorial

goes back to A. Capelli (1893) and L. Toscano (1939), respectively.[6] Graham, Knuth and Patashnik[7] propose to pronounce these expressions as "x to the m rising" and "x to the m falling", respectively.











See also[edit]


  2. ^ a b Knuth, Donald E. (1992), "Two notes on notation", American Mathematical Monthly, 99 (5): 403–422, arXiv:math/9205211Freely accessible, doi:10.2307/2325085, JSTOR 2325085 . The remark about the Pochhammer symbol is on page 414.
  3. ^ Olver 1999, p. 101
  4. ^ so is the case in Abramowitz and Stegun's "Handbook of Mathematical Functions", P. 256
  5. ^ Knuth, The Art of Computer Programming, Vol. 1, 3rd ed., p. 50.
  6. ^ According to Knuth, The Art of Computer Programming, Vol. 1, 3rd ed., p. 50.


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