Falling and rising factorials
Falling and rising factorials are polynomial functions that generalize the ordinary factorial to arbitrary real or complex arguments, playing key roles in combinatorics, finite difference calculus, and the theory of special functions. The falling factorial of x to the power n, denoted (x)_n, is defined as the product x(x-1)(x-2)\cdots(x-n+1) for positive integer n, with (x)_0 = 1.[1] The rising factorial, also known as the Pochhammer symbol and denoted x^{(n)} or (x)_n in some contexts, is defined as x(x+1)(x+2)\cdots(x+n-1), again with the base case 1 for n=0.[2] The falling factorial coincides with the standard factorial for positive integer n, as n! = (n)_n. The rising factorial satisfies n! = 1^{(n)}.[1][2] The falling and rising factorials are closely related through the identity (x)_n = (-1)^n (-x)^{(n)}, which highlights their symmetry and allows interconversion in analytic expressions.[1] Both can be expressed using the gamma function for non-integer extensions: the rising factorial satisfies x^{(n)} = \frac{\Gamma(x+n)}{\Gamma(x)}, while the falling factorial follows analogously as (x)_n = \frac{\Gamma(x+1)}{\Gamma(x-n+1)}.[2] Notation varies across literature; for instance, the Pochhammer symbol (x)_n typically denotes the rising factorial in special functions theory, but some texts use it for the falling version, necessitating caution.[3] Basic properties include recurrence relations, such as (x)_{n+1} = x(x)_n - n(x)_n for the falling factorial, and similar forms for the rising one.[1] In combinatorics, falling factorials appear in the binomial theorem's generalization, (x+y)_n = \sum_{k=0}^n \binom{n}{k} (x)_k (y)_{n-k}, counting permutations and combinations with repetitions.[1] Their generating function is (1+t)^x = \sum_{k=0}^\infty \frac{(x)_k}{k!} t^k, linking them to exponential series.[1] Rising factorials are essential in hypergeometric series expansions, where they parameterize confluent and generalized hypergeometric functions, such as {}_pF_q(a_1,\dots,a_p;b_1,\dots,b_q;z) = \sum_{k=0}^\infty \frac{(a_1)_k \cdots (a_p)_k}{(b_1)_k \cdots (b_q)_k} \frac{z^k}{k!}.[2] In finite differences, falling factorials form a basis for polynomials analogous to powers, enabling discrete analogs of derivatives via forward differences \Delta (x)_n = n (x)_{n-1}.[4] These applications extend to statistics for modeling polynomial regressions and in numerical analysis for interpolation and splines.[4][5]Definitions and Notation
Falling Factorial
The falling factorial, denoted (x)_n, is a polynomial in x of degree n defined for any real or complex number x and non-negative integer n.[1] It extends the concept of the ordinary factorial from positive integers to broader domains and appears frequently in combinatorics, algebra, and analysis.[6] For n \geq 1, the falling factorial is given by the product formula (x)_n = x(x-1)(x-2) \cdots (x-n+1), which consists of n successive terms decreasing by 1 starting from x.[1] For n=0, the empty product convention yields (x)_0 = 1.[1] This definition aligns with the notation introduced in seminal works on discrete mathematics, where the falling factorial serves as a basis for polynomial interpolation and finite differences.[7] When x is a positive integer k with k \geq n, the falling factorial simplifies to the permutation formula (k)_n = \frac{k!}{(k-n)!}, representing the number of ways to arrange n distinct items from k available ones, though here it is presented purely as an algebraic reduction.[8] This special case connects directly to the ordinary factorial k!, which is recovered when n = k as (k)_k = k!.[8] The falling factorial also admits a recursive definition: (x)_n = (x)_{n-1} (x - n + 1) for n \geq 1, with the base case (x)_0 = 1.[9] This recurrence facilitates computational evaluation and proofs involving the function.[9] As a counterpart, the rising factorial employs a product of increasing terms, providing a dual perspective in related mathematical contexts.[1]Rising Factorial
The rising factorial, denoted x^{(n)} and also known as the Pochhammer symbol (x)_n, generalizes the factorial to real or complex arguments x and nonnegative integers n. It is defined by the ascending product formula x^{(n)} = x(x+1)(x+2) \cdots (x+n-1) for n \geq 1, with the base case x^{(0)} = 1.[10] This construction emphasizes an increasing sequence of terms, in contrast to the falling factorial's decreasing sequence. When x is a positive integer, the rising factorial reduces to a ratio of factorials: x^{(n)} = \frac{(x+n-1)!}{(x-1)!}. This equivalence follows directly from the product form, as the terms x through x+n-1 form the upper portion of the expanded factorial (x+n-1)! after canceling the lower terms up to (x-1)!.[10] The rising factorial admits a recursive definition: x^{(n)} = x^{(n-1)} (x + n - 1), with x^{(0)} = 1, which mirrors the iterative multiplication in the product formula.[10] For analytic extension beyond integers, the rising factorial connects to the gamma function: x^{(n)} = \frac{\Gamma(x+n)}{\Gamma(x)}, valid when x is not a non-positive integer to avoid poles in the gamma function.[10] This relation, introduced by Leo August Pochhammer in the context of hypergeometric functions, enables evaluation for non-integer x and underpins its role in special functions and series expansions.Relation Between Falling and Rising
The falling factorial (x)_n and the rising factorial x^{(n)} are interconnected through identities that shift the argument or incorporate a sign change, allowing one to be expressed directly in terms of the other. A primary relation is given by the argument shift identity x^{(n)} = (x + n - 1)_n, which equates the product x(x+1)\cdots(x+n-1) for the rising factorial to the falling factorial starting at the raised base x + n - 1. The converse follows immediately as (x)_n = (x - n + 1)^{(n)}, reflecting the reverse ordering of factors in the products. Another fundamental link involves negation, with the identity (x)_n = (-1)^n (-x)^{(n)}, derived from expanding both sides and observing the sign pattern in the factors of the falling factorial at -x.[1] This relation facilitates analytic continuations and handles cases with negative or complex arguments. Both factorials feature prominently in generalized binomial expansions, where the falling factorial underlies the series for (1 + y)^x = \sum_{k=0}^\infty \binom{x}{k} y^k with \binom{x}{k} = (x)_k / k!, while the rising factorial appears in (1 - y)^{-x} = \sum_{k=0}^\infty \frac{x^{(k)}}{k!} y^k, linking them through argument adjustments like replacing y with -y and shifting x. The Pochhammer symbol (x)_n, standard for the rising factorial since its introduction by Leo August Pochhammer in studies of hypergeometric series, contrasts with falling factorial notation in contemporary texts, though earlier literature occasionally reversed these conventions.[3]Combinatorial Interpretations
Basic Examples
The falling factorial (x)_n and rising factorial x^{(n)} are defined as products of n consecutive terms, with the falling form decreasing from x and the rising form increasing from x.[11][1][2] For integer values, consider x = 5 and n = 3: (5)_3 = 5 \times 4 \times 3 = 60, while $5^{(3)} = 5 \times 6 \times 7 = 210. These computations follow directly from the product definitions.[11] For non-integer arguments, the definitions extend naturally. For example, with x = 1/2 and n = 2, \left(\frac{1}{2}\right)_2 = \frac{1}{2} \times \left(\frac{1}{2} - 1\right) = \frac{1}{2} \times \left(-\frac{1}{2}\right) = -\frac{1}{4}. The rising factorial in this case is \left(1/2\right)^{(2)} = (1/2) \times (3/2) = 3/4.[11] When n = 0, both the falling and rising factorials equal 1 for any x, as they represent the empty product.[11][1] The following table provides further computations for small integer values of x = 0, 1, 2, 3 and n = 1, 2, 3:| x | n | (x)_n (falling) | x^{(n)} (rising) |
|---|---|---|---|
| 0 | 1 | 0 | 0 |
| 0 | 2 | 0 | 0 |
| 0 | 3 | 0 | 0 |
| 1 | 1 | 1 | 1 |
| 1 | 2 | 0 | 2 |
| 1 | 3 | 0 | 6 |
| 2 | 1 | 2 | 2 |
| 2 | 2 | 2 | 6 |
| 2 | 3 | 0 | 24 |
| 3 | 1 | 3 | 3 |
| 3 | 2 | 6 | 12 |
| 3 | 3 | 6 | 60 |