In mathematics, the Pochhammer symbol, introduced by Leo August Pochhammer, Euclid, a famous Greek mathematician known as the father of geometry, is shown here in detail from The School of Athens by Raphael. ... Leo August Pochhammer (1841-1920) was a Prussian mathematician, known for his work on special functions. ...

is used in the theory of special functions to represent the "rising factorial" or "upper factorial" In mathematics, there is a theory or theories of special functions, particular functions such as the trigonometric functions that have useful or attractive properties, and which occur in different applications often enough to warrant a name and attention of their own. ... In mathematics, the factorial of a natural number n is the product of all positive integers less than or equal to n. ...

and, confusingly, is used in combinatorics to represent the "falling factorial" or "lower factorial" Combinatorics is a branch of mathematics that studies collections (usually finite) of objects that satisfy specified criteria. ...

To distinguish the two, the notations x⁽ⁿ⁾ and (x)_n are commonly used to denote the rising and falling factorials, respectively. They are related by a difference in sign:

where the left-hand side is a rising factorial and the right-hand side is a falling factorial. This notation will be used below.

Properties

The empty products x⁽⁰⁾ and (x)₀ are defined to be 1 in both cases. In mathematics, an empty product, or nullary product, is the result of multiplying no numbers. ...

The rising and falling factorials can be expressed in terms of a binomial coefficient: In mathematics, particularly in combinatorics, the binomial coefficient of the natural number n and the integer k is defined to be the natural number and (Here, for a natural number m, m! denotes the factorial of m. ...

Thus a large number of identities on the binomial coefficients carry over to the Pochhammer symbols.

It follows from these expressions that the product of n consecutive integers is divisible by n!. Furthermore, the product of four consecutive integers is a perfect square minus one. The term perfect square is used in mathematics in two meanings: a positive integer which is the square of some other integer, i. ...

The rising factorial can be extended to a real values of n using the Gamma function provided x and x + n are not negative integers: In mathematics, the real numbers are intuitively defined as numbers that are in one-to-one correspondence with the points on an infinite line—the number line. ... The Gamma function along part of the real axis In mathematics, the Gamma function extends the factorial function to complex and non integer numbers (it is already defined on the naturals, and has simple poles at the negative integers). ...

as can the falling factorial:

Alternate notations

Another, less common notation was introduced by Ronald L. Graham, Donald E. Knuth and Oren Patashnik in their book Concrete Mathematics. They define, for the rising factorial: Ronald L. Graham (born October 31, 1935) is a mathematician credited by the American Mathematical Society with being one of the principal architects of the rapid development worldwide of discrete mathematics in recent years[1]. He has done important work in scheduling theory, computational geometry, Ramsey theory, and quasi-randomness. ... Donald Knuth Donald Ervin Knuth (born January 10, 1938) is a renowned computer scientist and Professor Emeritus at Stanford University. ... Oren Patashnik (born 1954) is a computer scientist. ... Concrete Mathematics by Ronald L. Graham, Donald E. Knuth and Oren Patashnik is a textbook that provides its readers with mathematical background that can be especially useful in computer science. ...

and for the falling factorial:

Other notations for the falling factorial include P(x, n), ^xP_n, P_x,n, or _xP_n. (See permutation and combination). In mathematics, especially in abstract algebra and related areas, a permutation is a bijection from a finite set X onto itself. ... It has been suggested that Permutations and combinations be merged into this article or section. ...

Another notation of the falling factorial using a function is:

where −h is the decrement and k is the number of terms. The raising factorial is written:

Relation to umbral calculus

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 x^k in differential calculus. Note for instance the similarity of In mathematics, a polynomial is an expression in which constants and variables are combined using only addition, subtraction, multiplication, and positive whole number exponents (raising to a power). ... In mathematics, a difference operator maps a function f(x) to another function f(x + a) − f(x + b). ... In calculus, Taylors theorem, named after the mathematician Brook Taylor, who stated it in 1712, gives the approximation of a differentiable function near a point by a polynomial whose coefficients depend only on the derivatives of the function at that point. ... Calculus is a central branch of mathematics, developed from algebra and geometry. ... In mathematics, a finite difference is like a differential quotient, except that it uses finite quantities instead of infinitesimal ones. ...

and

(where D denotes differentiation with respect to x). The study of similarities of this type is known as umbral calculus. The general theory covering such relations, including the Pochhammer polynomials, is given by the theory of polynomial sequences of binomial type and by Sheffer sequences. In mathematics, the derivative is defined as the instantaneous rate of change of a function. ... In mathematics, before the 1970s, the term umbral calculus was understood to mean the surprising similarities between otherwise unrelated polynomial equations, and certain shadowy techniques that can be used to prove them. ... Definition In mathematics, a polynomial sequence, i. ... In mathematics, a polynomial sequence, i. ...

Note

Pochhammer actually used (x)_n to denote the binomial coefficient Knuth, ``Two notes on notation".