Pochhammer symbol
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In mathematics, the Pochhammer symbol, introduced by Leo August Pochhammer,
- <math>(x)_n\,<math>
is used in the theory of special functions to represent the "rising factorial" or "upperfactorial"
- <math>(x)_n=x(x+1)(x+2)\cdots(x+n-1)=\frac{(x+n-1)!}{(x-1)!}<math>
and, confusingly, is used in combinatorics to represent the "falling factorial" or "lower factorial"
- <math>(x)_n=x(x-1)(x-2)\cdots(x-n+1)=\frac{x!}{(x-n)!}<math>.
In the following, the notation of Ronald L. Graham, Donald E. Knuth and Oren Patashnik in their book Concrete Mathematics will be adopted. They use for the rising factorial:
- <math>x^{\overline{n}}=\frac{(x+n-1)!}{(x-1)!}<math>
and for the falling factorial:
- <math>x^{\underline{n}}=\frac{x!}{(x-n)!}<math>
The empty product (x)0 is defined to be 1 in both cases. Note that the falling factorial can be written as a binomial coefficient:
- <math>\frac{x^{\underline{n}}}{n!} = {x \choose n}<math>
- <math>\frac{x^{ \overline{n}}}{n!} = {x+n-1 \choose n}<math>
and thus a large number of identities on the binomial coefficients carry over to the Pochhammer symbols.
The rising factorial can be generalized to a continuous value of n using the Gamma function:
- <math>x^{\overline{n}}=\frac{\Gamma(x+n)}{\Gamma(x)}<math>
as can the falling factorial:
- <math>x^{\underline{n}}=\frac{\Gamma(x+1)}{\Gamma(x-n+1)}<math>
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 xk in differential calculus. Note for instance the similarity of
- <math>\Delta x^{\underline{k}} = k x^{\underline{k-1}}\,<math>
and
- <math>D x^k = k x^{k-1}\,<math>
(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.it:Fattoriale crescente fr:Symbole de Pochhammer