In
mathematics, the
Pochhammer symbol
- <math>(x)_n\,</math>
is used in the theory of
special functions to represent the
"rising factorial" or "upper
factorial"
- <math>(x)_n=x(x+1)(x+2)\cdots(x+n-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).</math>
The empty product (x)_{0} is defined to be 1 in both cases.
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
- <math>\Delta (x)_k = k (x)_{k-1}</math>
and
- <math>D x^k = k x^{k-1}</math>
(where
D denotes
differentiation with respect to
x).
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