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Fungrim entry: a5852d

(z)k=i=1k(zi+1)=i=0k1(zi)\left(z\right)^{\underline{k}} = \prod_{i=1}^{k} \left(z - i + 1\right) = \prod_{i=0}^{k - 1} \left(z - i\right)
Assumptions:zCandkZ0z \in \mathbb{C} \,\mathbin{\operatorname{and}}\, k \in \mathbb{Z}_{\ge 0}
TeX:
z \in \mathbb{C} \,\mathbin{\operatorname{and}}\, k \in \mathbb{Z}_{\ge 0}
Definitions:
Fungrim symbol Notation Short description
FallingFactorial(z)k\left(z\right)^{\underline{k}} Falling factorial
Productnf ⁣(n)\prod_{n} f\!\left(n\right) Product
CCC\mathbb{C} Complex numbers
ZZGreaterEqualZn\mathbb{Z}_{\ge n} Integers greater than or equal to n
Source code for this entry:
Entry(ID("a5852d"),
    Equal(FallingFactorial(z, k), Product(Parentheses(Add(Sub(z, i), 1)), Tuple(i, 1, k)), Product(Parentheses(Sub(z, i)), Tuple(i, 0, Sub(k, 1)))),
    Variables(z, k),
    Assumptions(And(Element(z, CC), Element(k, ZZGreaterEqual(0)))))

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Copyright (C) Fredrik Johansson and contributors. Fungrim is provided under the MIT license. The source code is on GitHub.

2019-08-21 11:44:15.926409 UTC