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Definition df-risefac 14718
Description: Define the rising factorial function. This is the function (𝐴 · (𝐴 + 1) · ...(𝐴 + 𝑁)) for complex 𝐴 and nonnegative integers 𝑁. (Contributed by Scott Fenton, 5-Jan-2018.)
Assertion
Ref Expression
df-risefac RiseFac = (𝑥 ∈ ℂ, 𝑛 ∈ ℕ0 ↦ ∏𝑘 ∈ (0...(𝑛 − 1))(𝑥 + 𝑘))
Distinct variable group:   𝑥,𝑛,𝑘

Detailed syntax breakdown of Definition df-risefac
StepHypRef Expression
1 crisefac 14717 . 2 class RiseFac
2 vx . . 3 setvar 𝑥
3 vn . . 3 setvar 𝑛
4 cc 9919 . . 3 class
5 cn0 11277 . . 3 class 0
6 cc0 9921 . . . . 5 class 0
73cv 1480 . . . . . 6 class 𝑛
8 c1 9922 . . . . . 6 class 1
9 cmin 10251 . . . . . 6 class
107, 8, 9co 6635 . . . . 5 class (𝑛 − 1)
11 cfz 12311 . . . . 5 class ...
126, 10, 11co 6635 . . . 4 class (0...(𝑛 − 1))
132cv 1480 . . . . 5 class 𝑥
14 vk . . . . . 6 setvar 𝑘
1514cv 1480 . . . . 5 class 𝑘
16 caddc 9924 . . . . 5 class +
1713, 15, 16co 6635 . . . 4 class (𝑥 + 𝑘)
1812, 17, 14cprod 14616 . . 3 class 𝑘 ∈ (0...(𝑛 − 1))(𝑥 + 𝑘)
192, 3, 4, 5, 18cmpt2 6637 . 2 class (𝑥 ∈ ℂ, 𝑛 ∈ ℕ0 ↦ ∏𝑘 ∈ (0...(𝑛 − 1))(𝑥 + 𝑘))
201, 19wceq 1481 1 wff RiseFac = (𝑥 ∈ ℂ, 𝑛 ∈ ℕ0 ↦ ∏𝑘 ∈ (0...(𝑛 − 1))(𝑥 + 𝑘))
Colors of variables: wff setvar class
This definition is referenced by:  risefacval  14720
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