Rising factorial: x^(n) = x(x+1)...(x+n-1). Falling factorial: x_(n) = x(x-1)...(x-n+1). Rising = Pochhammer symbol in analysis. Falling = Pochhammer in combinatorics (confusing!). Both connect to Γ: x^(n)=Γ(x+n)/Γ(x). Fundamental in special functions.
How to Use
Enter x and n:
- Rising: x^(n)
- Falling: x_(n)
- Expansion: Each factor
Key Connections
C(n,k) = n_(k)/k! (falling). Stirling numbers convert between ordinary and factorial powers. Hypergeometric series ₂F₁ uses ratios of Pochhammer symbols. Taylor series in finite calculus use falling factorials instead of ordinary powers.
Notation Warning
Confusingly: analysts write (x)_n for rising factorial. Combinatorialists write (x)_n for FALLING factorial. We use x^(n) for rising and (x)_n for falling to avoid ambiguity. Always check which convention an author uses!
Step-by-Step Instructions
- 1Enter x.
- 2Enter n.
- 3Compute rising.
- 4Compute falling.
- 5Compare.