Sum of Indices of Falling Factorial
Theorem
- $x^{\underline {m + n} } = x^{\underline m} \paren {x - m}^{\underline n}$
where $x^{\underline m}$ denotes $x$ to the $m$ falling.
Proof
| \(\ds x^{\underline {m + n} }\) | \(=\) | \(\ds \prod_{j \mathop = 0}^{m + n - 1} \paren {x - j}\) | Definition of Falling Factorial | |||||||||||
| \(\ds \) | \(=\) | \(\ds \prod_{j \mathop = 0}^{m - 1} \paren {x - j} \prod_{j \mathop = m}^{m + n - 1} \paren {x - j}\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \prod_{j \mathop = 0}^{m - 1} \paren {x - j} \prod_{j \mathop = 0}^{n - 1} \paren {x - {\paren {m + j} } }\) | Translation of Index Variable of Product | |||||||||||
| \(\ds \) | \(=\) | \(\ds \prod_{j \mathop = 0}^{m - 1} \paren {x - j} \prod_{j \mathop = 0}^{n - 1} \paren {\paren {x - m} - j}\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds x^{\underline m} \paren {x - m}^{\underline n}\) | Definition of Falling Factorial |
$\blacksquare$
Sources
- 1997: Donald E. Knuth: The Art of Computer Programming: Volume 1: Fundamental Algorithms (3rd ed.) ... (previous) ... (next): $\S 1.2.5$: Permutations and Factorials: Exercise $25$