Binomial Theorem for Negative Index and Negative Parameter
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Theorem
Let $n \in \Z_{\ge 0}$ be a positive integer.
Let $z \in \R$ be a real number such that $\size z < 1$.
Then:
\(\ds \dfrac 1 {\paren {1 - z}^{n + 1} }\) | \(=\) | \(\ds \sum_{k \mathop \ge 0} \binom {-n - 1} k \paren {-z}^k\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \sum_{k \mathop \ge 0} \binom {n + k} n z^k\) |
where $\dbinom {n + k} n$ denotes a binomial coefficient.
Proof
\(\ds \dfrac 1 {\paren {1 - z}^{n + 1} }\) | \(=\) | \(\ds \paren {1 + \paren {-z} }^{- n - 1}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \sum_{k \mathop \ge 0} \binom {- n - 1} k \paren {-z}^k\) | General Binomial Theorem | |||||||||||
\(\ds \) | \(=\) | \(\ds \sum_{k \mathop \ge 0} \dbinom {n + 1 + k - 1} k \paren {-1}^k \paren {-z}^k\) | Negated Upper Index of Binomial Coefficient | |||||||||||
\(\ds \) | \(=\) | \(\ds \sum_{k \mathop \ge 0} \dbinom {n + k} k z^k\) |
$\blacksquare$
Sources
- 1997: Donald E. Knuth: The Art of Computer Programming: Volume 1: Fundamental Algorithms (3rd ed.) ... (previous) ... (next): $\S 1.2.9$: Generating Functions: $(20)$