Summation Formula for Reciprocal of Binomial Coefficient/Proof 2
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Theorem
\(\ds \sum_{k \mathop \ge 0} \binom n k \dfrac {\paren {-1}^k} {k + x}\) | \(=\) | \(\ds \dfrac 1 {x \binom {n + x} n}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {n!} {x \paren {x + 1} \cdots \paren {x + n} }\) |
as long as the denominators are not zero.
Proof
Consider the value of $\map \Beta {x, n + 1}$, where $\Beta$ is the beta function.
We have:
\(\ds \map \Beta {x, n + 1}\) | \(=\) | \(\ds \int_0^1 t^{x - 1} \paren {1 - t}^n \rd t\) | Definition 1 of Beta Function | |||||||||||
\(\ds \) | \(=\) | \(\ds \int_0^1 t^{x - 1} \sum_{k \mathop \ge 0} \binom n k \paren {-t}^k 1^{n - k} \rd t\) | Binomial Theorem | |||||||||||
\(\ds \) | \(=\) | \(\ds \sum_{k \mathop \ge 0} \binom n k \paren {-1}^k \int_0^1 t^{x - 1} t^k \rd t\) | Linear Combination of Definite Integrals | |||||||||||
\(\ds \) | \(=\) | \(\ds \sum_{k \mathop \ge 0} \binom n k \paren {-1}^k \int_0^1 t^{x + k - 1} \rd t\) | Index Laws/Sum of Indices | |||||||||||
\(\ds \) | \(=\) | \(\ds \sum_{k \mathop \ge 0} \binom n k \paren {-1}^k \frac 1 {x + k}\) |
which equates to the left hand side of the identity.
We also have:
\(\ds \map \Beta {x, n + 1}\) | \(=\) | \(\ds \frac {\map \Gamma x \map \Gamma {n + 1} } {\map \Gamma {x + n + 1} }\) | Definition 3 of Beta Function | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {n! \map \Gamma x } {\map \Gamma {x + n + 1} }\) | Gamma Function Extends Factorial | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {n! \map \Gamma x } {\paren {x + n} \map \Gamma {x + n} }\) | Gamma Difference Equation | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {n! \map \Gamma x } {\paren {x + n - 1} \paren {x + n} \map \Gamma {x + n - 1} }\) | Gamma Difference Equation | |||||||||||
\(\ds \) | \(:\) | \(\ds \) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac {n! \map \Gamma x } {x \paren {x + 1} \cdots \paren {x + n} \map \Gamma x}\) | Gamma Difference Equation | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {n!} {x \paren {x + 1} \cdots \paren {x + n} }\) |
which equates to the right hand side of the identity.
Hence the result.
$\blacksquare$
Sources
- 1997: Donald E. Knuth: The Art of Computer Programming: Volume 1: Fundamental Algorithms (3rd ed.) ... (previous) ... (next): $\S 1.2.6$: Binomial Coefficients: Exercise $48$