Talk:Chu-Vandermonde Identity
Where should this go?
The Chu-Vandermonde Identity implies this relationship and vice-versa.
Help please. --Robkahn131 (talk) 04:27, 9 December 2022 (UTC)
- Haven't a clue. --prime mover (talk) 06:49, 9 December 2022 (UTC)
- I took the liberty of removing the source citation (link was broken, btw) as it did not contain the contents of this page. --prime mover (talk) 06:52, 9 December 2022 (UTC)
As you have noted below, you need more than just Chu-Vandermonde Identity, so instead I would try to give the identity a separate name. For example, it is very similar to the Binomial Theorem. So I suggest calling it like "Variant of Binomial Theorem", "Modified Binomial Theorem", "Rising Factorial Binomial Theorem" or something analogous. I also recommend transcluding it to the Binomial Theorem.--Julius (talk) 11:42, 9 December 2022 (UTC)
- Thank you!! --Robkahn131 (talk) 11:56, 9 December 2022 (UTC)
- Now I've had a chance to think about this, the logic does not really make sense. We know that the Chu-Vandermonde Identity is true, we don't need to hypothesise it. --prime mover (talk) 18:08, 9 December 2022 (UTC)
Theorem
Let $r, s \in \R, n \in \Z_{\ge 0}$.
- $\ds \sum_{k \mathop = 0}^n \binom r k \binom s {n - k} = \binom {r + s} n $
Implies the following identity:
- $\ds \leadsto \sum_{k \mathop = 0}^n \dbinom n k r^{\overline k} s^{\overline {n-k} } = \paren {r + s}^{\overline n}$
- A quick teachable moment: Help:Editing/House Style#Capital Letters begin Sentences. Bewilders me how this is such a common mistake. So many do it. --prime mover (talk) 06:49, 9 December 2022 (UTC)
Proof
From Rising Factorial as Factorial by Binomial Coefficient, we have:
| \(\ds r^{\overline k}\) | \(=\) | \(\ds k! \dbinom {r + k - 1} k\) | ||||||||||||
| \(\ds s^{\overline {n - k} }\) | \(=\) | \(\ds \paren{n - k}! \dbinom {s + n - k - 1} {n - k}\) | ||||||||||||
| \(\ds \paren{r + s}^{\overline n}\) | \(=\) | \(\ds n! \dbinom {r + s + n - 1} n\) |
Therefore:
| \(\ds \sum_{k \mathop = 0}^n \dbinom n k r^{\overline k} s^{\overline {n-k} }\) | \(=\) | \(\ds \sum_{k \mathop = 0}^n \paren {\dfrac {n!} {k! \paren{n - k}!} } \paren{ {k! \dbinom {r + k - 1} k} } \paren{ {\paren{n - k}! \dbinom {s + n - k - 1} {n - k} } }\) | Definition of Binomial Coefficients for Real Numbers | |||||||||||
| \(\ds \) | \(=\) | \(\ds n! \sum_{k \mathop = 0}^n {\dbinom {r + k - 1} k} \dbinom {s + n - k - 1} {n - k}\) | $k!$ and $\paren{n - k}!$ cancel | |||||||||||
| \(\ds \) | \(=\) | \(\ds n! \sum_{k \mathop = 0}^n {\dbinom {r + k - 1} {r - 1} } \dbinom {s + n - k - 1} {s - 1}\) | Symmetry Rule for Binomial Coefficients | |||||||||||
| \(\ds \) | \(=\) | \(\ds n! \sum_{k \mathop = 0}^n {\paren{-1}^k \dbinom {-r} {k} } \paren{-1}^{n - k} \dbinom {-s} {n - k}\) | Negated Upper Index of Binomial Coefficient/Corollary 2 | |||||||||||
| \(\ds \) | \(=\) | \(\ds {\paren{-1}^n n! \sum_{k \mathop = 0}^n \dbinom {-r} {k} } \dbinom {-s} {n - k}\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \paren{-1}^n n! \binom {-\paren{r + s} } n\) | Chu-Vandermonde Identity | |||||||||||
| \(\ds \) | \(=\) | \(\ds n! \dbinom {r + s + n - 1} n\) | Negated Upper Index of Binomial Coefficient | |||||||||||
| \(\ds \) | \(=\) | \(\ds \paren{r + s}^{\overline n}\) | Rising Factorial as Factorial by Binomial Coefficient |
$\blacksquare$