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My question may be really elementary, simple, or obvious, and yet the solution misses me.

My question is on a combinatorics answer to the question Proving $\sum\limits_{m=0}^M \binom{m+k}{k} = \binom{k+M+1}{k+1}$.

I follow most of the mathematics in the answer, such as the Chu-Vandermonde identity, identity $(\mathbf{1})$ below,

$$\binom{-n}{k}=(-1)^k\binom{n+k-1}{k}\tag{1}$$

and all the steps except for two, namely, how does $(\mathbf{6})$ follow from $(\mathbf{5})$?

$$ \begin{align} &=(-1)^{M-n}\binom{-k-n-2}{M-n}\tag{5}\\ &=\binom{M+k+1}{M-n}\tag{6}\\ \end{align}$$

The following shows my attempt to answer that question.

I noticed that $(\mathbf{5})$ takes the same form as the $(-1)^k\binom{n+k-1}{k}$ of identity $(\mathbf{1})$. For example, both have $(-1)$ raised to a power $k$, and multiplied by a binomial coefficient that is indexed by $k$. Now, if the expression $$(-1)^{M-n}\binom{-k-n-2}{M-n}\tag{5}$$

is somehow in a form compatible with identity $(\mathbf{1})$, then we can substitute $(\mathbf{5})$ into the RHS of $(\mathbf{1})$, and the LHS result of $(\mathbf{1})$ will hopefully equal $(\mathbf{6})$.

Still, it's not immediately obvious to me if or how $(\mathbf{5})$ can be transformed into the RHS of $(\mathbf{1})$, so that we can apply the identity $(\mathbf{1})$ to show $(\mathbf{5}) = (\mathbf{6}) $ .

What algebraic manipulations, renaming, substitutions, identities or otherwise, are needed to show $(\mathbf{5}) = (\mathbf{6})$?

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  • You can cheat a little, by realizing that $(-1)^{M-n}$ can only ever equal $-1$ or $1$. It could be easier to handle both cases separately, and then show they give the same result. – Newb Dec 09 '13 at 11:54
  • The following result is also an implication of Vandermonde's identity: $$\sum_{n=0}^\infty\left[\frac{(2n-3)!!}{(2n)!!}\right]^2=\frac4\pi$$ – Lucian Dec 09 '13 at 23:43

2 Answers2

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Notice how identity (1) takes the form:

$$\binom{-n}{k}=(-1)^k\binom{n+k-1}{k} = (-1)^k\binom{k-(-n)-1}{k}$$

Compare with

$$(-1)^{M-n}\binom{-k-n-2}{M-n}=(-1)^{M-n}\binom{(M-n)-(M+k+1)-1}{M-n}$$

It follows from (1) that

$$(-1)^{M-n}\binom{(M-n)-(M+k+1)-1}{M-n} = \binom{M+k+1}{M-n}$$

and hence

$$(-1)^{M-n}\binom{-k-n-2}{M-n}=\binom{M+k+1}{M-n}$$

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I hadn't noticed until today how straightforward it is to see how $\mathbf{(6,5)}$ are an instance of $(\mathbf{1})$. If you notice that the RHS of $$\binom{-n}{k}=(-1)^k\binom{n+k-1}{k}\tag{1}$$ hasn't many requirements, and allows for many possibilities to satisfy this form. The RHS allows any $k \in \mathbb{Z}^+$, has $k$ appear as the exponent to $(-1)$, and observe that $\binom{n+k-1}{k}$ only asks for some $n \in \mathbb{Z}$, and this imposes no relationship on $n, k$. So there isn't much special about $n \ \text{or}\ k$.

It does not take much to form an instance of $(\mathbb{1})$, so without much thought we can show that $(\mathbb{5})$ is an instance of the RHS of $(\mathbb{1})$ like so

Let $-n = M + k + 1, k = M - n$. Then $n + k - 1 = -(M+k+1)+(M-n)-1 = -k - n - 2$.

and without anything but variable assignments and substitution we have shown $(\mathbf 5, 6)$ are an instance of $(\mathbf{1})$. One might notice that the concluding statement $\binom{n+k+1}{k} = \binom{-k-n-2}{M - n}$ looks like a contradiction when we compare the binomial coefficients' indices, but keep in mind the $n$ defined in $(\mathbb{1})$ is not the same $n$ as in $(\mathbb{5,6})$, which is defined in the context of a larger $7$-step answer.

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