# A power series equality involving binomial coefficients

I believe the following identity holds for any non-negative integers $$m,k$$:

$$\binom{m+k}{k}^2 = \sum_{n=0}^m \binom{k}{m-n}^2\binom{2k+n}{n}.$$

It seems like there should be a slick double counting way of proving it, but I am not able to see it. I got this identity because I believe the following power series equality holds for $$|x| < 1$$,

$$(1-x)^{2k-1} \sum_{m=0}^\infty \binom{m+k-1}{k-1}^2 x^m = \sum_{j=0}^{k-1} \binom{k-1}{j}^2 x^j.$$

The binomial identity above comes from dividing by $$(1-x)^{2k-1}$$, applying the binomial theorem replacing $$k$$ with $$k+1$$ and comparing the coefficients of both sides.

But this power series equality doesn't seem any easier to prove than the binomial coefficient identity, since I don't really have a handle on the sums on either side.

Does anyone know a reference for an identity of this nature or have an idea for a double counting argument?

• Searching with Approach Zero you can find several references to this identity, for example here. You just need to tweak a little the index and variables. This is [Li Shanlan identity](en.wikipedia.org/wiki/… – BillyJoe Feb 19 at 8:56

## 1 Answer

Permit me to contribute an algebraic proof. We seek

$${m+k\choose k}^2 = \sum_{q=0}^m {k\choose m-q}^2 {2k+q\choose q}$$

Starting with the RHS we find

$$\sum_{q=0}^m {k\choose q}^2 {2k+m-q\choose m-q} \\ = \sum_{q=0}^m {k\choose q} [z^k] z^q (1+z)^k [w^{m}] w^q (1+w)^{2k+m-q}.$$

Now we may extend $$q$$ to infinity because the coefficient extractor $$[w^m]$$ enforces the upper limit. We get

$$[z^k] (1+z)^k [w^m] (1+w)^{2k+m} \sum_{q\ge 0} {k\choose q} z^q w^q (1+w)^{-q} \\ = [z^k] (1+z)^k [w^m] (1+w)^{2k+m} (1 + zw/(1+w))^k \\ = [z^k] (1+z)^k [w^m] (1+w)^{k+m} (1 + w + zw)^k$$

Re-expanding we find

$$[z^k] (1+z)^k [w^m] (1+w)^{k+m} \sum_{q=0}^k {k\choose q} w^q (1+z)^q.$$

We may set the upper limit of the sum to $$m.$$ (If $$k\lt m$$ the values $$k\lt q\le m$$ produce zero from the binomial coefficient and we may raise $$q$$ to $$m.$$ If $$k\gt m$$ the values $$m\lt q\le k$$ produce zero by the coefficient extractor $$[w^m]$$ and we may lower $$q$$ to $$m.$$) We get

$$[z^k] (1+z)^k [w^m] (1+w)^{k+m} \sum_{q=0}^m {k\choose q} w^q (1+z)^q \\ = \sum_{q=0}^m {k\choose q} {k+q\choose k} {k+m\choose m-q}.$$

Now observe that

$${k+q\choose k} {k+m\choose m-q} = \frac{(k+m)!}{k! \times q! \times (m-q)!} = {m+k\choose k} {m\choose m-q}.$$

This yields for our sum

$${m+k\choose k} \sum_{q=0}^m {k\choose q} {m\choose m-q}.$$

Using Vandermonde we obtain at last

$$\bbox[5px,border:2px solid #00A000]{ {m+k\choose k}^2.}$$

• Nice and instructive derivation. (+1) – Markus Scheuer Feb 19 at 22:07
• Thanks! This one worked using formal power series only so we were able to keep it simple. – Marko Riedel Feb 19 at 22:10
• Yes, I see. Nevertheless it's always both astonishing and a pleasure for me to see the nice binomial identities we derive in intermediate steps together with their generating functions. – Markus Scheuer Feb 19 at 22:14