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I was wandering if someone knows an elementary proof of the following identity:

$ \frac{(a)_n (b)_n}{(n!)^2} = \sum_{k=0}^n (-1)^k {1-a-b \choose k} \frac{(1-a)_{n-k}(1-b)_{n-k}}{((n-k)!)^2}\ , $ where $a,b$ are arbitrary real numbers, $(x)_0=1,\quad (x)_n:=x(x+1)\cdots(x+n-1) \quad \mbox{for $n\geq 1$} $ is the Pochhammer's symbol, and $ {x\choose 0}=1,\quad {x\choose k} = \frac{x(x-1)\cdots (x-k+1)}{k!} \quad \mbox{for $k\geq 1$}$ is a binomial coefficient.

The proof that I know uses the Hypergeometric differential equation. One has to continue analytically the solutions along a path connecting two singular points. This could be done by some well known integral representation of the Hypergeometric function.

I think that there should be a combinatorial proof. Since this is an identity between polynomials in $a$ and $b$, it is enough to prove it for $a$ and $b$ negative integers, i.e., we may assume that $a=-p$ and $b=-q$ where $p,q\in \mathbb{Z}_{\geq 0}$. The identity then turns into $ {p\choose n}{q\choose n} = \sum_{k=0}^n (-1)^k {1+p+q\choose k}{p+n-k\choose n-k}{q+n-k\choose n-k}. $ I did not try very hard to proof the above identity and I did not search the literature that much, but since it comes from an interesting subject I think that it is worth finding an alternative proof.

  • 1
    Put another way, we want to prove that $\binom{a+b+n-2}{n}{}_3 F_2\left({{-n,1-a,1-b}\atop{1,2-a-b-n}}\mid 1\right)=\binom{a+n-1}{n}\binom{b+n-1}{n}$2012-07-23

2 Answers 2

3

Here's a proof using generating functions.

We'll use the following repeatedly: $\displaystyle \sum_{k \ge 0} \binom{n}{k} x^k = (1+x)^n, \sum_{k \ge 0} \binom{k}{n} x^k = \frac{x^n}{(1-x)^{n+1}}$

We want to prove the following:

$ {p\choose n}{q\choose n} = \sum_{k=0}^n (-1)^k {1+p+q\choose k}{p+n-k\choose n-k}{q+n-k\choose n-k}. $

Let's attack the gullible left hand side first. Multiply by $x^n y^q$ and sum over all $n, q \ge 0$, and rearrange the summations to get $\sum_{n \ge 0} {p\choose n} x^n \sum_{q \ge 0} {q\choose n} y^q = \sum_{n \ge 0} {p\choose n} x^n \frac{y^n}{(1-y)^{n+1}} = \frac{1}{1-y} \left(1 + \frac{xy}{1-y} \right)^p = \frac{(1-y +xy)^p}{(1-y)^{p+1}}$

So much for the left hand side. Now to the right hand side.

Firstly, change $k \rightarrow n-k$ in the summation, so that our right hand side becomes:

$\sum_{k=0}^n (-1)^{n-k} {1+p+q\choose n-k}{p+k\choose k}{q+k\choose k} = \sum_{k=0}^n (-1)^{n-k} {1+p+q\choose n-k}{p+k\choose p}{q+k\choose k}$

Now, multiply first by $x^n$, sum over $n \ge 0$, and rearrange to get : (We'll multiply $y^q$ later)

$\sum_{k \ge 0} {p+k\choose p}{q+k\choose k} \sum_{n \ge 0} (-1)^{n-k} {1+p+q\choose n-k} x^n = \sum_{k \ge 0} {p+k\choose p}{q+k\choose k} x^k (1-x)^{1+p+q}$

Now comes $y$. Multiply by $y^q$, sum over $q \ge 0$, and rearrange to get :

$(1-x)^{1+p} \sum_{k \ge 0} {p+k\choose p} x^k \sum_{q \ge 0} {q+k\choose k} y^q (1-x)^{q} = (1-x)^{1+p} \sum_{k \ge 0} {p+k\choose p} \frac{x^k}{(1-y+xy)^{k+1}}$

Now, this is equal to $\displaystyle \frac{(1-x)^{1+p}}{(1-y+xy)} \frac{1}{(1-t)^{1+p}}$, where $\displaystyle t = \frac{x}{(1-y+xy)} \implies 1-t = \frac{(1-x)(1-y)}{(1-y+xy)}$, giving us that $\displaystyle \frac{(1-x)^{1+p}}{(1-y+xy)} \frac{1}{(1-t)^{1+p}} = \frac{(1-y +xy)^p}{(1-y)^{p+1}}$, which was exactly what was needed.

2

Suppose we seek to evaluate

$\sum_{k=0}^n (-1)^k {1+p+q\choose k} {p+n-k\choose n-k} {q+n-k\choose n-k}$ which is claimed to be ${p\choose n}{q\choose n}.$

Introduce ${p+n-k\choose n-k} = \frac{1}{2\pi i} \int_{|z_1|=\epsilon} \frac{(1+z_1)^{p+n-k}}{z_1^{n-k+1}} \; dz_1$

and ${q+n-k\choose n-k} = \frac{1}{2\pi i} \int_{|z_2|=\epsilon} \frac{(1+z_2)^{q+n-k}}{z_2^{n-k+1}} \; dz_2.$

Observe that these integrals vanish when $k\gt n$ and we may extend $k$ to infinity.

We thus obtain for the sum $\frac{1}{2\pi i} \int_{|z_1|=\epsilon} \frac{(1+z_1)^{p+n}}{z_1^{n+1}} \frac{1}{2\pi i} \int_{|z_2|=\epsilon} \frac{(1+z_2)^{q+n}}{z_2^{n+1}} \\ \times \sum_{k\ge 0} {1+p+q\choose k} (-1)^k \frac{z_1^k z_2^k}{(1+z_1)^k (1+z_2)^k} \; dz_2\; dz_1.$

This is $\frac{1}{2\pi i} \int_{|z_1|=\epsilon} \frac{(1+z_1)^{p+n}}{z_1^{n+1}} \frac{1}{2\pi i} \int_{|z_2|=\epsilon} \frac{(1+z_2)^{q+n}}{z_2^{n+1}} \\ \times \left(1-\frac{z_1 z_2}{(1+z_1)(1+z_2)}\right)^{p+q+1} \; dz_2\; dz_1$

or $\frac{1}{2\pi i} \int_{|z_1|=\epsilon} \frac{(1+z_1)^{n-q-1}}{z_1^{n+1}} \frac{1}{2\pi i} \int_{|z_2|=\epsilon} \frac{(1+z_2)^{n-p-1}}{z_2^{n+1}} (1+ z_1 + z_2)^{p+q+1} \; dz_2\; dz_1$

Supposing that $p\ge n$ and $q\ge n$ this may be re-written as $\frac{1}{2\pi i} \int_{|z_1|=\epsilon} \frac{1}{z_1^{n+1} (1+z_1)^{q+1-n}} \frac{1}{2\pi i} \int_{|z_2|=\epsilon} \frac{1}{z_2^{n+1} (1+z_2)^{p+1-n}} \\ \times (1+ z_1 + z_2)^{p+q+1} \; dz_2\; dz_1$

Put $z_2 = (1+z_1) z_3$ so that $dz_2 = (1+z_1) \; dz_3$ to get $\frac{1}{2\pi i} \int_{|z_1|=\epsilon} \frac{1}{z_1^{n+1} (1+z_1)^{q+1-n}} \frac{1}{2\pi i} \int_{|z_2|=\epsilon} \frac{1}{(1+z_1)^{n+1} z_3^{n+1} (1+(1+z_1)z_3)^{p+1-n}} \\ \times (1+ z_1)^{p+q+1} (1+z_3)^{p+q+1} \; (1+z_1) \; dz_3\; dz_1$

which is $\frac{1}{2\pi i} \int_{|z_1|=\epsilon} \frac{(1+z_1)^p}{z_1^{n+1}} \frac{1}{2\pi i} \int_{|z_2|=\epsilon} \frac{1}{z_3^{n+1} (1+ z_3 + z_1 z_3)^{p+1-n}} \\ \times (1+z_3)^{p+q+1} \; dz_3\; dz_1 \\ = \frac{1}{2\pi i} \int_{|z_1|=\epsilon} \frac{(1+z_1)^p}{z_1^{n+1}} \frac{1}{2\pi i} \int_{|z_2|=\epsilon} \frac{(1+z_3)^{n+q}}{z_3^{n+1} (1 + z_1 z_3 /(1+z_3))^{p+1-n}} \; dz_3\; dz_1$

Extracting the residue for $z_1$ first we obtain $\sum_{k=0}^n {p\choose n-k} \frac{(1+z_3)^{n+q}}{z_3^{n+1}} {k+p-n\choose k} (-1)^k \frac{z_3^k}{(1+z_3)^k}.$

The residue for $z_3$ then yields $\sum_{k=0}^n (-1)^k {p\choose n-k} {k+p-n\choose k} {n-k+q\choose n-k}.$

The sum term here is $\frac{p!\times (p+k-n)!\times (q+n-k)!} {(n-k)! (p+k-n)! \times k! (p-n)! \times (n-k)! q!}$ which simplifies to $\frac{p!\times n! \times (q+n-k)!} {(n-k)! \times n!\times k! (p-n)! \times (n-k)! q!}$ which is ${n\choose k} {p\choose n}{q+n-k\choose q}$ so we have for the sum ${p\choose n} \sum_{k=0}^n {n\choose k} (-1)^k {q+n-k\choose q}.$

To evaluae the remaining sum we introduce ${q+n-k\choose q} = \frac{1}{2\pi i} \int_{|v|=\epsilon} \frac{(1+v)^{q+n-k}}{v^{q+1}} \; dv$

getting for the sum ${p\choose n} \frac{1}{2\pi i} \int_{|v|=\epsilon} \frac{(1+v)^{q+n}}{v^{q+1}} \sum_{k=0}^n {n\choose k} (-1)^k \frac{1}{(1+v)^k} \; dv \\ = {p\choose n} \frac{1}{2\pi i} \int_{|v|=\epsilon} \frac{(1+v)^{q+n}}{v^{q+1}} \left(1-\frac{1}{1+v}\right)^n \; dv \\ = {p\choose n} \frac{1}{2\pi i} \int_{|v|=\epsilon} \frac{(1+v)^{q}}{v^{q-n+1}} \; dv = {p\choose n} {q\choose q-n}$

which is ${p\choose n} {q\choose n}.$

This concludes the argument.