Spring is here

It is amazing how a couple of days of sunshine and warmer weather erased most of the evidence that we had a major winter storm 2 weeks ago.

Winter storm Stella

Pi Day 2017

Happy Pi day (and a snowy one in the Northeastern US)! One of the early childhood mathematical memories I had was learning that π can be approximated by 22/7 and accurate up to 2 decimal places, a fact that was known since antiquity. In particular, Archimedes gave the first proof that 22/7 is strictly larger than π. The fifth century Chinese mathematician and astronomer Zu Chongzhi gave the approximation 355/113 that is accurate up to 6 decimal places. Today, sophisticated algorithms using Ramanujan's formula and its variants can compute trillions of digits of π.

Binomial coefficients

When studying binomial coefficients and how they form Pascal's Triangle, we learn that they are the coefficients of $(x+y)^n$, i.e.

$$(x+y)^n = \sum_{k=0}^n \left(\begin{array}{c}n\\k\end{array}\right)x^ky^{n-k}$$

Replacing $y$ with $-y$ we get:

$$(x-y)^n = \sum_{k=0}^n -1^k\left(\begin{array}{c}n\\k\end{array}\right)x^{n-k}y^{k}$$

For example, the coefficients of $(x+y)^4 = x^4 + 4x^3y + 6x^2y^2 + 4xy^3 + y^4$ are 1,4,6,4,1 which form the 5th row of Pascal's triangle.

What happens if the coefficients of the polynomial are repeated binomial coefficients such as 1,1,4,4,6,6,4,4,1,1? What is $$\sum_{k=0}^{2n+1} \left(\begin{array}{c}n\\\lfloor k/2\rfloor\end{array}\right)x^ky^{n-k}?$$

It is easy to show that this is equal to $y(x^2+y^2)^n + x(x^2+y^2)^n = (x+y)(x^2+y^2)^n$.

What about repeating coefficients with sign changes such as 1,1,-4,-4,6,6,-4,-4,1,1?

The same argument shows that 

$$\sum_{k=0}^{2n+1} -1^{\lfloor k/2\rfloor}\left(\begin{array}{c}n\\\lfloor k/2\rfloor\end{array}\right)x^{n-k}y^{k}$$

is equal to $y(x^2-y^2)^n + x(x^2-y^2)^n = (x+y)(x^2-y^2)^n$ which can be further simplified as $(x+y)^{n+1}(x-y)^n$.

If we now change the sign of every other coefficient as well, i.e. 1,-1,-4,4,6,-6,-4,4,1, -1,

we get 

$$\sum_{k=0}^{2n+1} -1^{\lfloor k/2\rfloor+n-k}\left(\begin{array}{c}n\\\lfloor k/2\rfloor\end{array}\right)x^{n-k}y^{k}$$

This can be shown to be equal to $x(x^2-y^2)^n -y(x^2-y^2)^n = (x-y)(x^2-y^2)^n$ which is equal to $(x-y)^{n+1}(x+y)^n$.

We can repeat the coefficients more than 2 times as well and show that

 $$\sum_{k=0}^{m(n+1)-1} \left(\begin{array}{c}n\\\lfloor k/m\rfloor\end{array}\right)x^ky^{n-k}$$

is equal to $\sum_{t=0}^{t=m-1}x^ty^{m-1-t}(x^m+y^m)^n$.