In this post I will mainly focus on proving the following nice result that a friend gave to me. Then I will list some other nice identities I have collected from books and the internet.

Let be n distinct numbers (they could even be complex-valued). Then

For example, for the case n = 3 we have

To prove (1), consider the following partial fraction expansion:

To find the , we multiply both sides of (2) by and then set to , for each . All terms on the right become zero except the one with :

Next, multiply both sides of (2) by to obtain

Taking the coefficient of of both sides of (4) (treated as a polynomial in z) gives

which after comparing (1) and (3) is what we wished to show. Note that taking coefficients of other powers of in both sides of (4) reveals more identities.

I found this result quite cute and it made me dig up other identities I have seen over the years. I will prove one more and the rest can be left as exercises for the interested reader.

If ,

An easy way of doing this eludes me. Adding the first and second terms inside the first pair of brackets gives

where in the last equality we used the fact that . Multiplying this by in the right bracket gives

Summing this cyclically over the other possible products gives

where the penultimate equality follows from one of the identities shown below.

Finally here is a list of interesting identities, with potential to grow. We assume they are defined for complex numbers for which any denominator shown is non-zero.

**Algebraic Identities**

- (elementary but very useful!)

- More generally,

- If is odd,

- Binomial theorem:

- In particular,

- Multinomial theorem:

where

(this implies if

- (follows from the previous result)

- (Lagrange’s identity)

Complex case: - Special case: (Brahmagupta-Fibonacci identity, or complex number multiplication)

- (proving that numbers of the form are closed under multiplication – see the Wikipedia entry on the Brahmagupta-Fibonacci identity)

- Euler’s four square identity (quaternion multiplication):

(used in Heron’s formula for the area of a triangle given side lengths )

- If ,

Here the or notation is used to denote a cyclic sum or product through the indices , so that the three permutations (x,y,z), (y,z,x) and (z,x,y) are used. It is also worth knowing that any symmetric polynomial can be written as a polynomial of elementary symmetric polynomials (which are sums of all possible products of a fixed number of the variables, for example , and ).

Yo CK man!

Comment by Radhika — November 23, 2010 @ 11:24 pm |

excellent master !!!

Comment by fernando — September 16, 2011 @ 2:42 pm |

fantastic altogether

Comment by babloo — November 24, 2012 @ 6:04 am |