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.
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.
- (elementary but very useful!)
- More generally,
- If is odd,
- Binomial theorem:
- In particular,
- Multinomial theorem:
(this implies if
- (follows from the previous result)
- (Lagrange’s identity)
- 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)
(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 ).