Converting a Product Function to a Summation

Date: 07/24/2004 at 03:25:12
From: Jay
Subject: convert product to summation

How do I convert the product of n terms to a summation?  For example,
if f(x) = a(x - a1)(x - a2)...(x - an), how do I get f(x) = the sum of
some series?

Date: 07/24/2004 at 10:43:06
From: Doctor Fenton
Subject: Re: convert product to summation

Hi Jay,

Thanks for writing to Dr. Math.  You would need to compute what are 
called the "elementary symmetric functions" of the roots a1,...,an.

If we define

   sigma_1(x(1),x(2),...,x(n)) = x(1) + x(2) + ... + x(n) 

   sigma_2(x(1),...,x(n)) = x(1)x(2) + x(1)x(3) + ... + x(n-1)x(n)

(you take all possible products of the form x(i)x(j), with i < j, and
add: for n=4, you would have x1x2 + x1x3 + x1x4 + x2x3 + x2x4 + x3x4,
for example)
 
   sigma_3(x(1),...,x(n)) = sum of all products x(i)x(j)x(k) , with
                            i < j < k ;

and so on, then

  (x - r(1))(x - r(2))...(x - r(n)) 

is equal to the sum

          n
         ---
         \
  x^n +  /   (-1)^i sigma_i(r(1),r(2),...,r(n))x^i   .
         ---
         i=1

For example, if your f(x) is a cubic, then

  (x - a1)(x - a2)(x - a3) = 

      x^3 - (a1 + a2 + a3)x^2 + (a1a2 + a1a3 + a2a3)x - a1a2a3 

If you have any questions or need more help, please write back and
show me what you have been able to do, and I will try to offer further
suggestions.

- Doctor Fenton, The Math Forum
  http://mathforum.org/dr.math/