View Single Post
Old 2017-07-09, 13:18   #6
Dr Sardonicus
Dr Sardonicus's Avatar
Feb 2017

3·1,787 Posts

Originally Posted by Nick View Post
It's a nice tale!
Thanks for the kind words!
Perhaps it would be a good idea to consider the link to the fundamental theorem of symmetric polynomials.
Hmm. The expressions (**) in the OP clearly are symmetric in r and r', so by that theorem, are expressible as polynomials in the coefficients (which, for a monic polynomial F(z)) are, up to sign, the elementary symmetric polynomials in the roots of F(z) = 0. (If F(z) isn't monic, you have to divide by the lead coefficient to get the elementary symmetric polynomials.)

The expression for Ln obviously generalizes to polynomials of any degree; the sum of the nth powers of the roots of a monic polynomial in Z[x] forms a sequence of integers with interesting divisibility properties; perhaps the best known case with degree greater than 2 is Perrin's sequence for x3 - x - 1.

The sum of the nth powers of the roots is the subject of Newton's identities, which I invite the interested reader to look up.

Last fiddled with by Dr Sardonicus on 2017-07-09 at 13:18
Dr Sardonicus is online now   Reply With Quote