[Dr Sardonicus]

Originally Posted by Nick

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 L_n obviously generalizes to polynomials of any degree; the sum of the n^th 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 n^th powers of the roots is the subject of Newton's identities, which I invite the interested reader to look up.