rsgerard Wrote:When I attempt to evaluate this recurrence, I see the result oscillate between 2 values when the principle root is used and -1 otherwise. So, I get the following solutions:

I believe these are the roots for

My two specific questions are:

1. Is this above correct?

2. Has something similar ever been shown in a more general case:

For example:

seems to be the roots for ???

First the roots of are

, .

So the roots of are at

, and :

If we look up the values , we indeed get that the roots of are and .

To tackle the general question we first have to know how a root at the complex domain is defined. More precisely which root is chosen from the possible roots of .

The general definition of the power in the complex domain is:

, where is the standard branch of the logarithm, i.e. is chosen such that the .

The oddity is that has a jump (not continuous) at the negative real axis. If we approach -1 from the upper plane we get and if we approach -1 from the lower plane we get . Both values are apart and if we repeat winding around 0 we get all the other branches of the logarithm .

So how applies this to the case of roots?

, where again is chosen. Taking the n-th root divides the argument/angle by n, in the way the angle is chosen it moves the point towards the positive real axis, like a scissor.

If we however consider (for simplicity for ) we first mirror at 0 and then divide the angle by towards the positive real axis. Mirroring at 0 means either to add for or to subtract for .

Say we start with a value in the upper halfplane, i.e. , then , is a point in the lower halfplane, is again in the upper halfplane and so on:

Generally

with and

with .

We dont know yet whether the sequences and have a limit, but if they have then the limit for must satisfy:

hence

, via :

Similarly the limit of would be

.

To be really complete one have to show that is strictly increasing for a starting value and striclty decreasing for a starting value , because then the existence of the limit is guarantied.

Ok, conclusion, if we compare above the roots of we see that the two limit points of are exactly the both roots of that are nearest the positive real axis ( and ). (That was also clear.)