[seqfan] Re: Extending signature sequences for transcendental numbers

franktaw at netscape.net franktaw at netscape.net
Mon Jun 21 18:36:04 CEST 2010

Interesting idea.

I've never seen anything on this, but I'm not in a position to do a 
literature search.

You can also look at S{n,k}: the ak in a0 + a1x + ... + an x^n. In 
general, it is interesting to look at S{n,k}(x) for 0 <= k <= n/2; 
after that, we have S{n,n-k}(x) = S{n,k}.(1/x).

Your T(x) sequence is going to be well-defined. It is only interesting 
if x > 1; for x < 1, it is all 1's. Similar to the above, you could 
call this T0(x), and extend to T1(x), T2(x), etc.

It is easier to see the meaning of your T(x) if you work with the 
decremented signature sequence: take a0 + a1x  for a0 and a1 
non-negative, instead of strictly positive. Sorting, the a0 sequence is 
the signature sequence for x with 1 subtracted from each term. This is 
exactly the same for the S{n,k} sequences. The similar T sequence is 
the a0's for the sorted a0 + a1x + a2x^2 + ... with all the ak's 
non-negative, and all but finitely many of them zero. Increment again 
to get your T sequence.

Franklin T. Adams-Watters

-----Original Message-----
From: Kerry Mitchell <lkmitch at gmail.com>

I've extended the idea of the signature sequence for transcendental
numbers.  To get the signature sequence of x, form y values:

y = a0 + a1x

for positive integers a0 and a1.  Sort the list by the value of y and 
sequence of a0's is the signature sequence.  If x is irrational, then 
sequence is a fractal sequence.

Transcendental numbers are not solutions to any polynomial equation with
integer coefficients.  I used that idea to create a series of sequences:

S1(x) is the sequence of a0's when sorting y = a0 + a1x.  (S1 is the
standard signature sequence.)
S2(x) is the sequence of a0's when sorting y = a0 + a1x + a2x^2.
S3(x) is the sequence of a0's when sorting y = a0 + a1x + a2x^2 + a3x^3.
Sn(x) is the sequence of a0's when sorting y = a0 + a1x + a2x^2 + ... +

Then, I defined T(x) to be the limit of Sn(x) as n goes to infinity.

For the few values I've investigated, it looks like T(x) is well 
defined and
is similar to, but different from, the regular signature sequence S1(x).

Is this a well known problem?  I didn't find any of my sequences in the
OEIS.  Can anyone provide some pointers?

Kerry Mitchell
lkmitch at gmail.com

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