Thread Rating:
  • 0 Vote(s) - 0 Average
  • 1
  • 2
  • 3
  • 4
  • 5
The balanced hyperop sequence
#1
Till now we always discussed right-bracketed tetration, i.e. with the mother law:
a[n+1](b+1)=a[n](a[n+1]b)

here however I will introduce that balanced mother law:

a[n+1](2b) = (a[n+1]b) [n] (a[n+1]b)

a major difference to the right-bracketing hyperopsequence is that we can only derive values of the form 2^n for the right operand. Though this looks like a disadvantage, it has the major advantage being able to uniquely (or at least canonicly) being extended to the real/complex numbers.

First indeed we notice, that if we set a[1]b=a+b and if we set the starting condition a[n+1]1=a then the first three operations are indeed again addition multiplication and exponentiation:

by induction



But now the major advantage, the extension to the real numbers. We can easily see that

for

for example , , . There and so , and .

Now the good thing about each is that it has the fixed point 1 () and we can do regular iteration there. For k>2, it seems .

Back to the operation we have
or in other words we define

.
I didnt explicate it yet, but this yields quite sure and also on the positive reals.

I will see to provide some graphs of x[4]y in the future.
The increase rate of balanced tetration should be between the one left-bracketed tetration and right-bracketed/normal tetration.

I also didnt think about zeration in the context of the balanced mother law. We have (a+1)[0](a+1)=a+2 which changes to a[0]a=a+1 by substituting a+1=a. However this seems to contradict (a+2)[0](a+2)=a+4. So maybe there is no zeration here.
Reply


Messages In This Thread
The balanced hyperop sequence - by bo198214 - 04/14/2008, 08:44 AM
RE: The balanced hyperop sequence - by andydude - 04/18/2008, 05:23 PM
RE: The balanced hyperop sequence - by bo198214 - 04/18/2008, 05:58 PM
RE: The balanced hyperop sequence - by bo198214 - 04/18/2008, 06:20 PM
RE: The balanced hyperop sequence - by andydude - 04/20/2008, 02:28 AM
RE: The balanced hyperop sequence - by bo198214 - 04/26/2008, 07:20 PM
RE: The balanced hyperop sequence - by bo198214 - 11/30/2009, 11:37 PM

Possibly Related Threads...
Thread Author Replies Views Last Post
  applying continuum sum to interpolate any sequence. JmsNxn 1 2,722 08/18/2013, 08:55 PM
Last Post: tommy1729
  A random question for mathematicians regarding i and the Fibonacci sequence. robo37 0 2,211 08/07/2011, 11:17 PM
Last Post: robo37
  find an a_n sequence tommy1729 1 2,287 06/04/2011, 10:10 PM
Last Post: tommy1729
  Name for sequence y=0[0]0. y=1[1]1, y=2[2]2, y= n[n]n, y=x[x]x, y=i[i]i, y=z[z]z etc? Ivars 10 9,801 05/03/2008, 08:58 PM
Last Post: GFR



Users browsing this thread: 1 Guest(s)