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+--- Thread: " x-theory " (/showthread.php?tid=1343)
" x-theory " - tommy1729 - 08/11/2021
"x-theory" is a preliminary name for some of my assorted ideas that do not belong anywhere else.
It is not standard calculus, geometry or even dynamics or tetration.
Basically it is subdivided in 2 main categories that are examplified by the following below :
consider 3 analytic functions f(z),g(z),h(z) and 9 distinct real numbers a1,a2,a3,.. such that :
f ' (z) = f(a1 z) + g(a2 z) + h(a3 z)
g ' (z) = f(a4 z) + g(a5 z) + h(a6 z)
h ' (z) = f(a7 z) + g(a8 z) + h(a9 z)
We have already considered the binary partition function before and its analytic asymptotic function F that satisfies F ' (z) = F(z/2).
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second example :
consider the sequence t(n) = 4*T(n-1) + 1 where T(n) is the n'th triangular number.
I like to define the sequence t(n) as above but it can also be defined as the centered square numbers.
(in elementary number theory) A centered square number is a centered figurate number that gives the number of dots in a square with a dot in the center and all other dots surrounding the center dot in successive square layers.
Or equivalently t(n) = n^2 + (n-1)^2.
The first few centered square numbers are:
1, 5, 13, 25, 41, 61, 85, 113, 145, 181, 221, 265, 313
Now consider the entire function f(z) defined by the taylor series :
f(z) = z + (z/5)^5 + (z/13)^13 + (z/25)^25 + (z/41)^41 + (z/61)^61 + ...
This functions has a special growth rate and a nice distribution of zero's.
Can you predict its growth rate or position of zero's before plotting or doing a lot of calculus ??
f(z) = sum (z/t(n)) ^ t(n).
I like to call this function f(z) names like Eisenstein-tommy function.
The closest to standard math is probably ' lacunary taylor series ' , ' lacunary polynomials ' , ' sparse polynomials ' and truncated taylor series.
And its connections to fake function theory might exist ...
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Although many tools probably exist to study these things , you do not see them during education or in books usually.
Correct me If I am wrong here, since I do not speak for all education and books around the world ofcourse.
I considered that these ideas and their variants have number-theoretic intepretations.
And ofcourse dynamics.
The 2 parts may be related.
And maybe gottfriends pxp function ideas are related as well.
***
I wonder what you guys think about it.
regards
tommy1729
RE: " x-theory " - tommy1729 - 08/12/2021
The first problem seems very matrix like ...
regards
tommy1729