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" 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).

***

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:

151325416185113145181221, 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 ...

***


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