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## REPLACED

This blog has been replaced. A new J Function blog should be referred to instead. The blog can be found here.

## The (old) J Function

The J function is a reasonably fast growing function. This blog will explain the structure of the function and will give various calculated values. The J Function will then be used by a brand new version of the Alpha function. Click here for more information about the Alpha Function.

## Introduction

For an introduction, it will be useful to refer to my other blogs on the Strong D function. The following notation rules will also help to understand the behaviour of Strong D Functions and the basic structure of the J Function.

Notation:

$$D(m_{[x]}) = D(m_1,m_2,...,m_x)$$

$$D(m_{[x]},n_{[y]}) = D(m_1,m_2,...,m_x,n_1,n_2,...,n_y)$$

$$D(1,0_{[y]}) = D(D(1_{[y]})_{[y]})$$

$$D(m,n_{[y]}) = D(m-1,D(m,n_{[y-1]},n_y-1)_{[y]})$$

Basic J Function Structure:

The J Function consists of a series of 3-tuple parameters. This simple example uses one 3-tuple only:

$$J(k,m,n) = D(k,0_{[m]},n)$$

Note that the equivalent Strong D Function will have m+2 parameters in total, all parameters except for the first and last will be zero.

Basic J Function Examples:

$$J(2,0,3) = D(2,3)$$

$$J(3,1,1) = D(3,0,1)$$

$$J(4,2,0) = D(4,0,0,0)$$

$$J(n,n-1,n) = D(n,0_{[n-1]},n) >> f_{\omega^2}(n)$$

## Definition

The J Function consists of a series of 3-tuple parameters. Each 3-tuple is simply 3 grouped input parameters. 3-tuples are delimited from each other with a semi-colon, instead of a comma. Here is a general example:

$$J(1,2,3;4,5,6;7,8,9;...;...;...)$$

J Function behaviour follows similar rules to the Strong D Function. The rules are as follows:

$$J(k,m,n) = D(k,0_{[m]},n)$$

$$J(k,m,n) = J(k-1,m,J(k,m,n-1))$$

$$J(k,m,0) >> J(k-1,m,J(k-1,m,k))$$    Explanation will be provided

$$J(1,m,0) >> J(J(1,m-1,2^m-1),m-1,J(1,m-1,2^m-1))$$    Explanation will be provided

Note that:

$$J(0,m,n) = J(0,0,n) = J(n) = D(n)$$ Because of the Leading Zero rule: L1

and

$$J(0,0,0) = J(0) = D(0) = 1$$

When more than one 3-tuple is used we get more complex behaviour. The general rule is:

$$J(f,g,h;k,m,n) = J(f,g,h;k-1,m,J(f,g,h;k,m,n-1))$$

But it will be easier to follow the examples when f = g = 0 < h:

$$J(h;k,m,n) = J(h;k-1,m,J(h;k,m,n-1))$$

and then these rules follow from the definitions above.

$$J(h;k,m,0) >> J(h;k-1,m,J(h;k-1,m,k))$$

and

$$J(h;1,m,0) >> J(h;J(h;1,m-1,2^m-1),m-1,J(h;1,m-1,2^m-1))$$

Some simpler evaluations are:

$$J(h;0,0,n) = J(h;n) = J(h;J(h;n-1))$$

and

$$J(h;0,0,0) = J(h-1;J(h-1;h,h,h),J(h-1;h,h,h),J(h-1;h,h,h))$$

and

$$J(1;0,0,0) = J(0;J(0;1,1,1),J(0;1,1,1),J(0;1,1,1))$$

$$>> f_{\omega^2}(J(1,1,1)) = f_{\omega^2}(D(1,0,1)) >> f_{\omega^2}(f_3(6))$$

## Calculated Examples up to J(1;0,0,n)

$$J(1;0,0,1) = J(1;1) = J(1;J(1;0)) >> f_{\omega^2}(f_{\omega^2}(f_3(6))) >> f_{\omega^2}^2(f_3(6))$$

and

$$J(1;0,0,n-1) >> f_{\omega^2}^{n}(f_3(6)) >> f_{\omega^2+1}(n)$$ when $$n < f_3(6)$$

or

$$J(1;0,0,n) >> f_{\omega^2+1}(n)$$ when $$n < f_{\omega^2}(f_3(6))$$

## Calculated Examples up to J(1;n,0,n)

$$J(1;1,0,0) = J(1;0,0,J(1;0,0,1)) >> f_{\omega^2+1}(f_{\omega^2}^2(f_3(6)))$$

and

$$J(1;1,0,1) = J(1;0,0,J(1;1,0,0)) >> f_{\omega^2+1}^2(f_{\omega^2}^2(f_3(6)))$$

and

$$J(1;1,0,n-1) >> f_{\omega^2+1}^n(n) = f_{\omega^2+2}(n)$$ when $$n < f_{\omega^2}^2(f_3(6))$$

or

$$J(1;1,0,n) >> f_{\omega^2+2}(n)$$ when $$n < f_{\omega^2+1}(f_{\omega^2}^2(f_3(6)))$$

then

$$J(1;2,0,0) = J(1;1,0,J(1;1,0,2)) >> f_{\omega^2+2}(f_{\omega^2+1}^2(f_{\omega^2}^2(f_3(6))))$$

and

$$J(1;2,0,n) >> f_{\omega^2+3}(n)$$

and

$$J(1;3,0,n) >> f_{\omega^2+4}(n)$$

then

$$J(1;n,0,n) >> f_{\omega^2+\omega}(n)$$

## Calculated Examples up to J(1;n,n,n)

$$J(1;1,1,0) = J(1;J(1;1,0,1),0,J(1;1,0,1)) >> f_{\omega^2+\omega}(J(1;1,0,1))$$

and

$$J(1;1,1,1) = J(1;1,1,J(1;1,1,0)) >> f_{\omega^2+\omega}^2(J(1;1,0,1))$$

then

$$J(1;1,1,n) >> f_{\omega^2+\omega}^n(n) = f_{\omega^2+\omega+1}(n)$$

and

$$J(1;n,1,n) >> f_{\omega^2+\omega+n}(n)$$ = f_{\omega^2+\omega.2}(n)\)

then

$$J(1;1,2,n) >> f_{\omega^2+\omega.2}^n(n) = f_{\omega^2+\omega.2+1}(n)$$

and

$$J(1;n,2,n) >> f_{\omega^2+\omega.2+n}(n)$$ = f_{\omega^2+\omega.3}(n)\)

then

$$J(1;n,3,n) >> f_{\omega^2+\omega.4}(n)$$

and

$$J(1;n,n,n) >> f_{\omega^2+\omega.n}(n) >> f_{\omega^2.2}(n)$$

## Calculated Examples up to J(n;0,0,0)

$$J(2;0,0,0) = J(1;J(1;2,2,2),J(1;2,2,2),J(1;2,2,2)) >> f_{\omega^2.2}(J(1;2,2,2))$$

and

$$J(2;0,0,1) = J(2;0,0,J(2;0,0,0)) >> f_{\omega^2.2}^2(J(1;2,2,2))$$

then

$$J(2;0,0,n) >> f_{\omega^2.2+2}(n)$$

and

$$J(2;n,0,n) >> f_{\omega^2.2+\omega}(n)$$

and

$$J(2;n,n,n) >> f_{\omega^2.3}(n)$$

or

$$J(3;0,0,0) >> f_{\omega^2.3}(n)$$

then

$$J(n-1;n,n,n) >> f_{\omega^2.n}(n)$$

or

$$J(n;0,0,0) >> f_{\omega^3}(n)$$

## Calculated Examples up to J(1,0,n;n,n,n)

$$J(f,g,h;k,m,n) = J(f,g,h;k-1,m,J(f,g,h;k,m,n-1))$$

then

$$J(1,0,0;k,m,n) = J(1,0,0;J(1,0,0;k,m,n-1),J(1,0,0;k,m,n-1),J(1,0,0;k,m,n-1))$$

and

$$J(1,0,0;0,0,0) = J(0,0,1;Z,Z,Z) = J(1;Z,Z,Z)$$

where $$Z = J(0,0,1;D(1,0),D(1,0),D(1,0))$$ Explanation will be provided

or $$Z = J(1;D(1,0),D(1,0),D(1,0)) = J(1;3,3,3)$$

then

$$J(1,0,0;0,0,0) = J(1;J(1;3,3,3),J(1;3,3,3),J(1;3,3,3))$$

$$>> f_{\omega^2.2}(J(1;3,3,3))$$

and

$$J(1,0,0;0,0,1) >> f_{\omega^2.2}^2(J(1;3,3,3))$$

$$J(1,0,0;0,0,n) >> f_{\omega^2.2+1}(n)$$

$$J(1,0,0;n,0,n) >> f_{\omega^2+\omega}(n)$$

$$J(1,0,0;n,n,n) >> f_{\omega^3}(n)$$

then

$$J(1,0,1;0,0,0) >> J(1,0,0;J(1,0,0;4,4,4),J(1,0,0;4,4,4),J(1,0,0;4,4,4))$$ Explanation will be provided

$$>> f_{\omega^3}(f_{\omega^3}(4))$$

and

$$J(1,0,1;0,0,n) >> f_{\omega^3+1}(n)$$

$$J(1,0,1;n,0,n) >> f_{\omega^3+\omega}(n)$$

$$J(1,0,1;n,n,n) >> f_{\omega^3+\omega^2}(n)$$

then

$$J(1,0,2;0,0,n) >> f_{\omega^3+\omega^2+1}(n)$$

$$J(1,0,2;n,0,n) >> f_{\omega^3+\omega^2+\omega}(n)$$

$$J(1,0,2;n,n,n) >> f_{\omega^3+\omega^2.2}(n)$$

then

$$J(1,0,3;n,n,n) >> f_{\omega^3+\omega^2.3}(n)$$

and

$$J(1,0,n;n,n,n) >> f_{\omega^3.2}(n)$$

## Calculated Examples up to J(2,0,n;n,n,n)

Without proof, calculations should continue with:

$$J(2,0,0;0,0,n) >> f_{\omega^3.2+1}(n)$$

$$J(2,0,0;n,0,n) >> f_{\omega^3.2+\omega}(n)$$

$$J(2,0,0;n,n,n) >> f_{\omega^3.2+\omega^2}(n)$$

and

$$J(2,0,1;n,n,n) >> f_{\omega^3.2+\omega^2.2}(n)$$

$$J(2,0,2;n,n,n) >> f_{\omega^3.2+\omega^2.3}(n)$$

$$J(2,0,h-1;n,n,n) >> f_{\omega^3.2+\omega^2.h}(n)$$

and

$$J(2,0,n;n,n,n) >> f_{\omega^3.3}(n)$$

## Calculated Examples up to J(n,n,n;n,n,n)

Without proof, calculations should continue with:

$$J(3,0,n;n,n,n) >> f_{\omega^3.4}(n)$$

$$J(f,0,n;n,n,n) >> f_{\omega^3.f}(n)$$

and

$$J(n,0,n;n,n,n) >> f_{\omega^4}(n)$$

then

$$J(1,1,0;0,0,n) >> f_{\omega^4+1}(n)$$

$$J(1,1,0;n,0,n) >> f_{\omega^4+\omega}(n)$$

$$J(1,1,0;n,n,n) >> f_{\omega^4+\omega^2}(n)$$

$$J(1,1,n;n,n,n) >> f_{\omega^4+\omega^3}(n)$$

$$J(n,1,n;n,n,n) >> f_{\omega^4.2}(n)$$

$$J(n,g,n;n,n,n) >> f_{\omega^4.g}(n)$$

and

$$J(n,n,n;n,n,n) >> f_{\omega^5}(n)$$

## Growth Rate of the J Function

$$J(3;0,0,0) >> f_{\omega^3}(3) = f_{\omega^{\omega}}(3)$$

$$J(4,0,4;4,4,4) >> f_{\omega^4}(n) = f_{\omega^{\omega}}(4)$$

$$J(5,5,5;5,5,5) >> f_{\omega^5}(n) = f_{\omega^{\omega}}(5)$$

The growth rate of the J Function therefore appears to be:

$$J(n,n,n;n,n,n;...;n,n,n) >> f_{\omega^{\omega}}(n)$$ with n+1 parameters in the J function

Each group of 3 parameters of value n represents one 3-tuple in the J Function. If we use the notation N to represent the 3-tuple n,n,n, then the growth rate of the J function may reach:

$$J(N_{[(n+1)/3]}) >> f_{\omega^{\omega}}(n)$$ with n+1 parameters in the J function

$$J(N_{[n]}) = J(N;N;...;N) = J(n,n,n;n,n,n;...;n,n,n) >> f_{\omega^{\omega^2}}(n)$$

## Some calculations for n=3

$$J(4,0,1) >> f_{\omega}(3)$$

$$J(3,1,1) >> f_{\omega.2}(3)$$

$$J(3,2,1) >> f_{\omega^2}(3)$$

$$J(3;0,0,0) >> f_{\omega^{\omega}}(3)$$

$$J(3;3,3,3;3,3,3) >> f_{\omega^{\omega.2}}(3)$$

If we use the notation T to represent the 3-tuple 3,3,3, then we can continue:

$$J(3;T;T;T) = J(3;3,3,3;3,3,3;3,3,3) >> f_{\omega^{\omega^2}}(3)$$

Look forward to any comments and questions. If anybody is interested, the J Function was named by my wife. The full name is the Juki Function.

Cheers B1mb0w.

The J Function