P Function Algorithm

A page about the p func algorithm.

Some tuple, $S$, has $\ell$elements. The ith element is $S_{i}$.

Find a function $f(\lambda)$ that equals $S_{\lambda}$ when evaluated at some integer between $0 \geq \lambda \geq \ell$.

The bound, $a(\ell)$or simply a, on the number of terms for the P function is given by:

$$a(\ell) = \lceil \log_{2}(\log_{2}(\ell)+1)+1\rceil$$

The P function:

$$P(a, n) = \frac{1}{4}\prod_{q=1}^{2^{a-1}} \cos^{2}(\frac{n \pi}{2^{q}})$$

When the p function is evaluated it produces a single 1 followed by 2^(2^(a - 1) - 1) - 1 zero then the pattern repeats. This will be used to turn on-and-off terms in $S$ because the number of zeros will always be greater than the number of elements in $S$.

1-D Case

There exist a function $f(\lambda)$ that equals $S_{\lambda}$ when evaluated at some integer between $0 \geq \lambda > \ell$.

$$f(\lambda) = \frac{1}{4}\sum_{i=0}^{\ell} S_{i}\prod_{q=1}^{2^{a-1}} \cos^{2}(\frac{(\lambda - i) \pi}{2^{q}})$$

N-Dimensional Generalization

This can be generalized into n-dimensions by summing different variables for each dimension.

The number of dimensions is N. The tuple being represented is labeled $S_{i j ..N}$, where each index corresponds to a different dimension. The tuple $\gamma$is the representation of a point inside an N-dimensional cube and the value in dimension d is $\gamma_{d}$.

$$f(\lambda_{1},\lambda_{2}, .. ,\lambda_{N}) = \sum_{i=0}^{\ell}\sum_{j=0}^{\ell}..\sum_{N=0}^{\ell}\frac{S_{i j .. N}}{4^{N}} \prod_{d=1}^{N} \prod_{q=1}^{2^{a-1}} \cos^{2}(\frac{(\lambda_{d} - \gamma_{d}) \pi}{2^{q}})$$

The N-dimensional f function has N sums each with its own label.

Example

For example the tuple:

$$S = ( 0, 2, 5 )$$

$$f(\lambda) = 2\cos(-1/8\pi + 1/8\pi \lambda)^2\cos(-1/4\pi + 1/4\pi \lambda)^2\cos(-1/2\pi + 1/2\pi \lambda)^2 + 5\cos(-1/4\pi + 1/8\pi \lambda)^2\cos(-1/2\pi + 1/4\pi \lambda)^2\cos(-\pi + 1/2\pi \lambda)^2$$

$\lambda$	$f(\lambda)$	$S_{\lambda}$
0	0	0
1	2	2
2	5	5

Application

The indefinite integral of this form

$$\int f(x)^{n} dx = \begin{cases} g(x)+c_1, & \text{if $n<0$}.\\ h(x)+c_2, & \text{otherwise}. \end{cases}$$

Can be rewritten in the form

$$\int f(x)^n dx = \alpha(n) g(x) + \beta(n) h(x) + c$$

Where $\alpha(n)$ and $\beta(n)$ are given by the p function with S equal to the tuple of length 2n created by the function:

$$s(q, n) = q^{2n} - q^{n}$$

This function returns q-1 n times and zero n times when written in base q. The tuple is generated by

$$S_{i}(q,n) = \frac{s(q, n)}{q^{i}} \mod q$$

Set q equal to two and the tuple becomes an on-off switch for mathematical functions. The functions a(n) and b(n) are inverses of each other.

$$\alpha(n) = \sum_{i=0}^{n} S_{i}(2, n) P(a(n), n - i)$$

$$\beta(n) = \sum_{i=0}^{n} S' P(a(n), n-i)$$

The function $S'$is the reverse of $S$. The length is known so simply changing the index of the sequence from $S_{i}$to $S_{2n-i}$.