#### Matt Benesi

Gold Member

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- 7

ETA. Read the bottom post first. Well, and..

Obviously mathematicians know this identity.

At the x=b=c=n=2 point, pi exists. There are also connections to the Wallis product (pi/2).

Anyway, I simplified it to the n=2 case. And re-remembered my fascination with the Pidentity, where x=n=b=c=2. The leverage point in the whole function system. Where all primes are included in order, on the same side of the coefficient quotient (all on one side).

n=2 (simplified equation)

k= number of radicals

b= [itex]\sqrt{2x}[/itex]

c= x^2-x

[tex]\\

Q_{x,k} = b^k \, \times \, \sqrt{x - \sqrt {c+\sqrt {c +...}}} \\

\\[/tex]

k-->infinity, with x and n of 2:

b= [itex]\sqrt{4}[/itex] =2

c= 2^2 -2 =2

Re-iterated: I'd like the coefficients for series associated with constants generated by the following function at the limit k-->infinity.

At x=2,n=2, the point at which the following function generates pi, the series are sine and cosine series (I know these coefficients, a_m = m!/2).

First, here is the cosine series I refer to, for k radicals:

[tex] 2 \times [\, 1- \frac {\pi^2}{2! \, 4^{1k}} + \frac {\pi^4}{4! 4^{2k}} -\frac {\pi^6}{6! \, 4^{3k}}+ \frac {\pi^8}{8!\, 4^{4k}} \,\, ...] =

\sqrt{2 +\sqrt{2+...}}[/tex]

Here is the function to generate constants (such as pi) with inputs x, n, and k as the number of radicals, x and n are reals>1:

[tex]Q_{x,n,k} =\sqrt[n] { \left( nx^{n-1} \right)^k \, \times \left[x - \sqrt[n]{x^n-x + \sqrt[n]{x^n-x + \sqrt[n]{x^n-x+...}}} \right]}[/tex]

Above function explained:

k=2, or 2 radicals:

[tex]Q_{x,n,2} =\sqrt[n] { \left( nx^{n-1} \right)^2 \, \times \left[x - \sqrt[n]{x^n-x} \right]}[/tex]

k=3, or 3 radicals:

[tex]Q_{x,n,3} =\sqrt[n] { \left( nx^{n-1} \right)^3 \, \times \left[x - \sqrt[n]{x^n-x+ \sqrt[n]{x^n-x}} \right]}[/tex]

As k approaches infinity, you get some value determined by x and n.

[tex]\gamma= q_{x,n} = Q_{x,n,k \to \infty}[/tex]

One of the values is pi at n=2, x=2:

[tex]\gamma= \pi \, = \, q_{2,2} \, = \, Q_{2,2,k \to \infty} =\sqrt { 4^k \, \times \left[2 - \sqrt[2]{2^2-2 + \sqrt{4-2 + \sqrt{2+\sqrt{2+...}}}} \right]}[/tex]

All of the constants have associated functions that allow "fractional iteration" of radical addition.

They provide continuity between iterative addition of radicals:

[itex]\sqrt[n] {x^n-x} [/itex] continuous to

[itex]\sqrt[n] {x^n-x + \sqrt[n]{x^n-x}} [/itex] to

[itex]\sqrt[n] {x^n-x + \sqrt[n]{x^n-x+ \sqrt[n]{x^n-x}}} [/itex].

There are exact formulas for the coefficients of terms in the functions that allow smooth iteration in this case. In the x=2, n=2 case, the coefficients are directly related to sine/cosine coefficients.

[itex]a_m[/itex] coefficients of specific terms in the expansions

[itex]b= nx^{n-1}[/itex]

[itex]\gamma= q_{x,n}[/itex]

one form of "Cosine like" function (k radicals):

[tex]x- \frac {(Q_{x,n,k})^n}{b^k} \, = \,x- \frac {\gamma^2}{a_2 \, b^{1k}} + \frac {\gamma^4}{a_4 \, b^{2k}}-\frac {\gamma^6}{a_6 \, b^{3k}}+ \frac {\gamma^8}{a_8 \, b^{4k}} \,\, ... =

\sqrt[n]{x^n-x +\sqrt[n]{x^n-x+...}}[/tex]

The above reduces to 2*cos(pi/4^(k-1)) in the x=2, n=2 case.

a form if the "sine like" function with gamma/2 (used with coefficients in spoiler below):

[tex]\frac {(Q_{x,n,k})^n}{b^k} \, = \, \frac {(\frac{\gamma}{2})}{a_1 \, b^{1k}} - \frac {(\frac{\gamma}{2})^3}{a_3 \, b^{2k}} + \frac {(\frac{\gamma}{2})^5}{a_5 \, b^{3k}} -\frac {(\frac{\gamma}{2})^7}{a_7 \, b^{4k}}+ \frac {(\frac{\gamma}{2})^9}{a_9 \, b^{5k}} \,\, ...

= \, x- \sqrt[n]{x^n-x +\sqrt[n]{x^n-x+...}}[/tex]

Using the functions, one can increase k with non-integer values, and traverse smoothly between iterations of radicals- it allows one to smoothly connect addition of radicals in a naturally arising way: infinitely nested radicals smoothly connect all radical addition.

[tex]{Q_{2,2,3}}^2 = \,\,\ \frac{ pi^2}{ 4^{0}} - \frac{pi^4}{12 \times 4^{3}} +\frac{ pi^6}{360 \times 4^{6}} - \frac{ pi^8}{20160 \times 4^{9}}\,\,\, \dots = \,\,\, 4^3 \, \times \left[2 - \sqrt{2 + \sqrt{2}} \right] [/tex]

so

[tex]\frac{{Q_{2,2,3}}^2}{4^3} = \,\,\ \frac{ pi^2}{ 4^{3}} - \frac{pi^4}{12 \times 4^{6}} +\frac{ pi^6}{360 \times 4^{9}} - \frac{ pi^8}{20160 \times 4^{12}}\,\,\, \dots = \,\,\,2 - \sqrt{2 + \sqrt{2}} [/tex]

so

[tex]2 - \,\,\ \frac{ pi^2}{ 4^{3}} - \frac{pi^4}{12 \times 4^{6}} +\frac{ pi^6}{360 \times 4^{9}} - \frac{ pi^8}{20160 \times 4^{12}}\,\,\, \dots = \sqrt{2 + \sqrt{2}} [/tex]

In the x=1.5, n=2 case we have [itex]\gamma = 6.42675515[/itex] b=3, for k roots of .75:

[tex]x-

\frac{\gamma}{3^k} + \frac {\gamma^2}{6 \, \times \, 3^{2k} }

- \frac {\gamma^3}{72 \, \times \, 3^{3k}} + \frac {\gamma^4}{1404\, \times \, 3^{4k} }-

\frac {\gamma^4}{(673920/17 \, \times \, 3^{5k} } \\

+\frac {\gamma^6}{1510080 \, \times \, 3^{6k} }

-\frac {\gamma^7}{\frac{178092794880}{2407} \, \times \, 3^{7k} }

+\frac {\gamma^8}{\frac{18248946075360}{4051} \, \times \, 3^{8k} } \\

-\frac {\gamma^9}{ \frac{112670561875981824}{8887225} \, \times \, 3^{9k} } = \\

k roots

\\

\sqrt{.75+\sqrt{.75+...}}

[/tex]

These are at first glance, continuous functions for addition of radicals.

They are also periodic functions at n=2, x=2 for k<0.

I am interested in the semiperiodic properties of the other infinitely nested radical constants ([itex]\gamma = q_{x,n}[/itex]) and functions at x and n other than 2.

For which, I require a generating function for a_m-- or a function f(x,n,k) that multiplied by cosine( g(x,n,k) ) gives the values I need.

I'd prefer both (because I want to check the a_m expansions against what I am given).

Thanks for any

for n=2:

a_1 is 1

a_2 is (2x)(2x-1)/2

a_3 is (2x)(2x-1)

a_4 is [itex] \frac{\left( 2 x-1\right) \, \left( 2 x\right) \, \left( 2 x-1\right) \, \left( 2 x\right) \, \left( 2 x+1\right) }{2} [/itex]

General cases n=2, in "order":

x=2 : ... yup

x=2.5 : 1, 10, 20, 600, 1200, 62000/3, 148800, 58032000/131, 32240000...

x= 3: 1, 15, 30, 3150, 50400, 812700/11, 2889600....

Obviously mathematicians know this identity.

At the x=b=c=n=2 point, pi exists. There are also connections to the Wallis product (pi/2).

Anyway, I simplified it to the n=2 case. And re-remembered my fascination with the Pidentity, where x=n=b=c=2. The leverage point in the whole function system. Where all primes are included in order, on the same side of the coefficient quotient (all on one side).

k= number of radicals

b= [itex]\sqrt{2x}[/itex]

c= x^2-x

[tex]\\

Q_{x,k} = b^k \, \times \, \sqrt{x - \sqrt {c+\sqrt {c +...}}} \\

\\[/tex]

k-->infinity, with x and n of 2:

b= [itex]\sqrt{4}[/itex] =2

c= 2^2 -2 =2

Re-iterated: I'd like the coefficients for series associated with constants generated by the following function at the limit k-->infinity.

At x=2,n=2, the point at which the following function generates pi, the series are sine and cosine series (I know these coefficients, a_m = m!/2).

First, here is the cosine series I refer to, for k radicals:

[tex] 2 \times [\, 1- \frac {\pi^2}{2! \, 4^{1k}} + \frac {\pi^4}{4! 4^{2k}} -\frac {\pi^6}{6! \, 4^{3k}}+ \frac {\pi^8}{8!\, 4^{4k}} \,\, ...] =

\sqrt{2 +\sqrt{2+...}}[/tex]

Here is the function to generate constants (such as pi) with inputs x, n, and k as the number of radicals, x and n are reals>1:

[tex]Q_{x,n,k} =\sqrt[n] { \left( nx^{n-1} \right)^k \, \times \left[x - \sqrt[n]{x^n-x + \sqrt[n]{x^n-x + \sqrt[n]{x^n-x+...}}} \right]}[/tex]

Above function explained:

[tex]Q_{x,n,2} =\sqrt[n] { \left( nx^{n-1} \right)^2 \, \times \left[x - \sqrt[n]{x^n-x} \right]}[/tex]

k=3, or 3 radicals:

[tex]Q_{x,n,3} =\sqrt[n] { \left( nx^{n-1} \right)^3 \, \times \left[x - \sqrt[n]{x^n-x+ \sqrt[n]{x^n-x}} \right]}[/tex]

As k approaches infinity, you get some value determined by x and n.

[tex]\gamma= q_{x,n} = Q_{x,n,k \to \infty}[/tex]

One of the values is pi at n=2, x=2:

[tex]\gamma= \pi \, = \, q_{2,2} \, = \, Q_{2,2,k \to \infty} =\sqrt { 4^k \, \times \left[2 - \sqrt[2]{2^2-2 + \sqrt{4-2 + \sqrt{2+\sqrt{2+...}}}} \right]}[/tex]

All of the constants have associated functions that allow "fractional iteration" of radical addition.

They provide continuity between iterative addition of radicals:

[itex]\sqrt[n] {x^n-x} [/itex] continuous to

[itex]\sqrt[n] {x^n-x + \sqrt[n]{x^n-x}} [/itex] to

[itex]\sqrt[n] {x^n-x + \sqrt[n]{x^n-x+ \sqrt[n]{x^n-x}}} [/itex].

There are exact formulas for the coefficients of terms in the functions that allow smooth iteration in this case. In the x=2, n=2 case, the coefficients are directly related to sine/cosine coefficients.

[itex]a_m[/itex] coefficients of specific terms in the expansions

[itex]b= nx^{n-1}[/itex]

[itex]\gamma= q_{x,n}[/itex]

one form of "Cosine like" function (k radicals):

[tex]x- \frac {(Q_{x,n,k})^n}{b^k} \, = \,x- \frac {\gamma^2}{a_2 \, b^{1k}} + \frac {\gamma^4}{a_4 \, b^{2k}}-\frac {\gamma^6}{a_6 \, b^{3k}}+ \frac {\gamma^8}{a_8 \, b^{4k}} \,\, ... =

\sqrt[n]{x^n-x +\sqrt[n]{x^n-x+...}}[/tex]

The above reduces to 2*cos(pi/4^(k-1)) in the x=2, n=2 case.

a form if the "sine like" function with gamma/2 (used with coefficients in spoiler below):

[tex]\frac {(Q_{x,n,k})^n}{b^k} \, = \, \frac {(\frac{\gamma}{2})}{a_1 \, b^{1k}} - \frac {(\frac{\gamma}{2})^3}{a_3 \, b^{2k}} + \frac {(\frac{\gamma}{2})^5}{a_5 \, b^{3k}} -\frac {(\frac{\gamma}{2})^7}{a_7 \, b^{4k}}+ \frac {(\frac{\gamma}{2})^9}{a_9 \, b^{5k}} \,\, ...

= \, x- \sqrt[n]{x^n-x +\sqrt[n]{x^n-x+...}}[/tex]

Using the functions, one can increase k with non-integer values, and traverse smoothly between iterations of radicals- it allows one to smoothly connect addition of radicals in a naturally arising way: infinitely nested radicals smoothly connect all radical addition.

[tex]{Q_{2,2,3}}^2 = \,\,\ \frac{ pi^2}{ 4^{0}} - \frac{pi^4}{12 \times 4^{3}} +\frac{ pi^6}{360 \times 4^{6}} - \frac{ pi^8}{20160 \times 4^{9}}\,\,\, \dots = \,\,\, 4^3 \, \times \left[2 - \sqrt{2 + \sqrt{2}} \right] [/tex]

so

[tex]\frac{{Q_{2,2,3}}^2}{4^3} = \,\,\ \frac{ pi^2}{ 4^{3}} - \frac{pi^4}{12 \times 4^{6}} +\frac{ pi^6}{360 \times 4^{9}} - \frac{ pi^8}{20160 \times 4^{12}}\,\,\, \dots = \,\,\,2 - \sqrt{2 + \sqrt{2}} [/tex]

so

[tex]2 - \,\,\ \frac{ pi^2}{ 4^{3}} - \frac{pi^4}{12 \times 4^{6}} +\frac{ pi^6}{360 \times 4^{9}} - \frac{ pi^8}{20160 \times 4^{12}}\,\,\, \dots = \sqrt{2 + \sqrt{2}} [/tex]

In the x=1.5, n=2 case we have [itex]\gamma = 6.42675515[/itex] b=3, for k roots of .75:

\frac{\gamma}{3^k} + \frac {\gamma^2}{6 \, \times \, 3^{2k} }

- \frac {\gamma^3}{72 \, \times \, 3^{3k}} + \frac {\gamma^4}{1404\, \times \, 3^{4k} }-

\frac {\gamma^4}{(673920/17 \, \times \, 3^{5k} } \\

+\frac {\gamma^6}{1510080 \, \times \, 3^{6k} }

-\frac {\gamma^7}{\frac{178092794880}{2407} \, \times \, 3^{7k} }

+\frac {\gamma^8}{\frac{18248946075360}{4051} \, \times \, 3^{8k} } \\

-\frac {\gamma^9}{ \frac{112670561875981824}{8887225} \, \times \, 3^{9k} } = \\

k roots

\\

\sqrt{.75+\sqrt{.75+...}}

[/tex]

These are at first glance, continuous functions for addition of radicals.

They are also periodic functions at n=2, x=2 for k<0.

I am interested in the semiperiodic properties of the other infinitely nested radical constants ([itex]\gamma = q_{x,n}[/itex]) and functions at x and n other than 2.

For which, I require a generating function for a_m-- or a function f(x,n,k) that multiplied by cosine( g(x,n,k) ) gives the values I need.

I'd prefer both (because I want to check the a_m expansions against what I am given).

Thanks for any

*actual*help. Anyone who suggests groots instead of kroots will be grooted.a_1 is 1

a_2 is (2x)(2x-1)/2

a_3 is (2x)(2x-1)

a_4 is [itex] \frac{\left( 2 x-1\right) \, \left( 2 x\right) \, \left( 2 x-1\right) \, \left( 2 x\right) \, \left( 2 x+1\right) }{2} [/itex]

General cases n=2, in "order":

x=2 : ... yup

x=2.5 : 1, 10, 20, 600, 1200, 62000/3, 148800, 58032000/131, 32240000...

x= 3: 1, 15, 30, 3150, 50400, 812700/11, 2889600....

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