Mathematics regarding infinite series, hypergeometric function, gamma function, pi formula, universal constants by Cetin Hakimoglu-Brown
New misc sums:
using:
and
obtain
or you can taking multiple derivatives of the double sum with respect to z to get the same expression. Or take the first term of the double sum and take derivatives with respect to c and let z be zero.
GENERAL CLOSED FORM SUMS
Interestingly, a detailed proof shows the 2k+1 value in the numerator
is the only value permissible for the series to have a closed form sum.
note: for reasons that are not so obvious w must be positive and |z^2x/w^2| less than one.
This expression can be used to generate infinite series for certain angles that cannot otherwise be expressed in terms of radicals
Such as:
Squaring the result seems deceptivly easy, but requires tedious manipulation of the hypergeometric function
BBP Formulua arctan(1/3) and arctan (1/7)
Using the later identity and this arctangent relation obtain
The proof begins by finding the arctangent and Ln components for the expression below for g=(1,2,3,4,5,6,7)and solving various linear equations
Other stuff:
Gives pi to 25 places using a 6 degree hypergeometric transformation imposed on a quadratic transformation for a singular elliptic value:
Gives pi to 18 places using a quadratic and quartic transform on an existing cubic identity to make a much faster infinite series, with the first term giving:
Gives 62 digits of pi using a cubic modular identity and applying quadratic, cubic and quartic transformation:
The 24th degree hypergeometric transformation:
This was derived by imposing a quadratically transformed hypergeometric function onto two other quradratic transformations and then imposing that result on a cubic transformation.
Hence, 2x2x2x3 = 24
This cubic identity from mathworld was used :
When you let a=7/16 you can see how it begins to come together.
Funny quote:
"no practical system for calculating with numbers is able to express pi exactly" -wikipedia
A Rapidly converging formula for the circumference of an elipse that gives log (1024/27((a+b)/(a-b))^8)
digits per term :
Then Define:
It is derived using 12th degree hypergeometric transfomation. You begin with a first degree hypergeometric function for the elliptic integral of the second kind Then you perform two quadratic & a cubic transform on it to make it converge faster while the values V1, V2, Q, and r are still rational. others:
The last formula is equal to
pending work:
1. elegant zeta-like infinite series
2. cotangent integrals to approximate odd integer values of the zeta function
4. binomial sums & rapidly converging infinite series (see paper)
5. 2f1 hypergeometric transformations & formulas for values of gamma function
6. 24th degree 2f1 hypergoemtric transformation
7. primer on deriving Ramanujan type pi formulas
8. approximating complete elliptic integrals of first and second kind though hypergeometric transformations and subsequent pi approximations (quadratic and quartic modular identities)
9. cubic modular equations & accelerating a 3F2 hypergeometric function to derive extremely efficient pi approximations
10. stock market energy level equations, volume, and 'v' shaped recoveries
11. rapidly converging infinite series to calculate perimeter of an ellipse
obtain
