Eisenstein Series, Alternative Modular Bases and Approximations of $1/pi$

where (a) n = Γ(n+a) Γ(a) = a(a + 1)(a + 2) . . . (a + n -1), and |w| < 1.

2. The Legendre function (see [1])

3. Let also q = e -π √ r , where r positive real. The cubic theta functions are (see [6])

(2) c(q) := ∞ m,n=-∞ q (m+1/3) 2 +(m+1/3)(n+1/3)+(n+1/3) 2 (3) and the cubic singular modulus is given by

The function

is called cubic elliptic function and is similar in properties to those of the elliptic integral of the first kind K(w) (see [1], [4], [8], [9], [10]):

We set m r to be the solution of

and call the k r = (m r ) 1/2 singular moduli.

As with the elliptic integral K, using the cubic elliptic theory we give evaluations of the singular modulus α r for certain r (see [5], [6]). Also we derive a formula for finding

We also consider and treat with u(w) := P -1/6 (w) = 2 F 1 1 6 , 5 6 ; 1; w as in the cubic case. The β r are solutions of the equation

The cases 2 F 1 1 2 , 1 2 ; 1; w and 2 F 1 1 4 , 3 4 ; 1; w are trivial and easily can studied. Moreover the first is K itself. There are also other hypergeometric functions that can be evaluated with K. For example

Where E is the complete elliptic integral of the second kind.

In most of our results we use Mathematica and in some cases we have no direct proofs.

If we set q 1 = e 2πiτ , τ = √ -r then (see [5],[6]):

where

(10)

or from the Theory of Eisenstein Series holds

where ζ(s) is the Riemann’s Zeta function and B n are the Bernoulli numbers, (see [1], [9]). Hence

Hence

Now using the above relations we have

or Proposition 1.

In the appendix I have construct a table with the Mathematica of a r for r = 1, 2, . . . , 37, and r = 40, 44, 49, 59. We set z 3 := 2 F 1 (1/3, 2/3; 1; α 3r ) and we begin to search relations between F := 2 π K and z, and u.

It is known that (see [4] pg. 456), (here

Hence the number

according to Ramanujan is algebraic, for r positive rational. The function a(r) is called elliptic alpha function (see [8]). Hence because

Proof.

Use (20),( 22)

In [4] chapter 21 one can see a very large number of hidden identities very useful for the evaluation of a(n). More precisely using ( 24) and ( 25) we can evaluate

is the multiplier of the modular equation of n-th degree, in the base K. Examples are (see [4] pg. 460 Entry 3.):

Where m 3 is root of the polynomial

The system of equations:

have solutions (a 3r , z 3 ) which can expressed in radicals of t 1 and t 2 .

Proof.

Use Mathematica to solve the system with the command ‘Solve’. The solution is very complicated to present it here.

Theorem 3.

Note.

For how we arrive to this identity, observe that if q = e -π √ r , then 1 + 240

Example.

u(β 16 ) = √ 2 177 + 1242

Note.

1. Relations (a1),( 20),( 21),( 22),( 23) may lead us to the concluding result that exists a number ξ such that 1 -504

n 5 q 2n 1 -q 2n = (1 + ξβ r )u (β r ) 6 .

By setting values one can see that ξ = -2.

2. Using the triplication formula for α (see [6])

and in view of Theorem 4 we have if

Also in [6] one can find the 2-degree modular equation for α r which is

3 Ramanujan type 1/π series When q = e 2πiτ , the modular j-invariant is defined by

Also we define

For the a(r) function see relation (25).

Theorem 5.(see [7], [8]) 3) For r = 5 we have

132566687 Some Results. The duplication formula for k r is

8. For r = 108

9. For r = 288

10. For r = 1728 we set

11. From (31) we have

1/3 and A = 1 + P 3 8(-69 + P ) 3 1/3 then α 216 = P 3 8(69 -P ) 3 and

12. From the relations (k432), (a432), (α144) we get

If we set the values J 432 and T 432 in Theorem 5 we get a formula that gives 53 digits per term.

13. It is

Set

then from the triplication formula (31) we get

Setting the values of T 1728 and J 1728 in the relation (36) of Theorem 5 we get a formula which gives 110 digits per term. Using the 2,3,5,7 degree modular equations for the cubic base (see [6]) we can evaluate higher order values of α r .