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A General Method for Constructing RamanujanType Formulas for Powers of 1êp N. D. Bagis This article discusses the theoretical background for generating Ramanujan-type formulas for 1 ê p p and constructs series for 1 ë p4 and 1 ë p6 . We also study the elliptic alpha function, whose values are useful for such evaluations.

‡ Introduction The standard definitions of the complete elliptic integrals of the first and second kind (see [1], [2], [3], [4]) are respectively: pê2

KHxL = ‡

0

1 2

dt, 2

1 - x sinHtL pê2

EHxL = ‡

(1)

1 - x2 sinHtL2 dt.

0

In Mathematica, these are EllipticK@x^2D and EllipticE@x^2D. We also have KHxL =

p 2

2 2 F1 I1 ê 2, 1 ê 2, 1; x M =

p 2

¶

‚ n=0

H1 ê 2Ln H1 ê 2Ln x2 n H1Ln

n!

(2)

and (see [5], [6]): d KHkL EHkL KHkL ° K HxL = = . k dk kI1 - k2 M

(3)

The Mathematica Journal 15 © 2013 Wolfram Media, Inc.

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N. D. Bagis

The elliptic singular moduli kr is defined to be the solution of the equation 1 - w2

K

=

KHwL

(4)

r.

In Mathematica, kr is computed NomeQ@E^H-Pi Sqrt@rDLD.

w = k@rD^2 = InverseEllipticÖ

using

The complementary modulus is given by Hk 'r L2 = 1 - kr2 . (For evaluations of kr see [7], [8], [9]). We need the following relation satisfied by the elliptic alpha function (see [7]): aHrL =

p 4 K Hkr L2

-

r

EHkr L KHkr L

-1 .

(5)

Our method requires finding derivatives of powers of the elliptic integrals K and E that can always be expressed in terms of K, kr , and aHrL. This article uses Mathematica to carry out these evaluations. The function aHrL is not widely known (see [7, 10]). Like the singular moduli, the elliptic alpha function can be evaluated from modular equations. The case aH4 rL is given in [7] Chapter 5: aH4 rL = H1 + kr L2 aHrL - 2

(6)

r kr .

In view of [7], [11], and [5], the formula for aH9 rL is a H9 rL r

- k92 r

= 1-

k9 r kr 3 M3 r

-

k '9 r k 'r 3 M3 r

-

1 3 M3 r

-

1 3 M32 r

+

1

aHrL

M32 r

r

-

kr2 3

,

(7)

where M3 r is a root of the polynomial equation 27 M34 r - 18 M32 r - 8 I1 - 2 kr2 M M3 r - 1 = 0.

(8)

In the next section, we review and extend the method for constructing a series for p-p based on aHrL. These Ramanujan-type formulas for 1 ê p p , p ¥ 4 are presented here for the first time. The only formulas that were previously known are of orders 1, 2, and 3 ([12], [13]). There are few general formulas of order 2 and only one for order 3, due to B. Gourevitch (see references [14], [15], [5], [16], [17], [18]: ¶

‚ n=0

2n n

7

1 220 n

I1 + 14 n + 76 n2 + 168 n3 M =

32 p3

.

In the last section we prove a formula for the evaluation of aH25 rL in terms of aHrL.

The Mathematica Journal 15 © 2013 Wolfram Media, Inc.

(9)

A General Method for Constructing Ramanujan-Type Formulas for Powers of 1/p

3

‡ The General Method and the Construction of Formulas for 1 ë p4 and 1 ë p6 We have (see [16]): 4 K2 fHxL = 3 F2

1 1 1 , , ; 1, 1; x = 2 2 2

1 2

I1 -

1-x M

(10) .

p2

This is the Mathematica definition. 1

4 EllipticKB 2 J1 -

f@x_D :=

1 - x NF

2

p2

Define c p HnL, p = 2, 4, 6, …, such that ¶

f p HxL = ‚

¶

64n

n=0

p

3

2n n

xn

= ‚ c p HnL xn .

(11)

n=0

It turns out that c2 HnL =

1 26 n

n

‚ s=0

2s s

3

2n-2s n-s

3

,

n

c4 HnL = ‚ c2 HsL c2 Hn - sL,

(12)

s=0 n

c6 HnL = ‚ c4 HsL c2 Hn - sL, s=0

…. Here are the Mathematica definitions for c p HnL for p = 2, 4, 6. c2@n_D := 1 2 ^ H6 nL

n

‚ Binomial@2 s, sD ^ 3 Binomial@2 n - 2 s, n - sD ^ 3 s=0

n

c4@n_D := ‚ c2@n - sD c2@sD s=0

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N. D. Bagis

n

c6@n_D := ‚ c4@n - s3D c2@s3D s3=0

Consider the following equation for the function f p HxL: Ap xp

d p fHxL p dx

p

+ A p-1 x p-1

d p-1 fHxL p dx

p-1

HxL + ... + A1 x

¶ n

p

‚ c p HnL x IC p n + C p-1 n

p-1

+ ... + C1 n + C0 M =

n=0

d fHxL p dx g

pp

+ A0 fHxL p HxL = (13)

.

Set x = 1 - H1 - 2 wL2 ; then w = kr2 and g, for suitable values of A j , is a function of kr and aHrL, so g is an algebraic number when r œ N. The A j and C j can be evaluated from (13). Higher values of r œ N and kr give more accurate and faster formulas for 1 ë p4 and 1 ë p6 .

· Series for 1 ê p4 The general formula produced by our method for 1 ë p4 is ¶

‚ c4 HnL Hkr k 'r L2 n AA4 n4 + HA3 - 6 A4 L n3 + n=0

HA2 - 3 A3 + 11 A4 L n2 + HA1 - A2 + 2 A3 - 6 A4 L n + A0 E =

g p4

(14) .

This computes the polynomial in the variable n in the sum (13). npoly@p_, A_D := A@0D + Collect@Table@Product@n - k, 8k, 0, K