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

N. D. Bagis

A General Method for Constructing Ramanujan-Type Formulas for Powers of 1/π
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This article discusses the theoretical background for generating Ramanujan-type formulas for and constructs series for and . 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:

(1)

In Mathematica, these are and .

We also have

(2)

and (see [5], [6]):

(3)

The elliptic singular moduli is defined to be the solution of the equation

(4)

In Mathematica, is computed using .

The complementary modulus is given by . (For evaluations of see [7], [8], [9]).

We need the following relation satisfied by the elliptic alpha function (see [7]):

(5)

Our method requires finding derivatives of powers of the elliptic integrals and that can always be expressed in terms of , , and . This article uses Mathematica to carry out these evaluations.

The function is not widely known (see [7, 10]). Like the singular moduli, the elliptic alpha function can be evaluated from modular equations. The case is given in [7] Chapter 5:

(6)

In view of [7], [11], and [5], the formula for is

(7)

where is a root of the polynomial equation

(8)

In the next section, we review and extend the method for constructing a series for based on . These Ramanujan-type formulas for , 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]:

(9)

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

The General Method and the Construction of Formulas for and

We have (see [16]):

(10)

This is the Mathematica definition.

Define , , such that

(11)

It turns out that

(12)

Here are the Mathematica definitions for for .

Consider the following equation for the function :

(13)

Set ; then and , for suitable values of , is a function of and , so is an algebraic number when . The and can be evaluated from (13). Higher values of and give more accurate and faster formulas for and .

Series for

The general formula produced by our method for is

(14)

This computes the polynomial in the variable in the sum (13).

To find the , the function Arules replaces by and by and sets all the Taylor expansion coefficients with respect to to 0.

Choose M large enough to get a solution for all the for . (Here and .)

Now that we have the A[i], this computes the sum on the left-hand side of (13).

This computes the right-hand side of (13).

Example 1. From [19] and [7], for and , we have and . Hence we get the formula

(15)

We verify this numerically.

Example 2. Here is another example for that we verify numerically.

Series for

The coefficients of and the parameters for the formula are obtained using the same method as for . (The same can be done as well for , of course.) Higher values of and give more accurate and faster formulas for and .

For we get

(16)

This calculates the .

Example 3. For ,

(17)

Example 4. For , we have and ; then

(18)

We verify this numerically.

Example 5. For , we have and ; then

(19)

Evaluating the Elliptic Alpha Function

It is clear from the results in the previous section that getting rapidly convergent series for and its even powers requires values of the alpha function for large , say (see [14], [20], [5]). In this section we address this problem.

From (4), (7), and [2] pages 121-122, Chapter 21, if we set , , , , then

(20)

From the duplication formula

and

equation (20) becomes

(21)

Setting

(22)

gives the following proposition.

Proposition 1

(23)

This connects Ramanujan’s results of Chapter 21 in [2] with the evaluation of the alpha function and the evaluations of . Solving (23) with respect to gives

Equations (21), (22), and (23) give another interesting formula,

where

(24)

Entry 4 of [2], p. 436 is

(25)

where and .

Set

(26)

where is the Rogers-Ramanujan continued fraction (see [2], [21], [22]):

(27)

and

(28)

this gives

(29)

and hence the evaluation

(30)

But for the evaluation of the Rogers-Ramanujan continued fraction, from [22] we have

Proposition 2

If and is a positive real, then

(31)

with

(32)

Proposition 3

(33)

Proof

From (23), (28), and (31),

(34)

with (see [22])

(35)

In some cases, the next formula from [9] is very useful:

(36)

Here the function is , where , , and are as defined in [9] and is the iterate of .

The coefficient was defined in (24) and occurred in (32); also satisfies the equation

If we know and , we can evaluate from (31) and then we can evaluate .

The following conjecture is most compactly expressed in terms of the quantity

(37)

The function is the -invariant (see [23], [8]). For more properties of and see [24].

Conjecture (Algorithm for

)

Numerical results calculated with Mathematica indicate that whenever , then .

For a given and , , or , if the smallest nested root of is , then we can evaluate the Rogers-Ramanujan continued fraction with integer parameters.

1. When ,

(38)

with , where , , are positive integers.

2. When ,

a) If , then

(39)

where

(40)

and where is the positive integer solution of . Hence and is a positive integer. The parameter is a positive rational and can be found directly from the numerical value of .

b) If , then

(41)

where we set . Then a starting point for the evaluation of the integers , is

(42)

the square of an integer.

3. When , then we can evaluate .

The degree of is 8 and the minimal polynomial of is of degree 4 or 8 and symmetric. Hence the minimal polynomial can be reduced to at most a fourth-degree polynomial and so it is solvable. With the help of step 2, we can evaluate .

(43)

where , , , , , are integers, and

(44)

Here are some values of that can found with the Mathematica built-in function Recognize or by solving Pell’s equation and applying the conjecture.

Example 6. If , from (54) we have , hence

(55)

(56)

Hence

(57)

References

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[3]	J. V. Armitage and W. F. Eberlein, Elliptic Functions, New York: New York: Cambridge University Press, 2006.
[4]	E. T. Whittaker and G. N. Watson, A Course of Modern Analysis, 4th ed., Cambridge: Cambridge University Press, 1927.
[5]	N. D. Bagis and M. L. Glasser, “Conjectures on the Evaluation of Alternative Modular Bases and Formulas Approximating 1/,” Journal of Number Theory, 132(10), 2012 pp. 2353-2370.
[6]	N. D. Baruah and B. C. Berndt, “Eisenstein Series and Ramanujan-Type Series for ,” Ramanujan Journal, 23(1-3), 2010 pp. 17-44. link.springer.com/article/10.1007/s11139-008-9155-8.
[7]	J. M. Borwein and P. B. Borwein, Pi and the AGM, New York: Wiley, 1987.
[8]	D. Broadhurst, “Solutions by Radicals at Singular Values from New Class Invariants for .” arxiv.org/abs/0807.2976.
[9]	N. Bagis, “Evaluation of Fifth Degree Elliptic Singular Moduli.” arxiv.org/abs/1202.6246v1.
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[13]	N. D. Baruah, B. C. Berndt, and H. H. Chan, “Ramanujan’s Series for : A Survey,” American Mathematical Monthly, August-September, 2009 pp. 567-587. www.math.uiuc.edu/~berndt/articles/monthly567-587.pdf.
[14]	N. Bagis, “Ramanujan-Type Approximation Formulas.” arxiv.org/abs/1111.3139v1.
[15]	S. Ramanujan, “Modular Equations and Approximations to ,” Quarterly Journal of Pure and Applied Mathematics, 45, 1914 pp. 350-372.
[16]	W. Zudilin, “Ramanujan-Type Formulae for : A Second Wind?” arxiv.org/abs/0712.1332.
[17]	E. W. Weisstein, “Pi Formulas,” from Wolfram MathWorld—A Wolfram Web Resource. mathworld.wolfram.com/PiFormulas.html.
[18]	The Mathematics Genealogy Project. “Jesús Guillera.” (Jul 17, 2013) genealogy.math.ndsu.nodak.edu/id.php?id=124102.
[19]	E. W. Weisstein, “Elliptic Lambda Function” from Wolfram MathWorld—A Wolfram Web Resource. mathworld.wolfram.com/EllipticLambdaFunction.html.
[20]	B. C. Berndt and H. H. Chan, “Eisenstein Series and Approximations to ,” Illinois Journal of Mathematics, 45, 2001 pp. 75-90. www.math.uiuc.edu/~berndt/publications.html.
[21]	B. C. Berndt, Ramanujan’s Notebooks, Part V, New York: Springer-Verlag, 1998.
[22]	N. D. Bagis, “Parametric Evaluations of the Rogers-Ramanujan Continued Fraction,” International Journal of Mathematics and Mathematical Sciences, #940839, 2011. doi:10.1155/2011/940839.
[23]	B. C. Berndt and H. H. Chan, “Ramanujan and the Modular -Invariant,” Canadian Mathematical Bulletin, 42(4), 1999 pp. 427-440. cms.math.ca/10.4153/CMB-1999-050-1.
[24]	N. Bagis, “On a General Polynomial Equation Solved by Elliptic Functions.” arxiv.org/abs/1111.6023v1.
N. D. Bagis, “A General Method for Constructing Ramanujan-Type Formulas for Powers of ,” The Mathematica Journal, 2013. dx.doi.org/doi:10.3888/tmj.15-8.