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
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(2) |
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(3) |
The elliptic singular moduli is defined to be the solution of the equation
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(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]):
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(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:
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(6) |
In view of [7], [11], and [5], the formula for is
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(7) |
where is a root of the polynomial equation
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(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]:
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(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 ![](https://content.wolfram.com/sites/19/2013/08/Bagis_Math_27.gif)
We have (see [16]):
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(10) |
This is the Mathematica definition.
Define ,
, such that
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(11) |
It turns out that
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(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 ![](https://content.wolfram.com/sites/19/2013/08/Bagis_Math_47.gif)
The general formula produced by our method for is
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(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
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(15) |
We verify this numerically.
Example 2. Here is another example for that we verify numerically.
Series for ![](https://content.wolfram.com/sites/19/2013/08/Bagis_Math_65.gif)
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
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(16) |
This calculates the .
Example 3. For ,
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(17) |
Example 4. For , we have
and
; then
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(18) |
We verify this numerically.
Example 5. For , we have
and
; then
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(19) |
Evaluating the Elliptic Alpha Function ![](https://content.wolfram.com/sites/19/2013/08/Bagis_Math_83.gif)
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
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(20) |
From the duplication formula
and
equation (20) becomes
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(21) |
Setting
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(22) |
gives the following proposition.
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(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
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(24) |
Entry 4 of [2], p. 436 is
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(25) |
where and
.
Set
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(26) |
where is the Rogers-Ramanujan continued fraction (see [2], [21], [22]):
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(27) |
and
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(28) |
this gives
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(29) |
and hence the evaluation
![]() |
(30) |
But for the evaluation of the Rogers-Ramanujan continued fraction, from [22] we have
If and
is a positive real, then
![]() |
(31) |
with
![]() |
(32) |
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(33) |
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(34) |
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(35) |
In some cases, the next formula from [9] is very useful:
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(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].
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
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(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
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(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.
![]() |
(45) |
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(46) |
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(47) |
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(48) |
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(49) |
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(50) |
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(51) |
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(52) |
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(53) |
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(54) |
Example 6. If , from (54) we have
, hence
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(55) |
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(56) |
Hence
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(57) |
References
[1] | M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, New York: Dover, 1972. |
[2] | B. C. Berndt, Ramanujan’s Notebooks, Part III, New York: Springer-Verlag, 1991. |
[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/![]() |
[6] | N. D. Baruah and B. C. Berndt, “Eisenstein Series and Ramanujan-Type Series for ![]() 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 ![]() ![]() |
[9] | N. Bagis, “Evaluation of Fifth Degree Elliptic Singular Moduli.” arxiv.org/abs/1202.6246v1. |
[10] | E. W. Weisstein, “Elliptic Alpha Function” from Wolfram MathWorld—A Wolfram Web Resource. mathworld.wolfram.com/EllipticAlphaFunction.html. |
[11] | J. M. Borwein and P. B. Borwein, “A Cubic Counterpart of Jacobi’s Identity and the AGM,” Transactions of the American Mathematical Society, 323(2), 1991 pp. 691-701. www.ams.org/journals/tran/1991-323-02/S0002-9947-1991-1010408-0/S0002-9947-1991-1010408-0.pdf. |
[12] | N. D. Baruah and B. C. Berndt, “Ramanujan’s Series for ![]() |
[13] | N. D. Baruah, B. C. Berndt, and H. H. Chan, “Ramanujan’s Series for ![]() |
[14] | N. Bagis, “Ramanujan-Type ![]() |
[15] | S. Ramanujan, “Modular Equations and Approximations to ![]() |
[16] | W. Zudilin, “Ramanujan-Type Formulae for ![]() |
[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 ![]() |
[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 ![]() |
[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 ![]() |
About the Author
Nikos D. Bagis is a mathematician with a PhD in Mathematical Informatics from Aristotle University of Thessaloniki.
N. D. Bagis
Stenimahou 5 Edessa Pellas
58200 Greece
nikosbagis@hotmail.gr