0027: Part 6, Ramanujan's pi formulas and the hypergeometric function

Back to Index

Part 6: General Forms for Ramanujan’s Pi Formulas

by Tito Piezas III

Abstract: A simple overview of Ramanujan’s four types of formulas for 1/pi will be given in terms of general forms using only the hypergeometric function and Dedekind eta function.

I. Introduction

II. Eisenstein series and the Borweins’ general formula for p = 1

III. The hypergeometric function and general formulas for p = 1,2,3,4

IV. Examples

I. Introduction

In 1914, Ramanujan gave the unusual pi formula,

Note the not-so-coincidental,

e^π√58 ≈ 396⁴ – 104.0000001…

He gave many other examples of form,

where h_p is an expression involving factorials for four different p. However, it turns out there is a general formula for A and B in terms of C, hence Ramanujan’s formulas are dependent only on a single variable. For the example above, this variable is d = -4∙58 = 232, which has class number h(d) = 2, and is also the discriminant of the prime-generating polynomial,

P(n) = 2n²+29

which is prime for the consecutive values n = {0 to 28}. Furthermore, given the fundamental unit,

U₂₉ = (5+√29)/2

involved in the fundamental solution {x,y} = {5, 1} to the Pell equation x²-29y² = -4, then,

2⁶(U₂₉⁶+ U₂₉^(-6))² = 396⁴

II. Eisenstein series and the Borweins’ formula for p = 1

Before we go to the general formulas for all four types, in “Ramanujan-Type Series”, the Borwein brothers gave one for p = 1 as,

where,

In the link above, the Borweins defined q as the negative real number q = -exp(-π√t). However, one in fact can use the more general form q = exp(2πiτ) which, for appropriate τ, can yield both positive or negative real values. (A small typo involving the Eisenstein series E₆ has also been corrected by this author.)

Example 1. Let τ = (1+√-d)/2; d = 7, then a(τ) = 24, b(τ) = 189, j(τ) = -15³.

Example 2. Let τ = (√-d)/2; d = 28 = 4∙7, then a(τ) = 1296i, b(τ) = 21546i, j(τ) = 255³.

Thus,

(Note: The j-function j(τ) for the second example is positive and greater than 1728, hence the a(τ), b(τ) terms are complex. But since the LHS of the Borweins’ formula has √-j(τ), then the imaginary unit i will just cancel out.) The formula given in the introduction apparently does not have an equally simple expression in terms of Eisenstein series. But using the hypergeometric function, one can come up with analogous formulas for all four types.

III. The hypergeometric function and general formulas for p = 1,2,3,4

Ramanujan’s pi formulas can be given in the form,

where the h_p are any of the following factorial quotients, or its equivalent Pochhammer symbol products (a)_n,

In the formulas below, j(τ) is the j-function, η(τ) is the Dedekind eta function, and ₂F₁(a,b;c;z) for |z| < 1 is the hypergeometric function. They are easily calculated in Mathematica or in www.wolframalpha.com as j(τ) = 1728KleinInvariantJ[τ], η(τ) = DedekindEta[τ], and ₂F₁(a,b;c;z) = 2F1Hypergeometric[a,b,c,z]. Examples will then be given in the next section.

For p = 1:

For p = 2:

For p = 3:

For p = 4:

Thus, for a particular p, then A,B,C depend only on the r_p(τ) expressed in terms of the Dedekind eta function η(τ), which in turn is dependent on appropriately chosen d above a bound. (For example, one cannot use j((1+√-3)/2) as this is equal to zero.) The four r_p(τ) are just the first few of the “moonshine functions”, and an exceedingly interesting fact about them is that they span a linear space of dimension 163. (Of all numbers, it had to be that one!) See Part 1: The 163 Dimensions of the Moonshine Functions for more details.

IV. Examples

Using the formulas for {A,B,C} in the previous section, we find that,

Type I.

p = 1: Let τ = (1+√-d)/2; d = 7, then A = 189i, B = 24i, C = r₁(τ) = -15³.

p = 2: Let τ = (2+√-d)/4; d = 148 = 4∙37, then A = 85840i, B = 4492i, C = r₂(τ) = -14112².

p = 3: Let τ = (3+√-d)/6; d = 267 = 3∙89, then A = 28302i, B = 1654i, C = r₃(τ) = -300³.

p = 4: Let τ = (4+√-d)/8; d = 64 = 4∙16, then A = 48i, B = 8i, C = r₄(τ) = -2⁹.

Hence,

Of course, the imaginary unit i just factors out of the equations above. If we wish for r_p(τ) to be positive,

Type II.

p = 1: Let τ = (1/2)√-d, d = 28 = 4∙7, then A = 21546; B = 1296; C = r₁(τ) = 255³.

p = 2: Let τ = (1/4)√-d, d = 232 = 4∙58, then A = 844480√2, B = 35296√2, C = r₂(τ) = 396⁴.

p = 3: Let τ = (1/6)√-d, d = 60 = 4∙15, then A = 66√5, B = 8√5, C = r₃(τ) = 15³.

p = 4: Let τ = (1/8)√-d, d = 112 = 4∙28, then A = 168, B = 20, C = r₄(τ) = 2¹².

Hence,

The form of the argument τ for both types is easily seen to be connected to p, and there are other d such that r_p(τ) is an integer or, in general, an algebraic number. An interesting example involving the golden ratio φ = (1+√5)/2 is,

p = 4: Let τ = (1/8)√-d, d = 400, then A = 80∙6(5^1/4)φ⁶, B = 80(5^1/4)φ⁴, C = r₄(τ) = 2⁶φ²⁴.

So,

However, the second type of τ also has a role in what are called singular moduli to be discussed in the next article.

-- End --

© Tito Piezas III, Dec 2011

You can email author at tpiezas@gmail.com.

◄ Previous Page Next Page ►