**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} ≈ 396^{4} – 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) = 2n**^{2}+29

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

*U*_{29} = (5+√29)/2

involved in the fundamental solution {x,y} = {5, 1} to the Pell equation x^{2}-29y^{2} = -4, then,

**2**^{6}(U_{29}^{6}+ U_{29}^{(-6)})^{2} = 396^{4}

**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 *E*_{6} 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) = -15**^{3}.

*Example 2*. Let τ = (√-*d*)/2; *d* = 28 = 4∙7, then** a(τ) = 1296***i*, b(τ) = 21546*i*, j(τ) = 255^{3}.

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 _{2}F_{1}(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 _{2}F_{1}(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 = 189***i*, B = 24*i*, C = *r*_{1}(τ) = -15^{3}.

*p* = 2: Let τ = (2+√-*d*)/4; *d* = 148 = 4∙37, then **A = 85840***i*, B = 4492*i*, C = *r*_{2}(τ) = -14112^{2}.

*p* = 3: Let τ = (3+√-*d*)/6; *d* = 267 = 3∙89, then **A = 28302***i*, B = 1654*i*, C = *r*_{3}(τ) = -300^{3}.

*p* = 4: Let τ = (4+√-*d*)/8; *d* = 64 = 4∙16, then **A = 48***i*, B = 8*i*, C = *r*_{4}(τ) = -2^{9}.

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 = ***r*_{1}(τ) = 255^{3}.

*p* = 2: Let τ = (1/4)√-*d*, *d* = 232 = 4∙58, then **A = 844480√2, B = 35296√2, C = ***r*_{2}(τ) = 396^{4}.

*p* = 3: Let τ = (1/6)√-*d*, *d* = 60 = 4∙15, then **A = 66√5, B = 8√5, C = ***r*_{3}(τ) = 15^{3}.

*p* = 4: Let τ = (1/8)√-*d*, *d* = 112 = 4∙28, then **A = 168, B = 20, C = ***r*_{4}(τ) = 2^{12}.

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})*φ*^{6}, B = 80(5^{1/4})*φ*^{4}, C = *r*_{4}(τ) = 2^{6}*φ*^{24}.

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*.