**Part 4: Watson’s Triple Integrals and Ramanujan-Type Pi Formulas**

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**By Tito Piezas III**

*Abstract*: Infinite series using the reciprocals of the *j-function* and other moonshine functions can be used to express the three *Watson’s triple integrals*.

I. Introduction

II. Moonshine Functions

III. McKay-Thompson series 1A

IV. McKay-Thompson series 2A

V. McKay-Thompson series 3A

VI. McKay-Thompson series 4A

VII. Conjecture

VIII. Watson and Ramanujan

**I. Introduction**

In 1939, G.N. Watson considered the following beautiful triple integrals,

Interestingly, these have a simple closed-form expression in terms of the *gamma function* Γ(n),

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But it turns out these integrals can also be expressed in terms of values that appear in *Ramanujan-type pi formulas* of form 1/pi. To illustrate, first define the factorial quotients,

h_{1} = (6n)! / ((3n)! n!^{3})

h_{2} = (4n)! / (n!^{4})

h_{3} = (2n)!(3n)! / (n!^{5})

h_{4} = (2n)!^{3} / (n!^{6})

or, equivalently, the *Pochhammer symbol* products,

h_{1} = 1728^{n} (1/2)_{n }(1/6)_{n }(5/6)_{n} / (n!^{3})

h_{2} = 256^{n} (1/2)_{n }(1/4)_{n }(3/4)_{n} / (n!^{3})

h_{3} = 108^{n} (1/2)_{n }(1/3)_{n }(2/3)_{n} / (n!^{3})

h_{4} = 64^{n} (1/2)_{n }(1/2)_{n }(1/2)_{n} / (n!^{3})

where (a)_{n} = (a)(a+1)(a+2)…(a+n-1). Then,

Compare to the Ramanujan-type pi formulas,

where one can see the summation for both sets have *common *denominators and *h*_{i}. These are not isolated results, and apparently there are an infinite number of such formulas for the Watson triple integrals. The common values are given by a few *moonshine functions*, namely the *McKay-Thompson series of class 1A, 2A, 3A, 4A* with an appropriately chosen constant term, and will be discussed in the next section.

**II. Moonshine Functions**

In “*The 163 Dimensions of the Moonshine Functions*”, Conway, Norton, and Atkin showed that these functions, rather curiously, *span a linear space of dimension 163*. However, for the purposes of this article, we need only four, namely,

r_{1A}(τ) = j(τ) = 1/q + 744 + 196884q + 21493760q^{2} + 864299970q^{3} + … (A007240)

r_{2A}(τ) = (f_{2 }+ 64)^{2} / f_{2 }= 1/q + 104 + 4372q + 96256q^{2} + 1240002q^{3} + … (A101558)

r_{3A}(τ) = (f_{3 }+ 27)^{2} / f_{3 }= 1/q + 42 + 783q + 8672q^{2} + 65367q^{3} + … (A030197)

r_{4A}(τ) = (f_{4 }+ 16)^{2} / f_{4 }= 1/q + 24 + 276q + 2048q^{2} + 11202q^{3} + … (A097340)

with,

*q* = e^{2πiτ} = exp(2πiτ)

While the first is essentially the *j-function*, given in *Mathematica* as *j*(τ) = *N[12*^{3}KleinInvariantJ[*τ], n]*, for arbitrary precision *n*, the other three can also be conveniently defined in terms of quotients *f*_{p} of the *Dedekind eta function*, *η*(τ), where,

f_{2} = (*η*(τ)/*η*(2τ))^{24}

f_{3} = (*η*(τ)/*η*(3τ))^{12}

f_{4} = (*η*(τ)/*η*(4τ))^{8}

also easily calculated in *Mathematica* as *η*(τ) *= N[DedekindEta[**τ], n]*. This then gives a convenient way to calculate the *r*_{pA}(τ). Ramanujan’s pi formulas are simply of the form,

where *C* = *r*_{pA}(τ) for p = {1,2,3,4} and, for appropriate τ, then *A,B,C* are algebraic numbers. We conjecture and give heuristic evidence that Watson’s triple integrals are,

which, for carefully chosen τ, then *w* is also an algebraic number.

**Part III. McKay-Thompson Series 1A**

In the *Introduction*, it was shown that the j-function *j(√-4) = 66*^{3} appears in the Watson integral *I*_{1}. However, for carefully chosen τ, it turns out one can use *j*(τ) for *all* three triple integrals. (A few similar formulas were found by C.H.Brown, but he didn’t connect them to the *I*_{i}.) Recall that,

r_{1A}(τ) = j(τ) = 1/q + 744 + 196884q + 21493760q^{2} + 864299970q^{3} + …

and,

*h*_{1} = (6n)! / ((3n)! n!^{3}) = 1728^{n} (1/2)_{n }(1/6)_{n }(5/6)_{n} / (n!^{3})

then,

These uses the four arguments τ = {√-4, √-3,^{ }(1+3√-3)/2, √-6}, respectively. (However, there are many more formulas using j(τ) and I merely chose the simplest ones.)

**Part IV. McKay-Thompson Series 2A**

Likewise, the moonshine function r_{2A}(τ), at certain τ, can be used for all three Watson integrals. Recall that,

r_{2A}(τ) = (f_{2 }+ 64)^{2} / f_{2 }= 1/q + 104 + 4372q + 96256q^{2} + 1240002q^{3} + …

where,

f_{2} = (*η*(τ)/*η*(2τ))^{24}

For *I*_{1}: τ = {*i*,^{ }(1+3*i*)/2,^{ }(1+5*i*)/2}

For *I*_{2}: τ = {√-3}

For *I*_{3}: τ = {(1/2)√-6}

Since *h*_{2} **= (4n)! / (n!**^{4}) = **256**^{n} (1/2)_{n }(1/4)_{n }(3/4)_{n} / (n!^{3}**)**, then,

**Part V. McKay-Thompson Series 3A**

Define,

r_{3A}(τ) = (f_{3 }+ 27)^{2} / f_{3 }= 1/q + 42 + 783q + 8672q^{2} + 65367q^{3} + …

where,

f_{3} = (*η*(τ)/*η*(3τ))^{12}

For *I*_{1}: τ = *i*

For *I*_{2}: τ = {(1/6)√-3,^{ }(3+3√-3)/6, (3+5√-3)/6, (3+7√-3)/6}

For *I*_{3}: τ = (1/3)√-6

Since *h*_{3} **= (2n)!(3n)! / (n!**^{5}) = **108**^{n} (1/2)_{n }(1/3)_{n }(2/3)_{n} / (n!^{3}**)**, then,

**Part VI. McKay-Thompson Series 4A**

Lastly,

r_{4A}(τ) = (f_{4 }+ 16)^{2} / f_{4 }= 1/q + 24 + 276q + 2048q^{2} + 11202q^{3} + …

where,

f_{4} = (*η*(τ)/*η*(4τ))^{8}

For *I*_{1}: τ = (1/2)(1+√-4)

For *I*_{2}: τ = (1/2)√-3

For *I*_{3}: τ = (1/2)√-6

Since *h*_{4} **= (2n)!**^{3} / (n!^{6}) = **64**^{n} (1/2)_{n }(1/2)_{n }(1/2)_{n} / (n!^{3}**)**, then,

**VII. Conjecture**

It should be pointed out that not just any τ will do in expressing the Watson triple integrals *I*_{i} similar to the forms above. For example, we have,

*r*_{4A}((1+√-2)/2) = -2^{6}

and using this and *h*_{4} in an infinite series, it was recognized by *Mathematica* that,

or, the neat equivalent form found by Ramanujan using *double factorials*,

However, the ratio of *x* with any of the three *I*_{i} needs more than algebraic numbers. Thus,

Hence,

*Conjecture*: “For *p* = {1,2,3,4}, let the moonshine function *r*_{pA}(τ) and Pochhammer products *h*_{p} be as defined in the previous sections. Let C = *r*_{pA}(τ). If τ is chosen as,

for some integer *m*_{k}, and if the infinite series below converges to a Watson triple integral,

*then w is an algebraic number*.”

**VIII. Watson and Ramanujan**

As an afterword, it can be mentioned that Watson was very familiar with Ramanujan’s work. G. N. Watson (1886-1965) co-wrote with Whittaker the classic, *A Course of Modern Analysis* (1915) which influenced a generation of Cambridge mathematicians, including Littlewood and Hardy. (Hardy, of course, was the one who discovered Ramanujan.)

After Ramanujan’s early death, his notebooks, including the so-called Lost Notebook, found its way to Watson who attempted to organize them. He would subsequently spend many years on Ramanujan’s formulas, especially on mock theta functions and others.

**-- End --**

In the next section, *Part 5: Complete Elliptic Integral of the First Kind K(k) and the Moonshine Functions*, it turns out the observations here can be generalized to cover the *K*(k) as well.

© Tito Piezas III, Aug 2011

You can email author at *tpiezas@gmail.com*.