0025: Part 4, Watson's Triple Integrals

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

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),

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,

h₁ = (6n)! / ((3n)! n!³)

h₂ = (4n)! / (n!⁴)

h₃ = (2n)!(3n)! / (n!⁵)

h₄ = (2n)!³ / (n!⁶)

or, equivalently, the Pochhammer symbol products,

h₁ = 1728ⁿ (1/2)_n(1/6)_n(5/6)_n / (n!³)

h₂ = 256ⁿ (1/2)_n(1/4)_n(3/4)_n / (n!³)

h₃ = 108ⁿ (1/2)_n(1/3)_n(2/3)_n / (n!³)

h₄ = 64ⁿ (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 + 21493760q² + 864299970q³ + … (A007240)

r_2A(τ) = (f₂+ 64)² / f₂= 1/q + 104 + 4372q + 96256q² + 1240002q³ + … (A101558)

r_3A(τ) = (f₃+ 27)² / f₃= 1/q + 42 + 783q + 8672q² + 65367q³ + … (A030197)

r_4A(τ) = (f₄+ 16)² / f₄= 1/q + 24 + 276q + 2048q² + 11202q³ + … (A097340)

with,

q = e^2πiτ = exp(2πiτ)

While the first is essentially the j-function, given in Mathematica as j(τ) = N[12³KleinInvariantJ[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,

f₂ = (η(τ)/η(2τ))²⁴

f₃ = (η(τ)/η(3τ))¹²

f₄ = (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) = 66³ appears in the Watson integral I₁. 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 + 21493760q² + 864299970q³ + …

and,

h₁ = (6n)! / ((3n)! n!³) = 1728ⁿ (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) = (f₂+ 64)² / f₂= 1/q + 104 + 4372q + 96256q² + 1240002q³ + …

where,

f₂ = (η(τ)/η(2τ))²⁴

For I₁: τ = {i,(1+3i)/2,(1+5i)/2}

For I₂: τ = {√-3}

For I₃: τ = {(1/2)√-6}

Since h₂ = (4n)! / (n!⁴) = 256ⁿ (1/2)_n(1/4)_n(3/4)_n / (n!³), then,

Part V. McKay-Thompson Series 3A

Define,

r_3A(τ) = (f₃+ 27)² / f₃= 1/q + 42 + 783q + 8672q² + 65367q³ + …

where,

f₃ = (η(τ)/η(3τ))¹²

For I₁: τ = i

For I₂: τ = {(1/6)√-3,(3+3√-3)/6, (3+5√-3)/6, (3+7√-3)/6}

For I₃: τ = (1/3)√-6

Since h₃ = (2n)!(3n)! / (n!⁵) = 108ⁿ (1/2)_n(1/3)_n(2/3)_n / (n!³), then,

Part VI. McKay-Thompson Series 4A

Lastly,

r_4A(τ) = (f₄+ 16)² / f₄= 1/q + 24 + 276q + 2048q² + 11202q³ + …

where,

f₄ = (η(τ)/η(4τ))⁸

For I₁: τ = (1/2)(1+√-4)

For I₂: τ = (1/2)√-3

For I₃: τ = (1/2)√-6

Since h₄ = (2n)!³ / (n!⁶) = 64ⁿ (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) = -2⁶

and using this and h₄ 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.

You can email author at tpiezas@gmail.com.

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