### 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 hi. 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,   r1A(τ) = j(τ) = 1/q + 744 + 196884q + 21493760q2 + 864299970q3 + …       (A007240)   r2A(τ) = (f2 + 64)2 / f2 = 1/q + 104 + 4372q + 96256q2 + 1240002q3 + …       (A101558)   r3A(τ) = (f3 + 27)2 / f3 = 1/q + 42 + 783q + 8672q2 + 65367q3 + …                 (A030197)   r4A(τ) = (f4 + 16)2 / f4 = 1/q + 24 + 276q + 2048q2 + 11202q3 + …                 (A097340)   with,   q = e2πiτ = 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 fp 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 rpA(τ).  Ramanujan’s pi formulas are simply of the form,     where C = rpA(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 Ii.)  Recall that,   r1A(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 r2A(τ), at certain τ, can be used for all three Watson integrals.  Recall that,   r2A(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,   r3A(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,   r4A(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 Ii similar to the forms above.  For example, we have,   r4A((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 Ii needs more than algebraic numbers.  Thus,     Hence,   Conjecture:  “For p = {1,2,3,4}, let the moonshine function rpA(τ) and Pochhammer products hp be as defined in the previous sections.  Let C = rpA(τ).  If τ is chosen as,     for some integer mk, 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.     ◄
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