Part 6b: Complex Series for 1/Pi using all Roots of the Hilbert Class Polynomial
by Tito Piezas III
I. Hilbert Class Polynomial II. Conjecture III. Other Series
I. Hilbert Class Polynomial
It is wellknown that the jfunction j(τ), for the form τ = (1+√d)/2 or τ = √d, is an algebraic number with degree k equal to the class number h(d). Its minimal polynomial is called the Hilbert class polynomial, H_{d}.
Of course, the smallest degree such that H_{d} has complex roots is class number h(d) = 3 with the smallest d as d = 23. Let,
A_{u} be any root of x^{3 } 1334x^{2 }+ 69069x^{ } 1216171 = 0 (eq.1)
B_{v} be any root of y^{3 } 27y^{2 }+ 56y  37 = 0 (eq.2)
C_{w} be any root of z^{3 }+ 155z^{2 }+ 650z + 23375 = 0 (eq.3)
All three equations have only one real root and, following the root numbering system of Mathematica, we have C_{1} as the real root, and C_{2} and C_{3} as complex conjugates. (Likewise for A_{u} and B_{v}.) Explicitly,
C_{1}^{3} = j((1+√23)/2) ≈ 3.4932 x 10^{6}
Then, we have,
where,
The first series uses all real {A, B, C}, but the last two use the complex roots of (eqns.1, 2, 3), where their subscripts follow the root numbering system of Mathematica. For d = 47 which has class number h(d) = 5, using all five roots of the Hilbert class polynomial H_{47}, it can be shown there are five similar equalities. And so on, apparently, for other d.
II. Conjecture
“Given {A, B, C} such that,
where {A, B, C} are algebraic numbers and P(x), P(y), P(z) their minimal polynomials. Let C_{w} be any complex root of P(z) = 0. Using the appropriate complex roots of the same minimal polynomials P(x), P(y) equated to zero, then,
where m is a rational number.”
As usual, the h_{p} are the factorial ratios, or Pochhammer symbol products (a)_{n},
Note: The series in the section below no longer use the Hilbert class polynomial.
III. Other Series
In “Divergent” RamanujanType Supercongruences, J. Guillera and W. Zudilin discovered the first series for 1/π with complex coefficients,
which involves Pochhammer symbols for p = 4. Since then, cases p = 2, 3 have been found by H.H. Chan, J. Wan, and W. Zudilin, in “Complex Series For 1/π”. However, there is a case p = 3 found by this author,
where i is the imaginary unit. Equivalently, if we wish to simplify the terms as well as express the complex denominator on the RHS as a cube,
where the conjugate i can also be used. It is the only known convergent series using only rational complex numbers discovered so far, and was found using the formulas for p = 3 in the previous article, Part 6: General forms for Ramanujan's Pi Formulas. The denominator on the RHS can also be used in a complex series for the complete elliptic integral K(k_{2}) as,
See next article for more details.
 End 
© Tito Piezas III, Jan 2012 You can email author at tpiezas@gmail.com.
