0027b: Part 6b, Complex series for pi

 
 

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 well-known that the j-function 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, Hd. 

 

Of course, the smallest degree such that Hd has complex roots is class number h(-d) = 3 with the smallest d as d = 23.  Let,

 

Au be any root of x3 - 1334x2 + 69069x - 1216171 = 0   (eq.1)

 

Bv be any root of y3 - 27y2 + 56y - 37 = 0                       (eq.2)

 

Cw be any root of z3 + 155z2 + 650z + 23375 = 0           (eq.3)

 

All three equations have only one real root and, following the root numbering system of Mathematica, we have C1 as the real root, and C2 and C3 as complex conjugates.  (Likewise for Au and Bv.)  Explicitly,

 

C13 = j((1+√-23)/2) ≈ -3.4932 x 106

 

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 H47, 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 Cw 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 hp 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” Ramanujan-Type 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(k2) as,
 
    
See next article for more details.
 
-- End -- 

 

 

© Tito Piezas III, Jan 2012

You can email author at tpiezas@gmail.com.

 

 
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