[Paper Review] Proof of some conjectured formulas for 1/pi by Z.-W. Sun

Gert Almkvist, Alexander Aycock+1 more|arXiv (Cornell University)|Dec 14, 2011

Advanced Mathematical Identities2 references3 citations

TL;DR

This paper proves several conjectured formulas for $1/\pi$ by Z.-W. Sun through a transformation method linking Ramanujan-type hypergeometric series to new identities. It establishes a general transformation formula that converts known $1/\pi$ series into new ones via modular invariants and hypergeometric functions, proving identities including one involving $163$ and the Ramanujan-Sato series.

ABSTRACT

Recently Z.W.Sun found over hundred conjectured formulas for 1/pi. Many of them were proved by H.H.Chan, J.Wan andW.Zudilin (see [3], [8] in the paper). Here we show that several other formulas in [6] are simple transformations of known formulas for 1/pi.

Motivation & Objective

• To prove several conjectured formulas for $1/\pi$ proposed by Z.-W. Sun, particularly those involving large algebraic numbers and class number one discriminants.
• To establish a general transformation framework connecting Ramanujan-like $1/\pi$ series to new identities via modular functions and hypergeometric functions.
• To verify that complex-looking formulas, such as the one with $262537412640769728$, are algebraic transformations of known series like Chudnovsky's.
• To provide a systematic method for generating new $1/\pi$ identities from known ones using the involution $x \mapsto -x/(1-Mx)$ and differential equations.
• To confirm supercongruences for certain divergent series, such as the one modulo $p^3$ for $s=1/3$, linking them to $1/\pi$ identities.

Proposed method

• Derives a general transformation: $\sum_{n=0}^\infty A_n x^n = \frac{1}{\sqrt{1-Mx}} \sum_{n=0}^\infty a_n \left(-\frac{x}{1-Mx}\right)^n$, where $A_n$ is a hypergeometric sum and $a_n$ is a known Ramanujan-type coefficient.
• Applies the transformation to known $1/\pi$ series of the form $\sum a_n (a + bn) x_0^n = \frac{1}{\pi}$, generating new identities with transformed parameters $w_0$, $A$, and $B$.
• Uses differential equations to verify equivalence: both sides of the transformation satisfy the same third-order ODE, and agreement of first four coefficients confirms identity.
• Employs modular functions and singular moduli: for $s=1/6, 1/3, 1/4$, the parameter $M$ corresponds to $1728, 108, 256$, linking to class number one $j$-invariants.
• Applies the involution property: the transformation is self-inverse, enabling bidirectional mapping between $a_n$ and $A_n$ series.
• Uses symbolic computation (Maple) to verify the transformation in specific cases like $s=1/6$, checking differential equations and initial terms.

Experimental results

Research questions

• RQ1 Can the monstrous formula involving $163$ and $262537412640769728$ be derived as a transformation of Chudnovsky’s $1/\pi$ series?
• RQ2 Is there a general transformation that maps Ramanujan-type $1/\pi$ series to new identities with different parameters and convergence behavior?
• RQ3 Do the conjectured formulas in Sun’s list, especially those with algebraic numbers and large denominators, arise from known $1/\pi$ series via modular invariance?
• RQ4 Can the transformation method be used to prove supercongruences, such as the $p^3$-congruence for $s=1/3$?
• RQ5 What is the role of singular moduli and hypergeometric functions in generating new $1/\pi$ identities from known ones?

Key findings

• The formula $\sum_{n=0}^\infty A_n P(n) \frac{1}{262537412640769728^n} = \frac{13803981511092062440689}{\pi \sqrt{163}}$ is a transformation of Chudnovsky’s formula with $M=1728$.
• For $s=1/3$, the transformation yields 8 new identities with $w_0 = 1/300, 1/1836, 1/8748, \dots$, all satisfying $\sum A_n (A+Bn) w_0^n = \frac{1}{\pi}$.
• For $s=1/4$, the transformation generates identities such as $\sum (1+8n)A_n \frac{1}{9^n} = \frac{9}{2\pi}$ and its dual $\sum (1-n)A_n \left(\frac{8}{9}\right)^n = -\frac{9}{\pi}$.
• For $s=1/6$, the identity with $\tau_0 = \sqrt{7}i$ leads to $\sum (A + Bn) A_n w_0^n = \frac{1}{\pi}$ and its dual $\sum (\hat{A} + \hat{B}n) A_n w_1^n = -\frac{7}{\pi}$, where $w_1 = \frac{1}{2} + \frac{171}{14450}\sqrt{1785}$.
• The method confirms a supercongruence: $\sum_{n=0}^{p-1} a_n (4+15n) \frac{1}{(-27)^n} \equiv 4p \binom{-3}{p} \mod p^3$, conjectured by Sun.
• The transformation is verified via differential equations: both sides satisfy the same ODE of order 3 with regular singularities at $x=0$, $x=1/1728$, and $x=\infty$.