Experimental Mathematics

Extremal modular lattices, McKay Thompson series, quadratic iterations, and new series for $\pi$

Kok Seng Chua

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Abstract

We give 20 new Ramanujan-type formulae and 20 quadratic approximations to {\small $\pi$}, parameterized by extremal modular lattices of minimal square norm 2 and 4, at the ten special levels corresponding to square-free orders of the Mathieu group {\small $M_{23}$}. An algorithm for uncovering rational relations between two given power series that we used to discover some of the explicit relations is also given. Explicit relations parameterized by modular lattices between McKay Thompson series for the group {\small $\Gamma_0(\ell)^+$} and {\small $\Gamma_0(2\ell)+p_j$}, where {\small $p_j$} range over all odd primes dividing the special level {\small $\ell$}, are uncovered.

Article information

Source
Experiment. Math., Volume 14, Issue 3 (2005), 343-357.

Dates
First available in Project Euclid: 3 October 2005

Permanent link to this document
https://projecteuclid.org/euclid.em/1128371759

Mathematical Reviews number (MathSciNet)
MR2172712

Zentralblatt MATH identifier
1084.11016

Subjects
Primary: 11F03: Modular and automorphic functions 11H31: Lattice packing and covering [See also 05B40, 52C15, 52C17] 11Y60: Evaluation of constants

Keywords
Modular lattices quadratic iterations McKay Thompson series $M_{23}$ Hauptmodul $\pi$

Citation

Chua, Kok Seng. Extremal modular lattices, McKay Thompson series, quadratic iterations, and new series for $\pi$. Experiment. Math. 14 (2005), no. 3, 343--357. https://projecteuclid.org/euclid.em/1128371759


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