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Three cubic trinomial series for 1/π1/\pi

A mix of human-written and AI-generated textHuman understanding: all partsmath.NT — Number Theory

Contributed by Khai-Hoan Nguyen-Dang ↗

Version 1 / Oct 29, 2026 / CC BY 4.0

Abstract

We prove the three identities in Sun's Conjecture VI, involving cubes of generalized central trinomial coefficients. A common double integral represents their generating functions. On two rational curves, this period reduces to quadratic expressions in Gauss hypergeometric functions associated with elliptic integrals. A first-order identity recovers the derivative transverse to each curve and supplies the linear weights in the three series. The constants are determined by an endpoint calculation and Bauer's identity. Classical modular equations of degrees three, five, and seven then complete the evaluations. We give explicit rational differential certificates for the period reductions and the transverse derivative identities.

Provenance statement

OpenAI ChatGPT/Codex (GPT-6-astra) assisted with symbolic calculations, literature checks. The main ideas were from the author. ChatGPT Astra pro assistance also included the developing of transverse derivative method. The October 2026 revisions used GPT-6 Astra Pro. Responsibility for the mathematical claims, references, and final text rests with the author.

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Version history

  1. v1Submitted by Khai-Hoan Nguyen-DangInitial depositCurrentOct 29, 2026