Three cubic trinomial series for $1/\pi$
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…
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