Fastest-converging 3F2-based π series (Equation 1.1)

Determine whether the π series given by equation (1.1) is the fastest-converging π formula among all 3F2 hypergeometric series of the form c·π = ∑_{n=0}^{∞} Q(n) (a + b n) z^n, where c is algebraic, a, b, z are rational, and Q(n) is an array of Pochhammer symbols.

Background

The paper develops an algorithm based on the Beta integral and hypergeometric identities to accelerate convergence of series for mathematical constants. A central result is the π series in equation (1.1), termed the '324 formula,' which the author shows converges at approximately 2.5 digits per term and arises from a 3F2 hypergeometric structure.

Within this context, the author explicitly conjectures that equation (1.1) attains the fastest convergence among π formulas restricted to a specific class: sums whose terms are a linear polynomial in n multiplied by an array of Pochhammer symbols corresponding to a 3F2 hypergeometric series, with rational parameters and an algebraic constant factor. The conjecture delineates the scope (3F2 series with particular term structure) and asks for confirmation of optimality within that class.