A Unified Framework of Hypergeometric Rapidly Convergent Series for Mathematical Constants: From the Lerch Transcendent to Systematic Study

Based on the Mellin-Barnes integral representation of the Lerch transcendent and Olde Daalhuis's terminating asymptotics, this paper establishes a unified mathematical framework for deriving hypergeometric rapidly convergent series representations of mathematical constants. We systematically analyze the structure of twelve mathematical constants: , e, , G, (3), (5), (7), , 2, 2, the golden ratio, and the plastic constant, revealing that transcendental constants can all be expressed as values or derivatives of generalized hypergeometric functions at rational points. Six equivalent forms are established: the Lerch transcendent form, the Bernoulli series form, the conjugate Bernoulli form, the hypergeometric series form, the Olde Daalhuis asymptotic form, and the Banerjee-Wilkerson Lambert series form, with rigorous transformation proofs. A unified generating function (a, b;c;z) =₍=₀^ (a) ₙ (b) ₙ (c) ₙ (2n) ! (n!) ²ⁿ2^{2n} is constructed, from which all transcendental constants can be represented. The convergence rates of the six forms are systematically compared, computational complexities are analyzed, and optimal parameter selection criteria are given. Complete numerical verification code and computational results are provided. All results are presented with complete derivations and rigorous mathematical proofs.