This paper establishes a unified expression for the Lerch transcendent at odd positive integers s = 2m + 1, based on the Mellin-Barnes integral representation and the large-z asymptotic theory of Olde Daalhuis .We prove that these series converge super-exponentially and provide rigorous error bounds using refined Stirling inequalities. A detailed comparison with the Guillera-Sun series leads to rational approximations with exponent θm = 1. The theory is extended to the p-adic setting, with corrected convergence conditions p ∤ (2m + 2). The paper culminates in Theorem 5.6, which unifies five equivalent forms: (A) Lerch transcendent, (B) Bernoulli series, (C) Hauss conjugate Bernoulli, (D) generalized GuilleraSun hypergeometric, and (E) Olde Daalhuis asymptotics. This unification reveals deep connections between special functions, number theory, and operator theory. Using these representations, we prove irrationality results for odd zeta values, including a new proof of Apery’s theorem and partial results toward ´ ζ(5). In the p-adic setting, we establish p-adic irrationality criteria and apply them to obtain new results on ζp(2m +1). The paper concludes with open problems and conjectures, including the algebraic independence of odd zeta values and connections to the Schanuel conjecture.
shifa liu (Wed,) studied this question.