Truncated Hypergeometric Representations of The Classical Apéry Numbers for ζ(2) and ζ(3)

This paper explores, via terminating hypergeometric sequences, the two classical sequences connected to Apéry's irrationality arguments for ζ(2) and ζ(3). Pochhammer symbol is the only symbol being allowed in the preliminary material; generalized hypergeometric notation, and finite termination are also restricted. The first part of the research is a direct comparison of the two Apéry sequences. It identifies the exact −𝑛, −𝑛, 𝑛 + 1 𝐵 = ₃F₂ ( ; 1) for the ζ(2) sequence and 𝐴 = ₄F₃ 𝑛 −𝑛, −𝑛, 𝑛 + 1, 𝑛 + 1 1,1 𝑛 ( 1,1,1 ; 1) for the ζ(3) sequence. The property to be discarded is given by the truncating value -n and each hypergeometric summand becomes the binomial product explicitly. The term-by-term identities are checked and the known three-term recurrences are tested for on the range 0 ≤ n ≤ 10 with exact computations. For the ζ(3) sequence, the computation also verifies Ap-1≡1 (mod p3) for p=5,7,11,13. The results provide a partial answer to the question of which Apéry sequence is generated by what hypergeometric function, and also allow a clear separation between the parametric truncation done in the case of n and the prime index truncation applied in the context of supercongruences.