In this paper, we develop a higher-order theory for the numbers \( y_{9,n}^{(\alpha)}(\lambda;a) \) and the associated polynomials \( y_{9,n}^{(\alpha)}(x,\lambda;a) \), extending several identities and interpolation formulas previously obtained for the case \( \alpha=1 \). Using the corresponding generating functions, we derive explicit representations, recurrence relations, and structural identities, and we establish connections with a broad range of classical and modern special numbers, including the Apostol--Bernoulli, Apostol--Euler, Euler--Frobenius, Fubini, and Stirling numbers. We further construct an interpolation function for the higher-order numbers and prove that its special values at negative integers reproduce these numbers up to an explicit normalization factor. A residue-class decomposition of this interpolation function is then obtained, leading naturally to a generalized hypergeometric Hurwitz--Lerch-type zeta function. Finally, we study several special cases and analytic properties of this zeta-type function, including its reduction to the classical Lerch transcendent and a differential identity.