Operational Mathematics of Hypergeometric Functions: Extending the Iteration Count to the Complex Domain

This work systematically transplants the core methodology of Operational Mathematics---the extension of the repetition count of fundamental operations from natural numbers to integers, rational numbers, real numbers, and ultimately complex numbers---onto a new class of binary operations: the hypergeometric operation ₂F₁ (a, b;c;z) and its inverse ₂F₁^-1 (a, b;c;z). A complete set of seven independent axioms is established, integer-order, fractional-order, real-order, and complex-order iterations are rigorously defined, and the existence of iterative roots at each level is proved by means of Schröder's equation, Abel's equation, and a suitably adapted Kneser construction. Uniqueness theorems under natural regularity conditions are provided. The singularity structure of complex-order hypergeometric iterations is analyzed in depth, revealing a fundamentally novel phenomenon: the presence of algebraic branch points (square-root type) at the critical values of the hypergeometric function, and logarithmic branch points at the images of the singular point z=1 (when c-a-b), at the zeros (when a or b is a negative integer), and at the essential singularity at infinity. The negative real axis (-, -1] is shown to be a natural boundary. Conditionally on the integrality of c-a-b, a secondary natural boundary appears on the vertical line (w) =- (c-a-b) ; under the Hypergeometric Riemann Hypothesis (proved unconditionally for the corrected function), this becomes a natural boundary as well. A fundamental structural discovery is rigorously proved: the hypergeometric operational hierarchy collapses completely for all levels n 2, leaving only the base operations at level n=1 and the collapsed family at level n=2. Fractional calculus and the fractional calculus of variations with hypergeometric kernels are shown to be special cases of the hypergeometric operational framework, thereby unifying discrete hyperoperations with continuous analysis. A categorical duality between the mathematics of numbers and the mathematics of hypergeometric operations is established, yielding a field isomorphism between the hypergeometric hyperfield and the complex numbers. A functorial relationship between the hypergeometric and Gamma hyperfields, reflecting the Euler integral representation, is constructed. The connection between hypergeometric iteration values and transcendental number theory is explored, with unconditional proofs of transcendence and algebraic independence of the values Z₀, ₁, ₂ (r) for rational r. A corrected hypergeometric function is defined using backward iterates, and the Hypergeometric Riemann Hypothesis is proved unconditionally via a Hilbert-Pólya self-adjoint operator construction. The paper is self-contained, and every essential statement is accompanied by a detailed proof. In particular, the Hypergeometric Riemann Hypothesis --- that all non‑trivial zeros of the corrected hypergeometric function ₂F₁ (z;a, b, c) lie on the line (z) =1/2 --- is proved unconditionally via an explicit self‑adjoint Hilbert–Pólya operator.