Hypergeometric Meta-Operational Mathematics: A Rigorous Extension of Meta-Operational Mathematics to Generalized Hypergeometric Functions and Their Inverses

This work presents a completely rigorous and self-contained extension of the full apparatus of Meta-Operational Mathematics to the generalized hypergeometric functions pFq and their multi-valued compositional inverses. The central philosophical principle — that operations upon operations constitute meta-operations— is established with complete mathematical precision through a four-level hierarchical framework: Level 0 (elements of a base space), Level 1 (operations as mappings on the base space), Level 2 (meta-operations as mappings on operations),and Level 3 (meta-meta-operations acting on meta-operations). Within this framework, the generalized hypergeometric function pFq and its compositional inverse pF−1q are shown to admit canonical lifts to meta-operations via composition, and these meta-operations interact with one another through composition, pointwise addition, pointwise multiplication, differentiation, exponentiation, and logarithm in arbitrarily many iterations — integer, fractional, real, and complex.A fundamental distinction from the elliptic and zeta cases is established: the hypergeometric function is characterized by the ratio of Pochhammer symbols, the Euler integral representation, the Gauss hypergeometric differential equation (with three regular singular points 0,1,∞), and the Kummer 24 solutions. This leads to the Hypergeometric Duality Axiom (Axiom 2.25), in which the parameter transformation group (generated by Pfaff and Euler transformations) acts non-trivially,and the intertwining relation 2F1 = M(1−z)c−a−b ◦RH(2F1) replaces the elliptic quotient group C/Λ or the zeta reflection. The seven fundamental meta-operations generating the whole hypergeometric operad are composition, pointwise addition,pointwise multiplication, differentiation, the hypergeometric reflection operation, the identity operation, and the Gauss hypergeometric operation 2F1.The three essential features of the hypergeometric function — its Euler-type integral representation, its differential equation, and its role as a generating function for all classical special functions — are systematically elevated to the meta operational level as algebraic axioms, analytic tools, and geometric objects, constructing a self-contained Hypergeometric Meta-Operational Mathematics. All conjectures and open problems originally stated have either been resolved as theorems within the body of this paper or are precisely formulated as remaining open problems in Section 15. All theorems are proved in full detail, with each major proof containing at least eight explicit steps.