Transcendental Operational Mathematics on Hypergeometric Function Families A Unified Iteration Theory from Single-Variable Hypergeometric Functions to Multivariable Appell Functions, q-Hypergeometric Functions, and Elliptic Hypergeometric Functions

This paper systematically establishes a complete theoretical system of transcendental operational mathematics on hypergeometric function families. Starting from single-variable hypergeometric functions (Gauss hypergeometric function 2F1, confluent hypergeometric function 1F1, generalized hypergeometric function pFq and their inverses), it progressively generalizes to multivariable cases (Appell function family F1, F2, F3, F4, Lauricella function family F(r)D , F(r)A, F(r)B, F(r)C, Kampe de F´eriet functions), as well as q-analogues (basic hypergeometric functions rϕs, bivariate q-hypergeometric functions, q-Appell functions) and elliptic analogues (elliptic hypergeometric functions 2E1). The core idea is to extend the iteration count of the translation operator Tv(z) = z + v from natural numbers to complex numbers, utilizing the series representations, integral representations, differential equations, and recurrence relations of each function family to establish a unified theory of fractional-order iterations. This paper proposes an axiomatic system consisting of twenty independent axioms, proves their consistency and independence; constructs single-variable and multivariable fractional-order translation iterations, derives fractional-order series expansions, integral representations, transformation formulas, and recurrence relations for each function family; analyzes the singularity structure of complex-order iterations (regular singular points, branch points, natural boundaries, singular manifolds); establishes a categorical duality system from single-variable to multivariable, proving categorical equivalences between each iteration category and the number category; proves the transcendence of fractionalorder hypergeometric values under Schanuel’s conjecture, and gives unconditional transcendence results in special cases. This paper provides complete numerical algorithms and verification results, transforming all open problems into rigorously proven theorems or well-posed conjectures.