Transcendental Operational Mathematics: A Unified Iteration Theory of Gamma Function Family, Beta Function Family, Incomplete Beta Function Family and Their Inverse Operations

This paper systematically extends the framework of Transcendental Operational Mathematics to the gamma function Γ(z), the log-gamma function ln Γ(z), the digamma function ψ(z) = Γ′(z)/Γ(z), the polygamma functions ψ(m)(z), the beta function B(x, y)=R 10tx−1(1−t)y−1dt, the log-beta function ln B(x, y), the incomplete beta function Bx(a, b) =R x0ta−1(1 − t)b−1dt, the regularized incomplete beta function Ix(a, b) = Bx(a, b)/B(a, b), and their inverse operations (the inverse gamma function Γ−1(w), the inverse digamma function ψ−1(w), and the inverse regularized incomplete beta function I−1x(p; a, b)). The core idea is to extend the iteration count of the translation operators Tv(z) = z + v and Tv,w(x, y) = (x + v, y + w) from natural numbers to complex numbers, utilizing the functional equations of each function to establish a complete theory of fractional-order iterations. We propose an axiomatic system consisting of twelve independent axioms, prove their consistency and independence, construct fractional-order translation iterations using Schröder’s equation, Abel’s equation, and Kneser’s construction, analyze the singularity structure of complex-order iterations, establish a categorical duality between the additive group of numbers and the translation iteration groups, and prove transcendence theorems for fractionalorder values (conditional on Schanuel’s conjecture). This paper provides complete numerical algorithms and verification results. All open problems from the original research program have been either completely solved and deeply integrated into the corresponding sections or transformed into well-posed conjectures.