Operational Mathematics over Quaternions: A Complete Extension of Hyperoperations, Fractional Iteration, Calculus, and Duality to a Non-Commutative Skew Field

This work presents a complete, rigorous, and exhaustive extension of Operational Mathematics—originally developed over the complex numbers—to the quaternion skew field H. We extend the number of repetitions of basic mathematical operations (addition, multiplication, exponentiation, tetration, and higher hyperoperations) from natural numbers to quaternion-valued iteration counts and quaternion bases. We establish a full axiomatic system that respects the non-commutative multiplication of quaternions, distinguishing left and right operations where necessary. Using quaternionic Schr¨oder’s equation, Abel’s equation, quaternionic Kneser construction, and transfinite induction on quaternionic sheets, we prove the existence of integer-order, fractional-order, real-order, and quaternion-order iterations. Uniqueness is proved under regularity conditions such as quaternionic convexity and slice regularity. We analyze the singularity structure of quaternionic tetration, proving the existence of logarithmic branch points at all negative integers and a natural boundary of real codimension 1 in H. We further prove that quaternionic fractional calculus (Riemann–Liouville and Caputo) and quaternionic fractional calculus of variations are natural subcases of this theory. A categorical duality is established between the additive group of quaternions and the iteration shift group over H. Exponentially convergent numerical algorithms are developed and verified with high-precision computations. All open problems from the complex version are transformed into rigorously proven theorems or precisely formulated conjectures over H.