This paper develops a complete and rigorous extension of Meta-Operational Mathematics to the theory of series, where a series is understood as an infinite iteration of additive or multiplicative meta-operations. The core philosophical principle—that operations upon operations constitute meta-operations—is systematically applied to infinite sums, infinite products, power series, Dirichlet series, Fourier series, Laurent series, and continued fractions. We establish a hierarchical framework (Level 0: base space elements; Level 1: operations; Level 2: meta-operations; Level 3: meta-meta-operations) and explicitly construct the space of series operations Series(F). A system of 17 axioms is formulated, and the category of series meta operations is shown to carry a Hopf algebra structure on its unary part, which is further endowed with a Hopf algebra morphism to the Connes–Kreimer renormalization Hopf algebra. Bornological convergence is introduced to handle infinite series operations, and applications to noncommutative geometry, cryptosystems, and the dynamics of the Riemann zeta function are presented. All classical special functions are shown to belong to the series-operational universe, and their fundamental identities become equations of meta-operations. Open problems resolved in this work include the rigidity of the series Hopf algebra, the complete classification of irreducible representations of the series Hopf algebra, a rigorous lower bound dimH J(ζ) ≥ 1 for the Julia set of the Riemann zeta function, and the existence of a repelling fixed point of ζ with multiplier ≤ 2. This work unifies analysis, algebra, geometry, topology, and quantum field theory under a single coherent language.
Liu S (2025) studied this question.