The Unified Integral

Hyperlogarithms and beta integrals are two pillars of special function theory that developed along separate lines. Hyperlogarithms generalize the logarithm through iterated integration on the punctured projective line — they encode periods, polylogarithmic identities, and motivic structure. Beta integrals generalize the Euler beta function through products of linear forms raised to complex powers — they encode Selberg-type integrals, conformal field theory correlators, and hypergeometric identities. Both families live on the same geometric space (the universal abelian covering of the punctured projective line) but address different computational traditions.

Iterated beta integrals unify these two families by allowing the integrand at each step to be either a logarithmic differential (as in hyperlogarithms) or a beta-type kernel (as in classical beta integrals). The key insight: the iterated structure — integrating one form, then another, with the result of each step feeding into the next — is the shared scaffold. What changes between hyperlogarithms and beta integrals is the choice of integrand at each step. Allowing both choices in the same iterated chain produces a strictly larger class that contains both as special cases.

The new class preserves the fundamental properties of both parents. The shuffle product structure of hyperlogarithms extends. The evaluation identities of beta integrals extend. The regularization procedures that handle divergences at the punctures extend. These aren't separate proofs — the unified framework makes them instances of the same mechanism.

The through-claim: the division between hyperlogarithms and beta integrals was an artifact of choosing one integrand type per iteration, not a structural distinction. The geometry of the punctured projective line supports both simultaneously. The unification doesn't compromise either family's properties because those properties were always properties of the iterated structure, not of the specific integrands.