Baran General B operator

Title: The Baran Generalized Infinite OperatorDescription: This note introduces a generalized infinite composition operator B, independently formulated by Abdullah Baran in June 2021. The operator is defined as a right-associative infinite nesting of a binary operation Δ with a sequence of context functions αₙ acting at each level. The classical operations of infinite summation (Σ), infinite products (Π), and continued fractions (K) are all special cases of B, obtained by choosing Δ and αₙ appropriately. When αₙ is the identity function I, the operator reduces to Σ or Π. When αₙ is a shift function, it produces continued fractions. Other choices of αₙ generate nested radicals, power towers, and entirely new infinite operations with no existing symbol. The note includes the finite and infinite forms of the operator, explicit examples, and a unified table showing how all classical operators emerge as children of B. Keywords: generalized operator, infinite composition, continued fraction, infinite series, infinite product, nested radicals, power tower, unification, operator notation. Abdullah Baran, Van-Çatak