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 α_n 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 α_n appropriately. When α_n is the identity function I, the operator reduces to Σ or Π. When α_n is a shift function, it produces continued fractions. Other choices of α_n 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