Fibonacci Numbers: Mathematical Properties, Generalizations, and Applications

Abstract

The Fibonacci sequence, defined by the recurrence relation F_n = F_{n-1} + F_{n-2} with initial conditions F_0 = 0 and F_1 = 1, represents one of the most studied sequences in mathematics. This comprehensive survey examines the fundamental mathematical properties of Fibonacci numbers, including their closed-form solutions (Binet's formula), divisibility properties, and relationships with the Golden Ratio (φ ≈ 1.618). We explore extensive generalizations including generalized Fibonacci polynomials, k-generalized Fibonacci numbers (k-nacci sequences), convolved Fibonacci numbers, q-Fibonacci numbers, and hybrid Fibonacci systems. Recent research applications span number theory, combinatorics, algebra (including Fibonacci octonions and quaternions), statistics (statistical convergence), cryptography (stream ciphers), and natural phenomena (phylotaxis, golden spirals). This paper synthesizes 20+ years of research developments from 2002 to 2026, presenting both classical results and cutting-edge generalizations that extend the Fibonacci framework into new mathematical territories.

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math.NT

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