2025/08/16 by G, Nagananda K, Kim, Jong Sung
#11B39 #11P84 #Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT)
paper · doi:10.48550/arxiv.2508.11877
In this paper, we show that the classical Cassini and Catalan identities for Fibonacci numbers arise naturally from a single quadratic theta-function identity of Ramanujan. Expanding the identity ψ(q)ψ(q3)=ψ(q4)φ(q6)+q φ(q2)ψ(q12) via the Jacobi triple product and equating coefficients yields the unified q-determinant Fn+r(q)Fn-r(q)-Fn(q)2=(-q) n-rFr(q)2, n≥ r≥ 1, where ψ(q) and φ(q) are Ramanujan's theta functions with q a complex parameter in the unit disc (| q | < 1) and Fn(q) denotes the Carlitz q-Fibonacci polynomials. The radial limit q→1- recovers Cassini's formula (r=1) and Catalan's one-parameter extension, while the same derivation with an auxiliary weight produces new partition-refined versions. The argument uses only standard q-series algebra (triple-product expansions, q-Pochhammer cancellations, and coefficient extraction), providing a transparent modular explanation of the alternating sign (-1) n-r in Catalan's identity through the level-6 provenance of φ and ψ. Beyond unifying Cassini\textendash Catalan in a single framework, the method lifts seamlessly to higher-order recurrences, giving a template for Tribonacci-type determinants and suggesting congruence phenomena obtained from modular dissections and root-of-unity limits. The results place familiar Fibonacci determinants within Ramanujan's analytic landscape, indicate routes to combinatorial bijections that mirror the analytic cancellations, and connect with themes in modern q-series\textemdash ranging from colored partition identities to quantum-modular and exactly solvable models\textemdash thereby highlighting both the explanatory power and the ongoing relevance of Ramanujan's theta identities.