Special Identities of the Stakhov Hyperbolic Functions

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Abstract
This paper establishes fundamental algebraic identities for Stakhov hyperbolic functions, a recent generalization of hyperbolic functions based on recurrence sequences with functional parameters. We derive Vajda-type additivity relations, d’Ocagne’s formulas, Catalan and Cassini-type multiplicative laws, Gelin–Cesàro–type identities, and generating functions through Binet’s formulas, and introduce a novel platinum matrix framework. The matrix methodology thus generates further identities—including Honsberger-type decompositions and shift formulas—thereby unifying discrete recurrences with continuous symmetries. Special cases recover classical identities for hyperbolic Fibonacci functions, and new results emerge for Pell, Jacobsthal, and Fermat-type generalizations. The unified framework bridges recurrence sequences and hyperbolic function theory and demonstrates applications in differential geometry.