Theory and Applications of the Conditional Type Hyperbolic Functions

In this paper, we introduce conditional and symmetric conditional hyperbolic sine and cosine functions by replacing a discrete variable with a continuous one and employing conditionally defined sequences. We establish fundamental relations between these functions and derive their generating and exponential generating functions, recurrence relations, sums and differences, double arguments, derivatives, De Moivre-type identities, and several key properties. Furthermore, we construct a quasi-sine conditional hyperbolic function by selecting an appropriate trigonometric function. Using these results, we develop three-dimensional conditional spiral and shofar structures. Finally, we present numerical and graphical illustrations to support the theoretical findings.