𝒲 α,β,ν trigonometric and hyperbolic generalized functions

This paper presents a novel extension of trigonometric and hyperbolic functions based on the Wα,β,ν-function recently introduced in the literature [Droghei. Properties of the multi-index special function W(α¯,ν¯)(z). Fract Calc Appl Anal. 2023;26(5):2057–2068. doi: 10.1007/s13540-023-00197-6]. The classical factorial is generalized to offer greater flexibility for applications in advanced mathematical fields such as combinatorics, special functions, and complex analysis. Using the Wα,β,ν-exponential function, we derive new expressions for generalized trigonometric and hyperbolic functions, including sine, cosine, hyperbolic sine, and hyperbolic cosine. These functions provide exact solutions for non-trivial fractional differential equations with variable coefficients. We discuss some relevant examples of applications to solve linear and nonlinear equations by using operational methods and separation of variables.