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𝒲α,β,ν trigonometric and hyperbolic generalized functions
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DOI：10.1080/10652469.2025.2577758.png)
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This paper presents a novel extension of trigonometric and hyperbolic functions based on the $ \mathcal {W}_{\alpha,\beta,\nu } $ W alpha,beta,nu-function recently introduced in the literature [Droghei. Properties of the multi-index special function $ \mathcal {W}<^>{( \bar \alpha, \bar \nu )}(z) $ W(alpha,nu)(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 $ \mathcal {W}_{\alpha,\beta,\nu } $ W alpha,beta,nu-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.
Keyword:
Generalized trigonometric functions
generalized hyperbolic functions
fractional hyper-Bessel operator
fractional differential equations
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1
论文数:
105
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0
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