Generalized Hyperbolic Functions

• Generalized hyperbolic functions are defined by nonlinear integrals that extend classical sinh and cosh using parameterized frameworks.
• They yield eigenfunctions and sharp inequalities for p-Laplacian equations, enriching spectral analysis and nonlinear differential theory.
• Operational generalizations via umbral calculus enable applications in quantum mechanics, geometry, and advanced special function theory.

Generalized hyperbolic functions constitute a family of extensions of the classical hyperbolic sine and cosine, unifying and generalizing their analytic, algebraic, and geometric properties via parameterized nonlinear differential, integral, and operational frameworks. Arising naturally in the context of $p$- and $(p, q)$-Laplacian equations, umbral analysis, and spectral problem representations, these functions interpolate between classical analytic functions, Bessel and Laguerre systems, and hypergeometric or even supersymmetric frameworks. Their study encompasses explicit construction, parameter monotonicity and convexity theory, functional inequalities, duality with generalized trigonometric counterparts, multidimensional variants, and applications in analysis, differential equations, and mathematical physics.

1. Foundational Definitions and Classes

The prototypical $p$-generalized hyperbolic functions are defined via the inverse of the nonlinear integral

$\arcsinh_p(x) = \int_0^x (1 + t^p)^{-1/p} dt \quad (x \geq 0,\; p > 1),$

with $\sinh_p$ as its inverse. The associated $p$-hyperbolic cosine is given by

$$\cosh_p(x) = \frac{d}{dx} \sinh_p(x) = (1 + [\sinh_p(x)]^p)^{1/p}.$$

These satisfy the $p$-hyperbolic identity

$$\cosh_p^p(x) - \sinh_p^p(x) = 1 \quad \forall x \in \mathbb{R},$$

and the system of differential equations

$$\frac{d}{dx} \sinh_p(x) = \cosh_p(x), \quad \frac{d}{dx} \cosh_p(x) = \sinh_p(x)^{p-1} \cosh_p(x)^{2-p}.$$

Two-parameter generalizations, denoted $(p, q)$0-hyperbolic functions, are constructed via

$(p, q)$1

An analogous $(p, q)$2-trigonometric system exists via integrals of $(p, q)$3, related to the hyperbolic system by analytic continuation and duality transformations (Baricz et al., 2013, Miyakawa et al., 2020, Miyakawa et al., 2022, Karp et al., 2024).

Significantly, operational generalizations based on umbral calculus yield families such as the Laguerre-hyperbolic and Bessel-Tricomi hyperbolic functions, constructed from generalized exponentials: $(p, q)$4 and higher-order/parameter variants thereof (Dattoli et al., 2017).

Additionally, the $(p, q)$5-hyperbolic functions, tailored for variable-coefficient or supersymmetric systems, are built from iterative generalized powers and their conjugates, leading to series and operational representations adapted to specific weight or ground-state functions (Ouellet et al., 2019).

2. Analytic Structure: Series, Integral Forms, and Eigenfunctions

Generalized hyperbolic functions admit convergent power series and hypergeometric representations. For $(p, q)$6-hyperbolic sine: $(p, q)$7 with $(p, q)$8 defined as its inverse. The Borel-type integral inverses connect the operational families to classical hyperbolic functions: $(p, q)$9 Weighted Mellin-Barnes or Dirichlet integrals and incomplete Beta-function representations further connect these objects to special function theory (Dattoli et al., 2017, Baricz et al., 2013, Jiang et al., 2013, Karp et al., 2024).

The functions $p$0 and $p$1 solve the nonlinear ODE

$p$2

and are eigenfunctions of the associated $p$3-Laplacian operators. The Laguerre and other operational generalizations are eigenfunctions of higher-order, deformed differential operators (e.g., generalized Laguerre derivatives), yielding complete orthogonal systems under appropriate weights (Dattoli et al., 2017, Klén et al., 2012).

3. Algebraic and Functional Relations

Generalized hyperbolic functions possess group-like addition, duplication, and De Moivre–type formulas, governed by nonstandard composition laws (e.g., Laguerre-sum, power/mean-based sums, or operational sum mappings). For $p$4-functions:

• Addition formulas inherit structure from their trigonometric analogues and are governed by duality relations (Miyakawa et al., 2020, Miyakawa et al., 2022).
• For Laguerre-exponential families, addition is realized via hybrid binomial polynomials: $p$5
• The Φ-hyperbolic functions exhibit binomial-type addition formulas mirroring the supersymmetric generalizations of binomial identities (Ouellet et al., 2019).

Unlike the classical case, most generalized families do not satisfy a simple Pythagorean-type identity $p$6, though supersymmetric analogs and modified hyperbolic identities exist (Ouellet et al., 2019, Dattoli et al., 2017).

4. Parameter Dependency: Monotonicity, Convexity, and Inequalities

With the introduction of parameters $p$7, the functional behavior is significantly enriched.

• Monotonicity: For fixed $p$8, the map $p$9 and $\arcsinh_p(x) = \int_0^x (1 + t^p)^{-1/p} dt \quad (x \geq 0,\; p > 1),$0 is strictly decreasing for $\arcsinh_p(x) = \int_0^x (1 + t^p)^{-1/p} dt \quad (x \geq 0,\; p > 1),$1; $\arcsinh_p(x) = \int_0^x (1 + t^p)^{-1/p} dt \quad (x \geq 0,\; p > 1),$2-monotonicity is more intricate (Karp et al., 2024).
• Convexity: $\arcsinh_p(x) = \int_0^x (1 + t^p)^{-1/p} dt \quad (x \geq 0,\; p > 1),$3 is log-convex in $\arcsinh_p(x) = \int_0^x (1 + t^p)^{-1/p} dt \quad (x \geq 0,\; p > 1),$4 for fixed $\arcsinh_p(x) = \int_0^x (1 + t^p)^{-1/p} dt \quad (x \geq 0,\; p > 1),$5 and $\arcsinh_p(x) = \int_0^x (1 + t^p)^{-1/p} dt \quad (x \geq 0,\; p > 1),$6, as shown by direct calculus on the parameter derivatives. Geometric convexity/concavity properties distinguish $\arcsinh_p(x) = \int_0^x (1 + t^p)^{-1/p} dt \quad (x \geq 0,\; p > 1),$7 (geometrically convex) from $\arcsinh_p(x) = \int_0^x (1 + t^p)^{-1/p} dt \quad (x \geq 0,\; p > 1),$8 (geometrically concave) (Jiang et al., 2013).
• Power-mean inequalities: For $\arcsinh_p(x) = \int_0^x (1 + t^p)^{-1/p} dt \quad (x \geq 0,\; p > 1),$9, $\sinh_p$0 obeys $\sinh_p$1-convexity on $\sinh_p$2: $\sinh_p$3 with broader regimes available for mixed orders (Baricz et al., 2013).
• Sharp analytic inequalities, including Adamović-Mitrinović, Wilker, Huygens, and Cusa-Huygens types, generalize classical results:
  • For $\sinh_p$4: $\sinh_p$5,
  • $\sinh_p$6 (Klén et al., 2012, Bagul et al., 2023).
  • Multiple exponential-type bounds hold for $\sinh_p$7, $\sinh_p$8, and $\sinh_p$9 (Bagul et al., 2023).
• Parameter limiting regimes connect generalized families to classical, linear, or exponential functions (as $p$0, $p$1, $p$2) (Bagul et al., 2023, Karp et al., 2024).

5. Duality, Transformations, and Special Cases

A full duality theory connects generalized trigonometric (e.g., $p$3, $p$4) and hyperbolic functions. Given structure constants, for example $p$5, dual formulas produce, for suitable $p$6,

$p$7

and conversely (Miyakawa et al., 2020, Miyakawa et al., 2022).

Special parameter choices recover notable analytic families:

• The classical case $p$8: $p$9, $$\cosh_p(x) = \frac{d}{dx} \sinh_p(x) = (1 + [\sinh_p(x)]^p)^{1/p}.$$0.
• Lemniscate case $$\cosh_p(x) = \frac{d}{dx} \sinh_p(x) = (1 + [\sinh_p(x)]^p)^{1/p}.$$1: functions relevant to lemniscatic and p-Laplacian spectral theory.
• In operational constructions: Laguerre/Bessel-based polynomials and Airy families emerge as parameter limits or under specific operational choices (Dattoli et al., 2017).

The analytic, algebraic, and functional properties of one family can be transferred to another via duality, enabling systematic derivation of inequalities, multiple-angle formulas, and reduction to special cases.

6. Applications and Broader Contexts

Generalized hyperbolic functions have several primary areas of application:

• Nonlinear analysis and PDE: they provide explicit eigenfunctions and basis functions for the Dirichlet $$\cosh_p(x) = \frac{d}{dx} \sinh_p(x) = (1 + [\sinh_p(x)]^p)^{1/p}.$$2-Laplacian, higher-order Laplacians, and nonlinear evolution equations. Their properties yield sharp constants in Sobolev or Moser–Trudinger type embeddings and nonlinear Gronwall inequalities (Klén et al., 2012, Karp et al., 2024, Bagul et al., 2023).
• Special function theory: They admit representations in terms of Gauss and generalized hypergeometric functions, incomplete Beta integrals, and series linked to classical orthogonal polynomials (Baricz et al., 2013, Karp et al., 2024).
• Supersymmetric quantum mechanics and spectral theory: $$\cosh_p(x) = \frac{d}{dx} \sinh_p(x) = (1 + [\sinh_p(x)]^p)^{1/p}.$$3-hyperbolic functions offer a powerful basis for the spectral parameter power series (SPPS) solution of (generalized) Schrödinger problems, allowing systematic construction of solution families for variable-coefficient ODEs (Ouellet et al., 2019).
• Integral and transform analysis: They provide new integral evaluations and facilitate the construction of generalized Borel, Mellin, and Volterra–composition transforms (Dattoli et al., 2017, Ouellet et al., 2019, Karp et al., 2024).
• Geometry, convexity, and inequalities: The geometric and analytic properties of these functions are directly applicable to quasiconformal mappings, geometric function theory, and abstract convexity (Jiang et al., 2013, Baricz et al., 2013, Karp et al., 2024).

7. Operational, Umbral, and Hierarchical Generalizations

Beyond the $$\cosh_p(x) = \frac{d}{dx} \sinh_p(x) = (1 + [\sinh_p(x)]^p)^{1/p}.$$4- or $$\cosh_p(x) = \frac{d}{dx} \sinh_p(x) = (1 + [\sinh_p(x)]^p)^{1/p}.$$5-families, operational generalizations define entire hierarchies via umbral calculus and deformed derivatives; e.g., the Laguerre-hyperbolic, Tricomi–Bessel, and Humbert–Bessel exponentials. These permit systematic construction of new function classes with group-like addition, explicit operational identities, and links to classical and modern special functions (Dattoli et al., 2017).

In sum, the study of generalized hyperbolic functions reveals a deep interplay between analytic, algebraic, and geometric function theory, leads to new sharp inequalities, power-convexity and monotonicity frameworks, and provides a toolkit for nonlinear differential equations, spectral problems, and advanced special function theory (Baricz et al., 2013, Jiang et al., 2013, Klén et al., 2012, Miyakawa et al., 2020, Miyakawa et al., 2022, Karp et al., 2024, Dattoli et al., 2017, Ouellet et al., 2019, Bagul et al., 2023).