Generalized Trigonometric Functions

Updated 11 December 2025

Generalized trigonometric functions are a family of special functions defined via p-Laplacian eigenproblems that generalize classical sine and cosine.
They exhibit rich analytic properties including power series, hypergeometric representations, and novel addition formulas extending traditional identities.
They play a crucial role in nonlinear spectral theory and applications in geometry, physics, and information theory, offering refined inequalities and eigenvalue estimates.

Generalized trigonometric functions (GTFs) constitute an extensive hierarchy of special functions that interpolate and generalize the classical sine and cosine, as well as their hyperbolic and elliptic analogues, by deforming the associated algebraic identities, periods, and differential equations. These functions arise naturally in the nonlinear spectral theory of $p$ -Laplace operators, extend to multi-parameter families with deep connections to special functions, algebraic equations, and informative inequalities, and possess analytic structures that unify diverse trigonometric phenomena under parameterized frameworks.

1. Foundational Definitions and Structural Properties

The one-parameter GTFs, introduced via the $p$ -Laplacian eigenvalue problem, generalize classical trigonometric functions by replacing the quadratic binomial in the Pythagorean identity with a $p$ -power. For $p>1$ , define

$\arcsin_p(y) = \int_0^y (1-t^p)^{-1/p}\,dt, \quad y \in [0,1]$

with $\pi_p = 2 \arcsin_p(1) = 2 \int_0^1 (1-t^p)^{-1/p}\,dt$ , and set the generalized sine as its inverse: $\sin_p(x) = \arcsin_p^{-1}(x), \quad x \in [0, \pi_p/2]$ The generalized cosine is given by differentiation: $\cos_p(x) = \frac{d}{dx}\sin_p(x) = (1-\sin_p^p(x))^{1/p}$ These satisfy the $p$ -Pythagorean identity: $|\cos_p(x)|^p + |\sin_p(x)|^p = 1$ and solve the nonlinear ODE

$(|u'|^{p-2} u')' + \lambda |u|^{p-2} u = 0, \qquad u(0) = 0,\, u(L) = 0$

with $\sin_p\left(\frac{\pi_p}{L} x\right)$ as the first eigenfunction and $\lambda_1 = (\pi_p / L)^p$ (Klén et al., 2012).

This framework extends to multi-parameter settings. For $p,q>1$ ,

$\arcsin_{p,q}(x) = \int_0^x (1-t^q)^{-1/p}\,dt$

and its inverse $\sin_{p,q}$ on $[0, \pi_{p,q}/2]$ yields

$\cos_{p,q}(x) = \frac{d}{dx}\sin_{p,q}(x),\quad |\cos_{p,q}(x)|^p + |\sin_{p,q}(x)|^q = 1$

The corresponding nonlinear eigenproblem has $u(x) = \sin_{p,q}(x)$ solving $-(|u'|^{p-2}u')' = (q/p^*)|u|^{q-2}u$ , $p^* = p/(p-1)$ (Takeuchi, 2019, Kobayashi et al., 2019, Bhayo et al., 2011, Karp et al., 20 Nov 2024, Takeuchi, 2016).

Generalized hyperbolic functions are defined analogously, with integrals involving $(1+t^q)^{-1/p}$ , and satisfy identities such as $|\cosh_{p,q}(x)|^p - |\sinh_{p,q}(x)|^q = 1$ .

2. Analytic, Differential, and Algebraic Properties

GTFs possess highly nontrivial analytic structures:

Power Series: The Maclaurin expansions for $\arcsin_p(x)$ , $\arcsin_{p,q}(x)$ , and their inverses involve generalized binomial coefficients, Gamma functions, Stirling and Bell numbers (Chebolu et al., 2021, Chebolu et al., 2021).
Differential Identities: For the single-parameter case, $d\sin_p(x)/dx = \cos_p(x)$ , $d\cos_p(x)/dx = -\cos_p(x)^{2-p}\sin_p(x)^{p-1}$ . For $(p,q)$ , $d\cos_{p,q}(x)/dx = -\frac{q}{p} \sin_{p,q}(x)^{q-1}\cos_{p,q}(x)^{2-p}$ .
Pythagorean-type Identities: All GTF families are governed by generalized algebraic identities, e.g., $|x|^p + |y|^q = 1$ defines the unit $(p,q)$ -circle (Chebolu et al., 2021, Kobayashi et al., 2019).
Inverse and Hypergeometric Representations: $\arcsin_{p,q}(x) = x\, {}_2F_1(1/p,1/q;1+1/q;x^q)$ , and closed expressions for inverse cosines, sines, and corresponding hyperbolic analogues in ${}_2F_1$ and ${}_3F_2$ forms (Karp et al., 20 Nov 2024).

3. Convexity, Inequalities, and Parameter Dependence

Significant effort has focused on parameter monotonicity and convexity properties:

$\sin_p(y)$ and $\cos_p(y)$ are strictly log-concave in $p$ for $y \in (0,1)$ . $\tan_p(y)$ is strictly log-convex for $p>1$ , with parallel claims for hyperbolic counterparts (Karp et al., 2014, Karp et al., 20 Nov 2024).
For the general $(p,q)$ -case, $p \mapsto \sin_{p,q}(y)$ and $q \mapsto \sin_{p,q}(y)$ are strictly concave on $p,q > 1$ , while $p \mapsto \sinh_{p,q}(y)$ is strictly log-convex.
These convexity results resolve Turán-type conjectures for GTFs, providing inequalities such as $[\sin_p(y)]^2 > \sin_{p-1}(y)\sin_{p+1}(y)$ and reversed inequalities for log-convex cases (Karp et al., 2014).

GTFs provide sharp generalizations of classical trigonometric inequalities:

Mitrinović–Adamović-type: $\cos_p(x)^{1/(p+1)} < \sin_p(x)/x < 1$ for $x \in (0, \pi_p/2)$ , best exponent $1/(p+1)$ (Klén et al., 2012). These extend naturally to $(p,q)$ and their hyperbolic counterparts (Miyakawa et al., 2022).
Wilker and Huygens-type: Two-parameter versions yield inequalities involving powers and combinations of $\sin_{p,q}$ , $\cos_{p,q}$ , and generalized tangents (Miyakawa et al., 2022, Klén et al., 2012).
Cusa–Huygens-type: Upper bounds for $\sin_{p,q}(x)/x$ in terms of $\cos_{p,q}(x)$ , optimal in various parameter regimes.

4. Addition, Multiplication, and Multiple-Angle Relations

Unlike the classical functions, GTFs admit highly nontrivial addition and double-angle formulas:

Double- and Multiple-Angle Formulas: The case $p=2$ or special rational $(p,q)$ are tractable, e.g.,

$\sin_{2,q}(2^{2/q}x) = 2^{2/q}\, \sin_{2,q}(x)\, \cos_{2,q}^{q-1}(x),\quad \cos_{2,q}(2^{2/q}x) = \cos_{2,q}^q(x)-\sin_{2,q}^q(x)$

Explicit double-angle formulas are available for cases such as $(p,q) = (2,3)$ and $(4/3,2)$ (Sato et al., 2019, Takeuchi, 2016).

Generalized Tangents and Addition Laws: For $T_{p,q}(x) := (\sin_{p,q}(x)/\cos_{p,q}(x))^{p/q}$ , $T_{p,q}(x+y)$ satisfies the standard tangent addition law, which leads to addition formulas for $\sin_{p,q}(x+y)$ and $\cos_{p,q}(x+y)$ in terms of $T_{p,q}$ (Miyakawa et al., 2022).
Algebraic Structure: For an arbitrary complex polynomial $P(x)$ , one can construct GTFs associated with $P$ , leading to higher-degree algebraic identities generalizing $\sin^2(x) + \cos^2(x) = 1$ (Yu, 2016).

5. Spectral, Geometric, and Information-Theoretic Applications

GTFs are intrinsically linked to spectral problems and nonlinear analysis:

Eigenfunctions of the $p$ -Laplacian: GTFs arise naturally as ground states and higher eigenfunctions in the Dirichlet problem for $p$ -Laplacian ODEs, with precise spectral constants and optimal Lyapunov inequalities captured by these functions (Takeuchi, 2019, Kobayashi et al., 2019, Bhayo et al., 2011).
Orthogonality and Fourier Analogues: While classical orthogonality holds only for $p=2$ , GTFs can support generalized spectral theory with weighted orthogonality for higher eigenfunctions (Chebolu et al., 2021, Takeuchi, 2019).
Information-Theoretic Densities: The family of Generalized Trigonometric Densities (GTDs) are extremal for tri-parametric functional inequalities (Fisher–Rényi–moment) and include (as special cases) stretched Gaussians, logistic, and raised-cosine densities. GTDs have precise scaling properties and tail behavior parametrized by GTFs (Puertas-Centeno et al., 31 Oct 2024).
Means Theory: Four new means ( $P_p$ , $T_p$ , $M_p$ , $L_p$ ) can be constructed using GTFs, generalizing Seiffert, logarithmic, and Neuman–Sándor means, and their relations are governed by GTF inequalities (Sándor et al., 2018).

Other notable applications include nonlinear pendulum equations with $p$ -Laplacian, self-similar solutions of fluid equations, and Wallis-type product and integral recursions that generalize classical elliptic and trigonometric integrals (Kobayashi et al., 2019, Dattoli et al., 2017).

6. Extensions: Algebraic, Umbral, and Associative Algebraic Frameworks

Algebraic GTFs: GTFs, including parabolic, Jacobi–type, and through the solution of higher-degree algebraic equations, provide radicals for cubics and quintics, with connections to Chebyshev radicals, hypergeometric series, and modular forms (Dattoli et al., 2011).
Umbral and Operational Generalizations: The Laguerre, Bessel, and Airy-type generalizations define new trigonometric–like functions via operational calculus (Laguerre derivative, umbral composition), with their own addition, duplication, and De Moivre formulas (Dattoli et al., 2017).
Associative Algebras: Over finite-dimensional commutative associative algebras, generalized exponential and sine/cosine functions can be formulated, with polar decompositions, generalized arguments, and determinant-based modulus, extending the Euler and De Moivre formulas to modules of arbitrary dimension (BeDell, 2017).
Analytic Domains and Continuation: GTFs can be analytically continued beyond their primary intervals; maximal univalence domains are determined by geometric and conformal mapping arguments, and the periodicity properties reflect deep connections to complex function theory (Ding, 2021).

7. Open Problems, Special Cases, and Future Directions

Open Problems: There is no full addition theorem in closed-form for general $(p,q)$ . The search for further closed double-angle and addition formulas for general parameters is ongoing (Takeuchi, 2016, Sato et al., 2019, Miyakawa et al., 2022).
Special Functions: The classical sine, cosine, hyperbolic, lemniscate, and Dixon’s elliptic sine arise as special cases for particular parameter choices, with deeper connections to elliptic integrals and modular forms (Dattoli et al., 2011, Sato et al., 2019).
Further Research: GTFs open avenues for expansions in nonlinear eigenfunction bases, refined inequalities for non-linear PDEs, heavy-tailed distribution modeling, and the systematic extension of classical trigonometric analysis into new non-Euclidean, algebraic, and probabilistic regimes (Karp et al., 20 Nov 2024, Puertas-Centeno et al., 31 Oct 2024, Bhayo et al., 2011).