Generalized Trigonometric Densities

• Generalized Trigonometric Densities (GTDs) are a three-parameter family of univariate probability densities defined using generalized trigonometric functions, unifying classical laws like Gaussian, logistic, and hyperbolic secant.
• They offer closed-form normalization and serve as extremal solutions in optimizing entropy and generalized Fisher information under moment constraints.
• The framework systematically interpolates between compact support, exponential decay, and heavy-tailed regimes, making it a versatile tool for both theoretical analysis and practical modeling.

Generalized Trigonometric Densities (GTDs) constitute a three-parameter family of univariate probability densities constructed using Drábek–Manásevich generalized trigonometric functions. GTDs subsume classical laws such as the Gaussian, stretched-$q$-Gaussian, hyperbolic secant, logistic, and raised-cosine densities as special cases. This family admits closed-form normalization, systematically interpolates between compact support, exponential, and power-law tails, and provides extremal solutions to entropy–information trade-offs under generalized moment constraints. GTDs provide a rigorous framework for characterizing heavy-tailed phenomena and optimizing generalized information-theoretic quantities (Puertas-Centeno et al., 2024).

1. Formal Definition and Analytical Structure

Let $p>0$ and set $p^* = \tfrac{p}{p-1}$. For real parameters $\beta, \lambda$, impose the conditions

$$1+\beta-\lambda \neq 0, \quad \text{and} \quad \mathrm{sign}(1-\lambda+\beta) = \mathrm{sign}\left(\frac{1-\lambda}{p}+\beta\right).$$

Define the "stretch" parameter

$$\kappa_{p,\beta,\lambda} = \left| \frac{\lambda-1}{1+\beta-\lambda} \right|^{1/p^*}, \quad v = \frac{\lambda-1}{\lambda-\beta}, \quad w = p^*.$$

The GTD is then given by

$c_{p,\beta,\lambda}(y) = a_{p,\beta,\lambda} \times \begin{cases} \bigl[\cos_{v,w}(\kappa_{p,\beta,\lambda} y)\bigr]^{1/(\lambda-\beta)}, & (\lambda > 1), \[2ex] \bigl[\cosh_{v,w}(\kappa_{p,\beta,\lambda} y)\bigr]^{1/(\lambda-\beta)}, & (\lambda < 1), \end{cases}$

where $\cos_{v,w}$ and $\cosh_{v,w}$ are generalized cosine and hyperbolic-cosine functions. The normalization constant is

$$a_{p,\beta,\lambda} = \frac{p^* \kappa_{p,\beta,\lambda}}{2}\, B\left( \frac{1}{p^*}, \frac{\beta\,\mathrm{sign}(1-\lambda+\beta)}{|1-\lambda|} + \frac{1}{p} 1_{\{1-\lambda>0\}} \right).$$

The domain of definition $p>0$0 guarantees normalization $p>0$1 using the generalized Pythagorean identity (for trigonometric or hyperbolic case as appropriate) (Puertas-Centeno et al., 2024).

2. Relation to Classical and Generalized Laws

The GTD construction encompasses a broad collection of standard and generalized probability densities:

• Stretched $p>0$2-Gaussian: For $p>0$3, the density reduces to the generalized stretched Gaussian:

$p>0$4

where $p>0$5 denotes the $p>0$6-exponential, widely used in nonextensive statistical mechanics.

• Logistic law: For $p>0$7, one obtains, up to normalization, $p>0$8.
• Hyperbolic secant law: $p>0$9 produces $p^* = \tfrac{p}{p-1}$0.
• Raised-cosine law: $p^* = \tfrac{p}{p-1}$1 yields $p^* = \tfrac{p}{p-1}$2 on $p^* = \tfrac{p}{p-1}$3.

This unifying structure allows analysis and interpolation between compactly supported kernels, sub-Gaussian, and heavy-tailed regimes, all within the same parameterization (Puertas-Centeno et al., 2024).

3. Generalized Entropic and Information-Theoretic Properties

The GTDs are extremal solutions for a variety of information-theoretic quantities:

• Rényi entropy: For order $p^* = \tfrac{p}{p-1}$4, $p^* = \tfrac{p}{p-1}$5,

$p^* = \tfrac{p}{p-1}$6

• Generalized Fisher information: Following Lutwak and Bercher, define

$p^* = \tfrac{p}{p-1}$7

A fundamental Stam-type inequality holds: for $p^* = \tfrac{p}{p-1}$8, $p^* = \tfrac{p}{p-1}$9, $\beta, \lambda$0,

$\beta, \lambda$1

Fixing $\beta, \lambda$2 shows that the GTDs $\beta, \lambda$3 uniquely minimize $\beta, \lambda$4, achieving minimal Rényi entropy at fixed generalized Fisher information (Puertas-Centeno et al., 2024).

4. Deformed Cumulants and Generalized Moments

GTDs provide a natural operator for the construction of generalized moments based on deformed cumulative distributions:

• Deformed CDF: For order $\beta, \lambda$5,

$\beta, \lambda$6

• Generalized moment of order $\beta, \lambda$7 (cumulative form):

$\beta, \lambda$8

This can also be written as

$\beta, \lambda$9

with the deformation parameter $$1+\beta-\lambda \neq 0, \quad \text{and} \quad \mathrm{sign}(1-\lambda+\beta) = \mathrm{sign}\left(\frac{1-\lambda}{p}+\beta\right).$$0 tuning the weight placed on the distribution's tails. These moments reduce to those of a deformed/regularized distribution and are key to the characterization of heavy-tailed behavior (Puertas-Centeno et al., 2024).

5. Heavy-tail Interpolation and Critical Finiteness Thresholds

The GTD family achieves a continuous interpolation between compact support, exponential decay, and power-law tails:

• For $$1+\beta-\lambda \neq 0, \quad \text{and} \quad \mathrm{sign}(1-\lambda+\beta) = \mathrm{sign}\left(\frac{1-\lambda}{p}+\beta\right).$$1 as $$1+\beta-\lambda \neq 0, \quad \text{and} \quad \mathrm{sign}(1-\lambda+\beta) = \mathrm{sign}\left(\frac{1-\lambda}{p}+\beta\right).$$2, all $$1+\beta-\lambda \neq 0, \quad \text{and} \quad \mathrm{sign}(1-\lambda+\beta) = \mathrm{sign}\left(\frac{1-\lambda}{p}+\beta\right).$$3 for every $$1+\beta-\lambda \neq 0, \quad \text{and} \quad \mathrm{sign}(1-\lambda+\beta) = \mathrm{sign}\left(\frac{1-\lambda}{p}+\beta\right).$$4 if and only if

$$1+\beta-\lambda \neq 0, \quad \text{and} \quad \mathrm{sign}(1-\lambda+\beta) = \mathrm{sign}\left(\frac{1-\lambda}{p}+\beta\right).$$5

This threshold marks the point at which the deformation parameter $$1+\beta-\lambda \neq 0, \quad \text{and} \quad \mathrm{sign}(1-\lambda+\beta) = \mathrm{sign}\left(\frac{1-\lambda}{p}+\beta\right).$$6 sufficiently suppresses the tails to ensure finiteness of moments. Thus, $$1+\beta-\lambda \neq 0, \quad \text{and} \quad \mathrm{sign}(1-\lambda+\beta) = \mathrm{sign}\left(\frac{1-\lambda}{p}+\beta\right).$$7 can be interpreted as the phase transition for moment finiteness in the presence of heavy tails. For $$1+\beta-\lambda \neq 0, \quad \text{and} \quad \mathrm{sign}(1-\lambda+\beta) = \mathrm{sign}\left(\frac{1-\lambda}{p}+\beta\right).$$8, the density is compactly supported; at $$1+\beta-\lambda \neq 0, \quad \text{and} \quad \mathrm{sign}(1-\lambda+\beta) = \mathrm{sign}\left(\frac{1-\lambda}{p}+\beta\right).$$9, moments become critical with exponential tails; for $$\kappa_{p,\beta,\lambda} = \left| \frac{\lambda-1}{1+\beta-\lambda} \right|^{1/p^*}, \quad v = \frac{\lambda-1}{\lambda-\beta}, \quad w = p^*.$$0, moments diverge and the law exhibits power-law tails (Puertas-Centeno et al., 2024).

6. Fundamental Properties and Optimization Principles

The structure of GTDs endows the associated generalized moments with several key properties:

• Monotonicity in $$\kappa_{p,\beta,\lambda} = \left| \frac{\lambda-1}{1+\beta-\lambda} \right|^{1/p^*}, \quad v = \frac{\lambda-1}{\lambda-\beta}, \quad w = p^*.$$1: For fixed $$\kappa_{p,\beta,\lambda} = \left| \frac{\lambda-1}{1+\beta-\lambda} \right|^{1/p^*}, \quad v = \frac{\lambda-1}{\lambda-\beta}, \quad w = p^*.$$2, the function $$\kappa_{p,\beta,\lambda} = \left| \frac{\lambda-1}{1+\beta-\lambda} \right|^{1/p^*}, \quad v = \frac{\lambda-1}{\lambda-\beta}, \quad w = p^*.$$3 is nondecreasing (by Hölder's inequality).
• Scaling behavior: For scaling $$\kappa_{p,\beta,\lambda} = \left| \frac{\lambda-1}{1+\beta-\lambda} \right|^{1/p^*}, \quad v = \frac{\lambda-1}{\lambda-\beta}, \quad w = p^*.$$4, the moment rescales as

$$\kappa_{p,\beta,\lambda} = \left| \frac{\lambda-1}{1+\beta-\lambda} \right|^{1/p^*}, \quad v = \frac{\lambda-1}{\lambda-\beta}, \quad w = p^*.$$5

where $$\kappa_{p,\beta,\lambda} = \left| \frac{\lambda-1}{1+\beta-\lambda} \right|^{1/p^*}, \quad v = \frac{\lambda-1}{\lambda-\beta}, \quad w = p^*.$$6.

• Extremality: Maximizing $$\kappa_{p,\beta,\lambda} = \left| \frac{\lambda-1}{1+\beta-\lambda} \right|^{1/p^*}, \quad v = \frac{\lambda-1}{\lambda-\beta}, \quad w = p^*.$$7 under constraints $$\kappa_{p,\beta,\lambda} = \left| \frac{\lambda-1}{1+\beta-\lambda} \right|^{1/p^*}, \quad v = \frac{\lambda-1}{\lambda-\beta}, \quad w = p^*.$$8 and $$\kappa_{p,\beta,\lambda} = \left| \frac{\lambda-1}{1+\beta-\lambda} \right|^{1/p^*}, \quad v = \frac{\lambda-1}{\lambda-\beta}, \quad w = p^*.$$9 leads, via the Lagrange and Euler–Lagrange formalism, to a solution of the GTD type:

$c_{p,\beta,\lambda}(y) = a_{p,\beta,\lambda} \times \begin{cases} \bigl[\cos_{v,w}(\kappa_{p,\beta,\lambda} y)\bigr]^{1/(\lambda-\beta)}, & (\lambda > 1), \[2ex] \bigl[\cosh_{v,w}(\kappa_{p,\beta,\lambda} y)\bigr]^{1/(\lambda-\beta)}, & (\lambda < 1), \end{cases}$0

The matching minimization of generalized Fisher information achieves a duality between entropy and information for this family.

These properties ensure that GTDs act as attractors or optimal distributions under a wide class of entropic and moment constraints (Puertas-Centeno et al., 2024).

7. Summary Table: GTDs and Special Cases

Family	GTD Parameters	Tail Type
Stretched $c_{p,\beta,\lambda}(y) = a_{p,\beta,\lambda} \times \begin{cases} \bigl[\cos_{v,w}(\kappa_{p,\beta,\lambda} y)\bigr]^{1/(\lambda-\beta)}, & (\lambda > 1), \[2ex] \bigl[\cosh_{v,w}(\kappa_{p,\beta,\lambda} y)\bigr]^{1/(\lambda-\beta)}, & (\lambda < 1), \end{cases}$1-Gaussian	$c_{p,\beta,\lambda}(y) = a_{p,\beta,\lambda} \times \begin{cases} \bigl[\cos_{v,w}(\kappa_{p,\beta,\lambda} y)\bigr]^{1/(\lambda-\beta)}, & (\lambda > 1), \[2ex] \bigl[\cosh_{v,w}(\kappa_{p,\beta,\lambda} y)\bigr]^{1/(\lambda-\beta)}, & (\lambda < 1), \end{cases}$2	Exponential/Power
Logistic	$c_{p,\beta,\lambda}(y) = a_{p,\beta,\lambda} \times \begin{cases} \bigl[\cos_{v,w}(\kappa_{p,\beta,\lambda} y)\bigr]^{1/(\lambda-\beta)}, & (\lambda > 1), \[2ex] \bigl[\cosh_{v,w}(\kappa_{p,\beta,\lambda} y)\bigr]^{1/(\lambda-\beta)}, & (\lambda < 1), \end{cases}$3	Exponential
Hyperbolic Secant	$c_{p,\beta,\lambda}(y) = a_{p,\beta,\lambda} \times \begin{cases} \bigl[\cos_{v,w}(\kappa_{p,\beta,\lambda} y)\bigr]^{1/(\lambda-\beta)}, & (\lambda > 1), \[2ex] \bigl[\cosh_{v,w}(\kappa_{p,\beta,\lambda} y)\bigr]^{1/(\lambda-\beta)}, & (\lambda < 1), \end{cases}$4	Exponential
Raised Cosine	$c_{p,\beta,\lambda}(y) = a_{p,\beta,\lambda} \times \begin{cases} \bigl[\cos_{v,w}(\kappa_{p,\beta,\lambda} y)\bigr]^{1/(\lambda-\beta)}, & (\lambda > 1), \[2ex] \bigl[\cosh_{v,w}(\kappa_{p,\beta,\lambda} y)\bigr]^{1/(\lambda-\beta)}, & (\lambda < 1), \end{cases}$5	Compact

GTDs unify these and interpolated behaviors in a single closed-form parameterization (Puertas-Centeno et al., 2024).

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Generalized Trigonometric Densities provide a unified analytic and information-theoretic framework for continuous probability densities, covering a broad range of decay types, optimizing entropy-information inequalities, and supporting rigorous characterization of heavy-tailed, sub-Gaussian, and compactly supported behavior (Puertas-Centeno et al., 2024).