*3.4. Entropy*

The Rényi entropy due to [36], is characterized as

$$I\_R(\gamma) = \frac{1}{1-\gamma} \log \left( \int\_0^\infty f^\gamma(x) dx \right),$$

Let us consider

*i*=1

 −

$$f^{\gamma}(\mathbf{x}) = (a\,\theta)^{\gamma} \, k^{\gamma}(\mathbf{x};\boldsymbol{\psi}) \, \overline{\mathcal{K}}(\mathbf{x};\boldsymbol{\psi})^{\gamma(-a-1)} \, \mathbf{e}^{-\gamma \left[\frac{1-\overline{\mathcal{K}}(\mathbf{x};\boldsymbol{\psi})^{a}}{\overline{\mathcal{K}}(\mathbf{x};\boldsymbol{\psi})^{a}}\right]}$$

$$\times \left\{ 1 - \mathbf{e}^{-\frac{1-\overline{\mathcal{K}}(\mathbf{x};\boldsymbol{\psi})^{a}}{\overline{\mathcal{K}}(\mathbf{x};\boldsymbol{\psi})^{a}}} \right\}^{\gamma(\mathcal{G}-1)} \,. \tag{12}$$

Expanding Equation (12) as in Section 3.1, the Rényi entropy reduces to

$$I\_R(\gamma) = \frac{1}{1-\gamma} \log \left\{ \sum\_{i=0}^{\infty} \sum\_{l=0}^{i} \zeta\_{i,l}^\* \int\_0^{\infty} k^\gamma(\mathbf{x}) \mathbb{X}^{-\{a(\gamma+i-l)-\gamma\}} d\mathbf{x} \right\},\tag{13}$$

$$\text{where } \mathbb{Z}\_{i,l}^{\*} = \frac{(-1)^{i+l}}{i!} (\mathfrak{a}\ \beta)^{\gamma} \binom{i}{l} \sum\_{j=0}^{i} (-1)^{j} (j+i)^{i} \binom{\gamma(\beta-1)}{j} \text{.}$$
