**5. Type I Quons**

As known, type I quons firstly studied in [23], are based on the asymmetric version of *q*-numbers

[*x*]*<sup>q</sup>* <sup>=</sup> *<sup>q</sup><sup>x</sup>* <sup>−</sup> <sup>1</sup> *<sup>q</sup>* <sup>−</sup> <sup>1</sup> , (43)

that is strictly related to the *q*-calculus introduced by Jackson [28]. In the present context, type I quons can be derived starting from the following deformed sum

$$\mathbf{x} \oplus \mathbf{y} = \mathbf{x} + \mathbf{y} + (q - 1)\mathbf{x} \,\mathrm{y}\,,\tag{44}$$

emerging also in the context of generalized statistical mechanics [47]. The deformation parameter *ξ* ≡ *q*, is restricted to *q* ≥ 0 and the *q*-sum recovers the standard sum in the *q* → 1 limit.

Equation (44) can be rewritten in

$$\text{tr}\boxplus y = \frac{\left[1 + \left(q - 1\right)x\right]\left[1 + \left(q - 1\right)y\right] - 1}{q - 1},\tag{45}$$

so that

$$1 + (q - 1) \left( \mathbf{x} \oplus \mathbf{y} \right) = \left[ 1 + (q - 1) \ge \right] \left[ 1 + (q - 1) \,\, \mathbf{y} \right],\tag{46}$$

which forces us to define the quantity

$$a^{-1}(\mathbf{x}) = \frac{1}{\ln q} \ln(1 + (q - 1)\mathbf{x}) \,, \tag{47}$$

and its inverse *a*(*x*) coincides with function defined in Equation (43), that is

$$a(\mathbf{x}) \equiv [\mathbf{x}]\_{\emptyset} \; . \tag{48}$$

Asymmetric *q*-numbers have been employed in [23–25] to introduce quon statistics of type I starting from the *q*-generalization of the quantum oscillator algebras of creation and annihilation operators, like for the type II.

However, the algebra of asymmetric *q* numbers differs from those of the symmetric one since

$$\left[\overline{\mathbf{x}}\right]\_q = -q^{-\mathbf{x}}\left[\mathbf{x}\right]\_{q\ \mathbf{y}}\tag{49}$$

being, in general 0∗ = 0 and

$$\overline{\mathbf{x}} = -\frac{\mathbf{x}}{1 + (q - 1)\,\mathbf{x}} \,'\,\tag{50}$$

that is, the opposite in A ≡ (⊕*q*, ) does not coincides with the opposite of the ordinary algebras in like in the symmetric case.

Within the *q*-algebra the generalized exponential reads *E*(*x*) ≡ exp*q*(*x*) and analogously the generalized logarithm reads *L*(*x*) ≡ ln*q*(*x*) where

$$\exp\_q(\mathbf{x}) = \left(1 + (q - 1)\mathbf{x}\right)^{1/\ln q},\tag{51}$$

$$\ln\_q(\mathbf{x}) = \frac{q^{\ln x} - 1}{q - 1} \,\_ {\prime} \tag{52}$$

that are strictly related to the *q*-exponential and *q*-logarithm introduced in the generalized statistical mechanics [47] and fulfill relations (23) and (25), respectively, with the *q*-sum given in (44).

The function *a*(*f*) given in Equation (48) defines univocally the function *b*(*f*) throughout Equations (18) or (19) with *c* = 1. However, in this case due to relation (49), we must separate the case of boson-like quons, with

$$b(f) = 1 + q \, [f]\_{q, \text{-} \,} \tag{53}$$

from the case of fermion-like quons, with

$$b(f) = 1 - q^{1-f} \left[ f \right]\_{\mathfrak{q}}.\tag{54}$$

*Entropy* **2019**, *21*, 841
