content_image/355588.jpg content_image/355589.jpg content_image/355592.jpg content_image/355593.jpg content_image/355594.jpg content_image/355595.jpg content_image/355590.jpg content_image/355591.jpg
where we employ the definition of ˜ U$_{q}$ given by 

Equation (A8) leads to 

<img src='content_image/355588.jpg'>

<img src='content_image/355589.jpg'>

With the use of Eqs. (A10) and (54), the physical temperature is given by [38] 

leading to 

where we employ the relation: ∂ ˜ X$_{q}$/∂ ˜ U$_{q}$ = q ˜ γ ˜ X$_{q}$ . 

## Appendix: B. Evaluations of averages by the exact approach 

## 1. The q -average 

We first discuss evaluations by the q -average given by 

by using the exact expressions for the gamma function [13, 48]: 

<img src='content_image/355592.jpg'>

<img src='content_image/355593.jpg'>

<img src='content_image/355594.jpg'>

<img src='content_image/355595.jpg'>

<img src='content_image/355590.jpg'>

<img src='content_image/355591.jpg'>

