content_image/342701.jpg content_image/342700.jpg content_image/342697.jpg content_image/342698.jpg content_image/342699.jpg content_image/342696.jpg content_image/342695.jpg
## B. Solution in the positive region ( x > 0) 

Let us study the other region in which x > 0, from Eq. 7 using the variable 1 /z = 1 + e a ( x − L ) and the transformation φ = z − $^{ω}$(1 − z ) − $^{ρ}$g ( z ), we obtain: 

<img src='content_image/342701.jpg'>

Following a line of thought similar to the one outlined in the previous subsection we obtain the mass function 

<img src='content_image/342700.jpg'>

The final result for the parameters ω, ρ and δ (the coefficient introduced requiring that the term multiplying the factor 1 / [ z (1 − z )] be a costant) is found to be: 

The differential equation of the function g ( z ) in Eq. (20) becomes: 

<img src='content_image/342697.jpg'>

<img src='content_image/342698.jpg'>

<img src='content_image/342699.jpg'>

<img src='content_image/342696.jpg'>

with the general solution ( d$_{1}$ and d$_{2}$ constants): 

<img src='content_image/342695.jpg'>

Let us briefly comment on the change of variables chosen in the negative region x ∈ [ −∞ , 0]: y = − e − a ( x + L ) and in the positive region x ∈ [0 , + ∞ ]: 1 /z = 1 + e a ( x − L $^{)}$. This implies clearly that y ∈ [ −∞ , − e − $^{aL}$] and z ∈ [(1 + e − $^{aL}$) − $^{1}$, 0] (which reduce to y ∈ [ −∞ , 0] and z ∈ [1 , 0] with the assumption aL ≫ 1). The reader might worry that the choice in the negative region is not appropriate because then the Hypergeometric series $_{2}$F$_{1}$ ( a, b, c ; y ) would not be convergent 

