content_image/336685.jpg content_image/336684.jpg content_image/336683.jpg content_image/336681.jpg content_image/336682.jpg
integral transformations: 

<img src='content_image/336685.jpg'>

with $^{P}$− $_{n}$( χ ) ≡ $^{P$_{n}$}$− $_{1}$( χ ). Note, that in (A11) we employ the standard notations adopted in [15, 22]. Namely, x$_{s}$ = 2 x − ξs (1 s $^{2}$) $_{y}$, s$_{i}$ , ( i = 1 , ... 4) stand for the four roots of the equation 2 x$_{s}$ − x 2 s − ξ 2 = 0 given by the following expressions: 

where 

<img src='content_image/336684.jpg'>

<img src='content_image/336683.jpg'>

y$_{0}$ and 1 y$_{1}$ are the two solutions of the equation s$_{1}$ = s$_{2}$ ; while y$_{1}$ and 1 y$_{0}$ are the two solutions of the equation s$_{3}$ = s$_{4}$ ; 

<img src='content_image/336681.jpg'>

<img src='content_image/336682.jpg'>

