38 DR GUSTAVE PLARR ON THE 



which is also equal to zero by 



r = §(n — 1 — s) , 

 which gives 



n = 2r'+s + l . 

 Then 



Y — {3 = r' +s+| — v — n— h + v 



= r'+s + l—n 



= — r' 



e' — (3 = r'+s+l — v — n—}i + v 



= r' + s + $ — n 



= r'+s + ^ — 2r' — s — l 



= —r'-\. 



Putting F = 1 in the second member of (HI.), we get 



(-/)*/+!( _ r '_ | )»/+1 



We have thus expressed the sum S(w, v) by a product under a monomial form. We 

 will now make use again of the function Il(x) = (x!) for the sake of simplification of the 

 factorials and their products. 



We have first (as above § II.) 



(-l) u 2 2v Il(2n-2v) 



- 2 n + s +^llvll(n - vjRln - s - 2u)IL(s +u- v)Ii(u - v) ' 

 Then 



( u+£-^)° /+1 



(«-< v +iyi +1 ' 



and, as the limits of v are and u, 



We introduce again the transformation 



(2a) 26 /- 1 = 2 26 a 6 /- 1 (a-J)"/- 1 . 

 Hence 



(d) (a -F'- n2a x n ( a - 6 ) 

 W ta *J ~2 26 n(2 a -26) X Ila ' 



This gives, identifying a with w — w, 



t> (- l)"II(2w- 2u)Tl(n-u-v) U(u-v) 

 2*»II(2n-2u-2v)n(n-u) X TLu - 



Hence the product 



. R ( --_l)«+»II(2w - 2v)II(2w- 2u) B, 



a k - 2«+«n??ii(w - ioriwii(?i - u) x no - s - 2%) • 



