( 351 ) 



It is rational to put in the integral of Kummer 



p z=z n — e, 

 assuming n to be an integer and 8 an arbitrary infinitesimal, and 

 then to determine the limit for 6 = 0. 



Let us therefore examine the limit of 

 Un-,= r(n + l— f)/(— w-1-6, ^>^)+ r(— n — l+8)62«+2-2ey(;i^2-f,6') 

 for « = 0. 

 Suppose 



r(n-|-l— 6) — A,-\-A,s-{-A^a^... 



ƒ (_,,-!_,, b^) = ?^ + B,-{-B,eJr-" 

 s 



r(_n-l+6) = ^ + C, + C,8 + .. 



8 



f{n-^2-s, b') = E,^E,B^E,B'-\-, 

 then 



and the limit 



U, = A,B, + A,B, + C,D,E, -\- C,D,E, + C,D,E,, 

 for we shall see that 



A,B, + C,D,E, = 0. 

 Let us now determine the various coefficients. 

 First we have 



r'(H-\-i) 

 r(n+i-E) = r(n4-i) - B r(n+i) ; "[ ; + . . . 



r(w-j-i) 



or if we put 



r(.r) ^^ ^ 



r(n + l-6) = n! [l-8ip (« + 1) +...], 

 thus 



A, = n! , 



^1 = — ?Jifj(«-[-l). 



To find ^0 and B^ we write 



1 * , J2n+2+2s 



"^ Ts=.o(/i+«+i)/(-^^+6)(-«+i+f)...(-i+f)(i+4-.(^+f) 



If 



