INFINITE SElil 



lES OF ANALYTIC FUNCTION'S. 

 is e<jual to 



J_,/_ ,\. + i _ y(y+l)...(y+n) 1 f 1 



/_ ,\. + i _ yy...yn 



(in.+i)(i*++2)...(ji+2+i)i^i?^ F -i. 



F^n+l.y + n, /, + 2it+l,l 



(27). 



Since both sides of the equation represent analytic functions for all values of y and 

 j>, unless y or p+l is a negative integer, the restriction tliat R(y) and K(y;+ 1-y) 

 are positive may now be removed. 



14. From Theorem I., we find that when is very large, 



F(-n, p + n, y, x) 

 is approximately equal to 



for all finite values of jc ; and 



is aj)proximately equal to 



rt}---i' ..... (29) 



for all finite values of t. 



The second term on the right side of (27), when n is large, is approximately 

 equal to 



/ 



)( n+ V 



'-f J 

 and vanishes or becomes infinite when increases indefinitely, according a 



- 1 + 2 yxc | $ j 2- 1 + 2 vt\ , 



