A Generalized Hypergeometric Function. 175 



where f(n) is independent of t and the summation is taken 

 over the various values of t, which make 



(t(n)*=z 9 



the series y 1 — l+ — -{ — |... being obtained by solving- in 



descending powers of t, the equation resulting from the 

 equation 



{^ + t^x){^ + t + na 1 -x)...{^ + t + nu n _ 1 -w)y 1 =t n y l) 



by the substitution of the appropriate value for oc. It is 

 evident that the coefficients a 1? a 2 ... are independent of 

 the particular nth root of z chosen. Clearly the relation 

 between the coefficients a 1? a 2 .. will involve more than two 

 consecutive ones ; we shall therefore not attempt to find any 

 but the first two. 



The equation may be written in the form 



yi(^ + i«i.«*" 1 + i«a<"" 1 ...+ia»-i* + 1 a ll ) + ^ 1 )( a a 1 «»- 1 



+ 2 a 2 ^-^..+ 2 an) + ...+^- 1 (yi)( w - X a«-i^4 ?l -iaJ 



+ ^(y 1 )=i>i, 



and the coefficients r a s can be found by induction and 

 inspection. In order to find only the first few coefficients 

 in the series of descending powers of v, only a selection of 

 the r a s need be deter mined. 



Equating the coefficients of t n ~\ t n ~ 2 , t n ~' s , on either side 

 to each other, we get the equations 



iai=0, 



!«! a x + x a 2 — 2 a i a i = 0, 



ia^u 2 + 1 a 2 o('\-{-ia z — ( 2 a 2 «i4 2 2 %a 2 ) + 3aj«i = 0. 



These equations determine x, a l and u 2 . As to the value 

 of the r a s coefficients involved, it is evident that 



_ n(n-l) 



% a l — n t 3 a l — 9 5--- 



lhe value or Y a x is -*— ^ =2,, where 2, represents the sum 



* 1 ' r 



of the terms 



—x, n^ — x, ... na n _ l — #, 



taken r at a time. Since iCLi = this gives 



•07= 2, a 



