Ordinary Linear Differential Equations of 2nd Order. 609 



These are two solutions, each written in two forms in order 

 to show all cases where the series are finite. The case of 

 zero factors in the denominators remains to be examined. 



A constant has been introduced whereby simple relations 

 hold between the functions for different values of a and n ; 

 the following are for the series u^ and u z ; the sign of n 

 must be changed for the series i* 3 and w 4 . 



U /t (a-l :#) = 

 ^—^vUn'iai^-aUnia;*)] (7) 



U ?l '(<Z— 1 ; x) = 



^[(^■-«)U»V;^+(J-«)UnCa;^)] (8) 



U n (a ;./,•) = 



^r^U ? /(a-l; t iO + (a-^)U 3l (a-l;^)l (9) 



U,/(a ;a?) = 



^pU.Wi;*)+(J-«)u«(«-i;^)J (10) 



p»-l(a— i ;#)== 

 ^[ut , (a;^H-^U»(a;^)] (11) 



U' n _|(a — i ;#) = 



^i + ^)u:(.;.) + {^-3u(.;-)] (12) 



U»(a ;.<■■) = 



^Eu^.-*;.)+{J-?%^}^.-*;.)] • • .(U) 



B»+i(a ;#) = 



<n-a)(» + a + lj [< ftt + 1 ) U ^ a '' ^ l^T^ + —f ?*.»■)] • (15) 



