Algebraic Theory of Kapteyn Series 



61 



The system (9) will be supposed to be supplemented by a preliminary 

 formula 



v + 2 



v.p-\-l 



-1. 



v+l.j' + 2 







forming the first sub-system. What have hitherto been called the 1st, 

 2nd, 3rd, .... sub-systems will now rank as the 2nd, 3rd, 4th, .... 

 sub-systems. The system (9) so extended can be represented by 



Xl= 0; 



Xl = 0, Xl = 0; 



Xl = 0, Xl = 0, Xl = 0; 



xl = 0, Xl = 0, xl=o, xt=o 



(17) 



These X's are interconnected (see Nielsen, I.e., p. 301). 

 Nielsen has shown, by a very elegant method, that 



XZ={p+2nrx:-\ X:-' = {u + 2nyx:-^ etc., 



so that Z;j= (i/+2w)^""2 xl . By proving that the first member Xl of 

 the X„ — sub-system in (17) vanishes, he proves at the same time that 

 X„^ , XJ, .... up to J¥^ also vanish. 



The character of his analysis can be appreciated most readily by 

 considering a particular example. Suppose, for instance, that we desire 

 to know a relation connecting Xl and Xl with Xl. We have 



Xt = 



v.v-\-l.v + 2.v-\-3 



- 3 



(^ + 2)^ 



j/.v-f 1.V + 2.J/ + 3 



- - 3 



V-\-l.V + 2.V-\-3.V+4: 



v-{-S.v+4:.v-^5.v-i-Q 



+ 3 



(i'-\-4:y 



v-h2.v-\'3.v-\-4:.v-\-5 



+ 3 



(»'+4)^ 



j'+l.j' + 2.j'+3.i'+4 u-\-2.v+S.v-\-4:.v-\-5 

 (j'+6)3 



v-\-3.v-{-4.v-\-5.v-{-Q 



Multiply Xs by (j/+6)-, where v+Q is the last number of the set 

 V, v-\-2, v+4, . . . that occurs in Xl, and subtract Xl from the product. 

 Hence 



