= 

 = 

 = 

 = 



Algebraic Theory of Kapteyn Series 



(»'+2)« 



53 



f+l 



1! 



72 



72+ — — — 74 



2!j'+1.j'+2 i'+3 2! 



»^^ {v+2Y 



3!j'+1.z^+2.j'+3 2!1!j^+3.i'+4 



, (.+4)4 (.+6)4 



72+— — ; 74— : — 76 



1!2!j'+5 



3! 



' + 2)f 



41.+ 1..+2..+3.V+4 3! l!.+3.i'+4.j'+5 



72 + 



■74- 



2!2!j'+5..+6 1!3!j'+7 



The solutions of these equations are 



1 1 1 



(.+6)« , (.+8)« 



76+ —— 78 



(5) 



4! 



72 = 



74 = 



76 = 



78^ 



1 



.+ 1 .+2.i'+3 ■" i'+3.i'+4.i'+5 ■ " .+4.i/+5.i'+6.j' + 7 



If these values are inserted in (5) the resulting equations can be written 

 in the form 



.+3 

 -2(. + 2) 

 + (j'+l) 



(^4-4)2 = 



I/+4..+5 

 -3. .+2. .+5 

 +3.J/+1..+4 

 - .+ 1..+2 



(f+4)4 

 (^+6)4 = 



(6) 



V+5..+6..+7 

 -4..+2.v+6.i'+7 

 +6..+ 1..+4..+7 

 -4..+ l.j'+2..+6 

 + .+ 1.V+2..+3 



{v-{-2Y 

 (»'+4)« 

 (^+6)« 

 (,.+8)« = 



It is to be remarked that the sum to the left of a vertical line is in each 

 case zero. These sums can be transformed into 



. + 2 J/+4 



- 2 



V. v-\-\.v-\-2 



V 



i'.. + l.j' + 2..+3 



+ 



= 



.+ 1.. + 2..+3 V + 2..+3..+4 



v + 2 , ^ .+4 



V+1.. + 2..+3..+4 



.+6 

 j'+3.j'+4..+5..+6 



f + 2 



+ 3 



. + 2..+3.j'+4.i/+5 



= 



I (7) 



..j/+l.. + 2..+3.»'+4 

 .+4 



6 



"* .+ 1.J/ + 2..+3..+4..+5 "^ 

 1/ + 6 .+8 



. + 2. ...+6 



.+3 . . . . + 7 



+ 



.+4 



.+8 



= 



