Algebraic Theory of Kapteyn Series 55 



These formulae can be verified by direct analysis, but we propose to 

 consider them as special cases of more general formulae, first remarking 

 that the two sets (7), (8), together with the supplementary formulae 

 just considered, can be blended into a more comprehensive system, for 

 which it is sufficient to write down in each case the first two terms. 

 This system is 



(i) 



• I -2 ^±^ ... =0 



v.v+\.v + 2 v-\-l.v-{-2.v+2, 



" _2 (- + ^)^ ... =0 



(ii) 



V . v + 2 



l/.V+l. 1^+2.^ + 3 v+\.v-^2.v + Z.v-\-4: 



= 



< 



-3 <-+^)' ... =0 



j,v-\-l.v+2.v-\-S v-\-l.v+2.v-]-d.v+4: 



!: -3 )!^:Lfi ... =0 W9) 



V , v-\-2 



p.v-]-l.v-{-2.v-\-3.v-^^ v+l.v+2.v-\-3.v-\-4:.v+5 



= 



..3 (v+2y 



-4 ^--^^ . . . =0 



(iii) i 



v.v+l.v +2.v+3.j'+4 V -{-l.v+2.v+3.v-\-4.v-\-5 



v' ^ {u-\-2Y 



v.v-\-l.v +2.j/+3.i'+4 J/ +1.J/+2.V+3.V+4.I/+5 



= 

 = 



v.v+l.v +2.1' -\-S.v+4: i'+l.j' + 2.i^+3.i/+4.v+5 



Two questions which suggest themselves immediately are: 

 (i) Why are the numerators of (9) always odd powers of v, v-\-2, 

 v+4, . . .'? 



(ii) Can the system (9) be still further extended by introducing 

 higher powers of v, v-\-2, v-\-4, . . . than those actually employed in the 

 sub-systems of 2, 3, 4, . . . formulae respectively? 



Let us first generalize the expressions on the left-hand side of the 

 first sub-system of (9) by replacing them by 



-2 -^ISIJ^ + -}^!-Lll (10) 



v.v + l.v-^2 V -\-l.v-{-2.v-\-d J/ +2.Z/ -f 3.j'+4 



where w = 0, 1, 2, 3, . . , . The expression, on reduction to a common 



