Algebraic Theory of Kapteyn Series 59 



cr • r W(m— 1) ^_9 . . , , ,0/11 1 



coemcient 01 v "is without the terms v\ v- (and has the 



2! 



value 480j^); the coefficient of ^^'-^~ ' ^'"^ 1 v""-^ is without the 



3! 



term v^ (and has the value — 18v-+36f — 24). It follows that the highest 

 power of v that actually occurs is the (m — 1)*^^. In other words there 

 are no terms in 1^"'+^ , v""^^ , v'"+^ , p"". 



The coefficients of v"", mv^~^ , v^~^ , etc., are in appear- 

 ance cubic polynomials. The conditions for the vanishing of the term 

 in v^ in the first four of these cubics are given by 



1.0"-4.r+6.2"-4.3''+1.4" = (a = 0, 1, 2, 3) 

 a special case of the general formula 



1.0''-7z.r + ^^^'— i-^ .2\.. .+l.w" = (a = 0, 1,2, 3,...,«-l) (15) 

 which can be deduced immediately from the identity 



l.e°^-7ze^+ ^J^?—^e'-^- . . .+(-!)« e""=(l-0'' 



where P„ (x) is a power series beginning with the term x". 

 It may be worth while to indicate one method of proof of the vanishing 

 of the terms in v^ in the second and third cubics, because it also depends 

 on (15). 

 We have to show that 



-4(2+6+7)2''+6(H-4+7)4''-4(l+2+6)6"+(H-2+3)8" (a=l, 2) 



This may be written 



6 [-4.2" + 6.4"-4.6"+1.8"] 



+ 3[-4.2".3+6.4".2-4.6M]. 



The first line vanishes by reason of (15), the second line is zero in virtue 

 of the identity 



-4.r.3 + 6.2".2-4.3M=0 (a=l, 2) 



which can be regarded as a special case of the general identity 



in«^ ^1-^ 1 I w(w-l) n(n-l) {n -2) ^ 



l.U .n~n.\ .n — l-\- . z .n — Z— .6 .n — o 



2! 3! 



+ . . . . +l.w".0 =0 (a = l,2, . . . ,w-2). 



This can be written 



