52 



Transactions of the Royal Canadian Institute 



By expanding the Ts on the right hand side of (1) and equating the 

 coefficients of the powers of x we are led to a series of equations (in which 

 a2n is replaced by a2„) 



1 = 



= 



r(v+i) "' 



,."+2 



(v + 2) 



v+2 



T{v + 2)''' T{v^S) "■'' 



,."+4 



(v + 2) 



v+i 



2\Tiv-hS)"'° T{p+4) 



0-2 + 



(.+4) 



y+4 



r(v+5) 



On 



(2) 



which are simplified by replacing a^ by /So , a^ by ^2, 



{v+^) 



^+4 





r(.+i) 



a4 by /34, etc. They thus take the form 



Viv+2,) 



= 



v+1 



/3o-/32 



= 



= 



2! v+l.j'+2 



„6 



ft - i^±?): ft+ft 



1! j'+S 



/3i 



(^ + 2)^ ,.+ ^^±l)!,.-,e 



3!j'+1.i^+2.v+3 2!v+3.v+4 1!j^+5 



(3) 



The first few values of the set of jS's are readily seen to be 



^0=1, ^2- 



^2 (J. + 2)'' 



. /34 = -- 



{v+^y 



. /36 = — 



{v+^y 



3! 1^+3.^+4.^+5 



1! I' + l 2! v+2. 1^+3 



They point to the general formula 



h, = ^ <-+^"^"" (4) 



w! v+w.v+w + 1 . . . .j'+2« — 1 



This formula can be established by mathematical induction (see 

 Neilsen, p. 301). We propose to study the equations (3) from another 

 point of view which will give u's an insight into the algebraic nature of 

 the foundations on which the structure of Kapteyn series rests. 



Replace ^2, by J^ (, + 25)2^-^2, , ^ = 1, 2, 3, 



and j8o by 1. The first four 7-equations are: 



