186 THE ROYAL SOCIETY OF CANADA 



In terms of the ^-function, Gegenbauer's formula becomes 

 R v f 2.v sin - ) = r („) 2 ( D+ 5 ) K,, s (cos 6) (^J 



^i>+it~ t^w RV} s ( A) ■ 



The expression on the left-hand side is, after multiplication by T(v-\-l), 

 the power-series 



sin-— sin 4 — sin 6 — 



1- t— TT^ 2 + ! xi ~ A c +...; (I) 



1.0+1 1.2.»+l.i>+2 1.2.3.0+1. w+2.»+3 



that on the right-hand side can be converted into a power-series by 

 using Schônholzer's expansion for J\ (x) in the form 



" u; ,:„ r 2 0<+/+i) V / /V2/ ! (( + 1) ' 



where ( , ) stands for the binomial coefficient „C r . In the first 

 place we get the double series 



( - 1)' /2!' + 2s + 2\ / ,v \ 2 '+ 2 

 + (»+l)^iï,P(, +J +2) C I )w 



(-1)' /2»+2*+4\ /*\ 



+»+2)g„ 2s s r. ( , +1+8) ( , ;(t; 



: ] 



t . \2H4 



+ 



which, in turn, gives us the power series 



rdor(*+i) \vK.,o pApj) 



,2 (o + 2) 



[ 



+(i) , {c+Mo-.(*r)«..} P Ji 



+ (f y |(^+2)x t ,. 2 -u-+i)^ 2ï, + 4 ^^.i + 



] 



