[harkness] GENERALIZED LEGENDRIAN COEFFICIENTS 187 



The general term within the brackets [ ] is 



The comparison of the coefficients in (I), (II) shows that 

 1 =K v ,o , 



a I 



2 2 sin 2 —- = 2K v ,o — — K v ,\ , 



1 _ 2»+3 2 1 ~ 



— 7 2 4 sin 1 — = — — - K v ,o — — A„,i H ; — TTT A»,2 , 



2 ! :. » + 1 v v{v+l) 



etc. 

 The general form for these identities is 



1 09I . ,, 6 1 r /2v+2s\ j. 



TT^ sm " T= .(.+1) (r+2) :..(•+,) I 'V , ; Aî "°- 



when »==§, the identity becomes 



^^^4-i53^[(»4i>'--»(?- + i> + 



5 C- + 2 >- ■]• 



We shall now proceed to show that Nielsen's identity is based on 

 the formulae that express the powers of sin 2 linearly in terms of the 

 generalized Legendrian coefficients. 



Expand both sides of 



2?„_ | (x sin 6) _ 1 x (v+2s) T(s + ^)T(v) 



^r- v 



(t) ! 



iC r ., (cos d) 



Rv+2s (A*) 



r(z-+25+i 



power-series in f — J and equate coefficients. After simple re- 

 :orresponding identities are 



ductions the corresponding identities are 

 1 = Kv,o 



sin 2 6 = T(v) 



