308 



whence the formula follows on differentiation. 

 {I J). n = m, /u :=: 



We must show that: 



(_)m 



I)[^ (r" Pn {cos &)) — ^-^ r"-"' P^„, (cos ^) . 2 cos m<p 

 Proof. Since (>' = ^^ we have 



r" P,, == i 



;, 2^P{pfy{n-2p)! 



Operating on this with Z)"' and observing that 



n / COS»— Sw-S/» ^ sin^P & 



K^^{cos»)=--sin^n^2{-)P 



"-"•^ ' 2'« y^' (n— 2m_2p)/22/'p/(m+p)/ 



the above writlen formula follows. An analogous formula holds of 

 course for the operator i>'". 



(III). m=jtO, fi^l 



Using (1^) and (II), (2j) is found. 



{IV). mz=0 



This is also verified without difficulty. 



