( 805 ) 



^ + '-1 + ^i + ...-^ = 



(n-}-3)(7i + l)^(«-hl)(«— 1)^ (« — l)(n-3) 3.1 



-] ^ 1 ^ \- . .. — = \ 71 even 



{n-\-b){n-\-d) ' (M + 3)(yi + l) ' (n+l)(w— 1) 5.3 



1 



(2n + l)(2n-l) ' (2n-l)(2n-3) (2«-3)(2w-5) (n + l)(n-l) 

 and 



^0 



(n + 4)(n + 2) (n4-2)(.0 ■ (n) (n-2) 5.3 



-1 ' i- \- . ^^ = \ /^ odd 



(n + 6)(n + 4) (n + 4)(w + 2) (?i-l-2)(n) 7.5 



(2??-fl)(2?^-l) ' (2M-l)(2n-3) ' (2w-3)(2n-5) (?« + 2)(w) 



By successive elimination of a,,, a^ . . . a^, a^ . . . we find from these 

 equations for the general form of the R functions : 



(n + 2)(n + l) „ , 0/ + 2)(n + l)(»)(n- l)^^^_^ 

 ^' 2.(2n + l) ^ 2 .4.(2M + l)(2n-l) ^^ 



and from this expression by dividing it by x'^ — 1 : 



^, 7i(n — l) n in — 1) (n — 2) \n — 3) 



U'n — .i.'" ^ A-"-'-^ H ^ — .t"'-'i — etc. . (10) 



2.(27rfl) ^ 2.4.(2»?-fl) (2n-l) ^ ^ 



The recurrent formula for bolli R and R' is : 



n(n 4-2) 



i?'„+, — w R'n A -^^—— R'n -1 = 



. ^ ^(2« + 3)(2?i-fl) 



and the functions are solutions of the diff. equations 



{.v^ - 1) --4- - {n + 2) {n i- 1) Rn^o = 

 ax' 



(x^-l) — ^ + A.v ~ (n-f 3) n En = 0. 



diV' d.v 



On comparing the expression for R'n with that for Q» it is readily 

 seen that the R' functions may be found by differentiation of the 

 (3n+i -function, so that: 



■n + l dx ^ ^ 



This might have been expected as the value: 



