( 806 ) 



K = k,t — 



clx 



satisfies the condition (8) 



I ,^'"+2 -^!^ dx = I A-"» f/.f, 7n <[ n 



j dx J dx 



which is easil}' proved by partial integration. 

 Therefore the series discussed here : 



n ■= 2 An Rn-^2 n =z . I . 2 . . . 



might also (but for a constant factor) be written thus : 



w = {x' — 1) 2 An -^^ « = 0.1.2 . . . 

 dx 



The calculation of the ^-constants is based upon the evident 

 property of the R functions that : 



ƒ' 



/?„_^2 R'm d-^ = 0, m different from ii 

 hence 



An^= ^ I It R'n dx 

 where : 



ji-l = lRn+-2 R'n (J'V = lRn+2 -V'' dx = j ,1'« {x' — 1) R'n dx z= 



or, by (11) 



/^— 1 = I X" {x^ — 1) !— dx 



n-{-lJ dx 



From the diff. equation of the 7^-function follows : 



1 \{x^ - 1) ^"+^1 = (n + 2) {n + 1) Qn+i 

 dx |_ dv J 



thence : 



^^=-~^-- \xn+lQ,^,dx 

 n + lj 



or by (8) : 



22n+i (n -f 2)/ 71.' n! n! 



f-i — 



(2n + 3) (2n + 1)/ (2n + 1).' 

 and An is calculated by the expression : 



An = ti 



nin—l) w(w— l)(«-2)(n-3) 1 



f*"-' + n . J.^wl .( f*"-'' - - etc. (12) 



2.(2n + l)' ' 2.4.(2n + l)(2n — 1) 



The negative sign of i? is due to our having chosen as geaeral 



