2J9 



n . /ƒ* (O - («-1) . //,^5i (O = //f ^'^0 . . - (10) 

 Let 



»»=",+«, + ••• + «11 ; »~i = «0 = 0; . . . (11) 

 then we define: 



/y,f ^ («) = ^i'^ ii k>l, n>-\ (12) 



s„ =^1"-' if n> - 1 . (12a) 



ö\^' = n . [aT - ^4[,^i] if /c > O, „ > O . . . . (13) 



Vvom (10), (i'2) and (13) we deduce: 

 P'inaliy we define 



Ou ^= An — A„^i it /: > 1 (14) 



,fk (^•) = 2 [^If - ^Ifli] . *•" if A- > O . . . . (15) 

 tlius 



co 



,fQ(,-)=2: a„.v" ■ . . (16a) 



1 



We prove the following identities:') 



Ak-\-l) 



fin [o J =: ö„ -j (16) 



71 



<(), (•<•) -f (1 - •^) . '/;. (•^) = 7 ■ 'f k-i (-"^ • • • • <' ^) 



(l-.i^.,f-(a^) = 2[ö%-a'J'^].x" .... (18) 

 " o 



Proof of (16). 

 üy (14) we liave : 



<h, + rt..-l + ■ . : + «1 = 



^ [^i' -"'- ^Lli] + . . . + \Af~'^ - 4*] + [^'-'\ 



= [.4'/-^^+ .vr^' + ... + ^'^-^'] - [4f' + ^'^^' + . . . + 4i\] 



=:n.^* -(«-!). ^!l+/^ 

 = n . An - n . An^l -r An-i 



lience : 



') We tacitly assume tliat the power series p, and pk-i ^""^ convergent if 

 — 1 < j: < -(- 1 : in our applications this will be the case. 



15 

 Proceedings Royal Acad. Anasterdara. Vol. XXVI. 



