( 659 ) 



,/• an<l V Itccomo iiilojïors, say .r=z(ti). i/=zf{ D, wliore (c mul ,^ are 

 |irinu' lo earli olliei', we liave 



~ = 2 ^ï" 7' — = 2 1) :£ /' - ] = -lJ) J' (O) — n. 

 y=o V •<■ / y=o V '^ / 



III a soiiK'W hal (liircvciil form tliis result is found in a |>a|ior l)y 



Stkhn M- a ^vllole scl of t'inictioiis ol' liie re(|iiii'e(l kind may l>o 



dcdneed in (|uitc tlic same way. Wc uidy have to iiutiee that the 



fiiiiclioji 



/';.H= :^'" — — ■ — , (^■>i) 



satisfies the Cnndamental rehition 



7=0 V «/ y=0 \ (t J 



wliere ajiaiji <c and ,i are prime lo each other. 

 Hence if we [uit 





(II) 



we ,Lret for ./' = a D, y = ,■? /9 



//=U V '^^ / «=o V ''f / 



that is 



In the functional i-eIalion (II) the lei-m l'\{ 'J is not easily 



e\aliial(Ml ; hence the series ]'\{ii) may lie suitably re|)lace(I hy the 

 laltcM' of" the Iwo series 





„^1 (2rr»)-^-i 



./•>/, (/0 = 2(-iy^~-i ^ 



.=.1 (2.t;/)2/^- 

 Indeed, if we denote the nernoidlian [)olyiiomial of order /// 



/<'«+l 1 !("' B^ /<"' -I B., . il'n- -3 



■ ' r» I ' / 1 \ I i I ' I I > \ I 1^ . . . . » 



(m-f-1)! 2 w' 2! (m — 1)! 4' {m-'6)\ 



l>y ƒ,«('/) the series //,« (//) is identical with /",a{ii) ^oi" all \alues of 

 // between zero and unity. Therefore, whatever may he //, the series 

 llm{if) nii\v he i-egarded as a jjolyiiomial of the ///''' degree in 



') '•Zui' Theorie der Function A' (./:)." Joiunal f. Malli., I0:>. p. '.i. 



45 

 Proceedings Koyul Acad. Anisleiiiiun. Vol. V. 



