136 Dr Kienast, Extensions of 



Thus we have established 



Theorem lO. //' l.a^x^ has unity as radius of convergence, 

 then 



S ancc" = t c. 



1 A+K=p ' 



00 



M = l 



.(f^) ^," 4. V 1 ,,(P) ,..« 



(13) 



4. Equation (13) has now to be considered when .r->l. To 

 the first terms on the right-hand side we apply Theorem 6, which 

 gives 



00 



2(n + l)...(n + x-2)r;;)^_^^-'^ ^» 



hm . = lim -Jl±h=l_ , 



T^^— jyi 2 (/i + 1) . . . (h +X- 1)^'* 



Again, by Theorem 6, 



CO 1 -. 



lim (1 - ^;) 2 -4t ^'I^^'" = lim - r^"* : 

 and finally Theorem 8 gives 



Theorem 11. If ta^x" has unity as radius of convergence, 

 and if 



\im~rl^^ = 0, 



J » 00 1 



then hm 2 a^ x" = lim 2 - j-^"' x" 



x^l 1 0,-^! p «(/?+!) " 



5. Furthermore, equations (7) and (12) lead to 



«'' 1 , , , , m-l n 



'"•"1 1 m—l 1 1 



P w + 1 " p 7z(w+l) ■" ■ ^m ^'^ •'' • 



Putting X = 1, it follows that 



-2 ^ 7^^'"^= 2 L^,>+i» , 1 ,.(p+i) 



