Abel's Theorem and 'its converses 



135 



But it is not difficult to see that the recurrence formulae (8) and 

 (9) still hold, if the number p of steps exceeds the index n. It is 

 only necessary to put ?-^" = whenever n > m. The form of the 

 relations (7), viz. 



is the cause why the coefficients of the remaining terms are not 

 influenced by the fact that some terms disappear. Thus 



'I'' A, K A, K, n 



1 A+K=p \n=\ 



+ S -a;^{7>'_7» I (11). 



n ' » n-V V ^ 



n=p "• 



To evaluate the first of these sums we have 



(A+k) 



(1 - xf s (« + 1 ) . . . {n + X - 2) r;;;;:, ^" 



n = \ 



in 



,»H+1' 



A, (C, 7J 



\,K,Vl + V ' 



say. Each of the i\(\ + l) terms contained in the second sum 

 has the form 



K{p + i)(p + 2)...{p + x-2)rl^;;!y+', 



{fi= V, V + 1, ...\; V = 1, 2, ...X; p = m — v + 1). 



Therefore, by Theorem 8, we have, for every j x \ < 1 and any 

 finite X, 



limS G,^^^^.r"^(l-a^^{n + l){n+2)...{n+X-2)rl^^;;^^x'\ 



«4-*-» 1 1 



The second sum in (11) gives 



'"- 1 . N , \ U-1 -\ 



vif,.(p)_^,.W }./= S r^' 



(p) 



n + 1 



;i -'^ ^ (p) m 



m «' 



= {\-x) S - - 7>' A'" + S ~ , 7>* X'' + i 7>^ .^'" 



p II + \ " p II (v + 1) " m '« 



(12); 



and again, by Theorem 8, we have, for every , x j < 1 and any 

 finite p. 



lim % x' {r^'^ - 9>\} = (1 ^ X) 2 — ^ 7>\*'" + ^—^ 

 m-» p *'' " ""' p» + l " ,n(n- 



+ 1) " 



,.(p) _.,."_ 



