205 



The ciimntities S^''\ Ti*' and IT^(^' are now also defined. The 

 following identity will be satisfied for p — i>l: 



S(f) T(<i) + Sii') Til) -i-... + S(p) 7'(9) = S'i'->) Tii+>) + . . . + *-07'(?+') (4) 



1 n ' 2 n—1 ' ' ri 1 1 n ' ' u 1 ^ ' 



This follows by induction: 



Sip) Til) 4- SW Til) + . . . f Sii>) Tig) — 



1 H ' 2 »— 1 ' ' n i 



= 50'-i)7'(9) + (50—1) 1 s(/--ij) Til) + ... + (S'l'-^^ +... + Sf/'-i))T(?>=: 

 = 5(/>-i)(7'(v) + ... -f T(?)) + 5(/'-i)(7'(9) + ... -[- TW)) 4- ... + 50'~i) 7'W) = 

 = a^-i)T(9+i) + «(z*-!) 7'(9+0 4- ... 4- S0'-«)7'(v+i). 



1 li ' 2 n—1 ' fl 1 



We also prove with induction: 



W(i'+'/) = Sip) Til) + S(p) Til) + . . . + Sir') Til) ... (5) 



n 1 n ' 2 n — 1 ' ' // 1 ^ ' 



for we have by (5) : 



Wip+<i+^)= Wip+1) + Wip+<i) + . . . + Wip+i) 



n n ' n—1 ' ' 1 



"^ 1 n ' ■ .1 1 J ' ' 1 ,1 I ' ' n -1 1 ' ' ' 1 1 



= Sip)\Ti<t) + .. . + Tii)] + Sip) \Ti</) +... + 7'(9;] +.._A- Sip)TW= 



I '- u ' ' 1 ' ' 2 L n—1 ' ' 1 J ' ' n 1 



= S(p)Ti9+^) + Sip)Tii+^) -i- ... Jr Sip)Tii+^) (5«) 



In 2 n — 1 n 1 



Finally we deduce from : » 



PF(/'+i) — S(i)7'(/') + 5(i)7'(;^) -f . .. -)- 5;(i)7'(^). . . (6) 



ti 1 n ' 2 n-l ' ' n 1 ^ ' 



with the aid of S'-^^i = a^ -\- az -\- . . . . -\- an 



lF(/'+i) = «1 Tip J -f (ai + ao) T(;^ + . . . + (aj f «2 + • •• + «-) Tip) 

 = ai [TO') f Tip}^ -f . . . + TO')] 4 «2 [ T<>^^ f . . . + TO')] + . . . 4-a„T0') 



or 



Tr(/'4-i)= ai Tip+i) f «2 7'(/'4-i) + . . . 4- «„ 7X/'4-i) ... (7) 



n N ji - 1 1 



The ?i''' mean-value of order p of the series «i -|- (12 + • ■ ■ • lespect- 

 ively bi -\- b'i -\- ■ ■ ■ ■ is defined as 



Sip+^) Tip+i) 



respectively 



n n 



If such a mean value of order p has a limit for ?i = 00 we call 

 the corresponding series siimmable of order p '). )iy a well-known 



') In our definition the fust term of a series has the index 1 and not zero as 

 is usually tlie case. 



14* 



