THE INDEX SYMBOL IN THE CALCULUS OF OPEEATIONS. 
31 
And in the same way as in the case of the ^^s, it may be shown that 
v(v,—(5y).. (v„—7nf3JY 
= 2{(as^+/3Xai£^-f /3i) .. 1) .. 
Again, for the intermediate terms of the system, 
u{v, — /3i )V = 2 { (a + /3a^)(w — ^) [(« + 1 )«! «,+ 1 + (^ — 1 )/3, a J } 
v{Ui-a,) = 2 { (a^ + — ?■ — l)a, (w — ^ + 1 )/3i «;_ , ] } 
Hence 
|w(vi— — a,)}V 
= 2 2 1 (aa'^' ■+ j3)(a 1 + 1 . 
Again, 
{ ^(I’l — /3, ) + v(lli — a ,) } (^f2 — 
=:22|(aa^+/3)(a,+/3ia~*)(w— ^)^■[(w— ^ — 2)a2«;+(^i— ? + l)/32«,_,]}^’”“*y, 
= 2 { (aa* + /3)(a. + )(w — 
Hence 
u, — a,)(V2 — 2/32) + )(?^ 2— 2^2 ) + — «i)K — 2^2) 
— -A 
= 32|(aa‘**+/3)(a,+/3ja ‘^‘)[(n—i)?cc.^a2(n—i—l)-\-(n~i-j-l)(n—i)j32ai_t(i—l)]jx’‘~’f 
r A ' _1 _A n 
= 32 1 (aa*' -3-i3)(a, H-i3,a *)(w — ?)(w — ? — 1 | ^”“*3/'. 
Again, for the general term, let 
_A _A A A 
„X^_i(a4-/3a (a„_p+/3„_pa'‘?‘)(a^-p+,£''*+^^-;,+,)..(a^_,£‘''+i3,„_,)(w—^).-(w—«’-w+iJ )?••(«— + 
m 
m p-i 1 ^-^dx"^-PdyP 
_A ® A A 
>V>(“ + ^^ '"*■)•• (“m-p-l +/3,„_;,_,r*)(a„,_pa''* +^m-p)”(«m-l — *) • • 0^ — « 
Also let 
K=(a4-/3a *■) .. («OT-p+/3;„_pa +/3m-p+i) •• (“m-iS*'+/3,„_,). 
F 2 
