266 THE REV. F. H. JACKSON 



In the equation 



n q (z)=-?- T b.- 1 \((i 1 + a 2 qx + a 3 qh?+ . . . + cg—V- 1 )^*) \ . . (28) 



q — 1 I / 



replacing FI 9 by the series ^c r x r we obtain 



^c,y= -V— A -1 \ (a^c + a 1 c ] x + a 1 c 2 x 2 + .... +a 1 r, m x m + ) 



+ (qa 2 c x + qa 2 c^x- + + qn 2 c m x m+l + . . . . ) 



+ ((] 2 a s c Q x 2 + q^CyX 2 -4 + q 2 a s c m x m+1 +...),. . (28a) 



+ {q n -'a n c^'- 1 + q n - l a n c^' + + q"- 1 a„c m ^ m+ "-')] ■ 



x m+ * 



Now A 1 (ax , ") = C + a— — =, (C being an arbitrary constant), therefore performing 

 the operation A _1 on the terms of this series (28a), we obtain 



1 (ft \ x , ( l 



2 /, / \ -v3 



Z c ■-'''' = c + tt\ a ^ x + Hhv iCi + ga * G vw\ + q^\v lC2 + qa -'' + ' y " ffl3 '' o y[3]r + 



and we find, by equating the coefficients of powers of x, that 



q 

 q-l 



1 



!= ?^i( ¥l+2 r 



Cr = -^-^(a 1 c r _ 1 + qa 2 c r _ 2 + q 2 a ;i c r _ 3 + + q n ' l a n c r _ n ) . . (29) 



Subject to the convergence of the series, we write therefore 



• n >i>«2> a ?( , x) = l + ^^x + ^ 2 _^ _ 1} | a* + (q - IK } 



+ (g s _ 1 K ^ l)( g - 1) { a * + <g " 1 > a i a » + (g2 " 1)a ^ + {q ~ 1)(g2 ~ 1)as } 



+ (E7&r) { «i 4 + (?- l)a 1 2 o 2 + (2 2 - ] K 2 «2 + ('i" l)('/-l)¥3 + (? 3 -l)«i 2 « 2 



+ (2-l)(2 8 -l)a 2 2 + (ff 8 -l)(2 2 -lKa 1 + (3)(2)(l)a 4 } + . (30) 



The recurrence relation connecting n + 1 successive coefficients being 



Cr = ( »• _ i ) 1 a '' v - 1 + IWr-l + 9 2 <¥V- 3 + +2"~ 1 <W-»|'j • • ( 31 ) 



the series is absolutely convergent for all values of x, in case | q | > 1, for the series is 

 seen to be the product of n absolutely convergent series formed as follows : — 



Resolve the polynomial (l+a 1 x + a 2 x 2 + . . . +a n x n ) into its factors (I + pfl) 

 (1 + p 2 x){l + Pz x) ... (I + Pn x). 



