ON g-FUNCTIONS AND A CERTAIN DIFFERENCE OPERATOR. 261 



We notice that since A i»n.„_ Io §g . * 



and A n E g (ax) = a n E g (ax), 



the equation A m s/ + a x A™- V + + a m _Ay + a m y = <f>(x) 



may be solved by methods similar to those adopted in the case of differential equations 

 of like form. It is obvious that by giving solutions of such equations as 



3>(A)u = *(x), 



in which <£ is so chosen that $ _1 (A) is capable of representation in such series as 

 £, X . A", a great variety of q series and relations may be formed : for example, from 

 such equations as 



(1 - 2 A)(1 - 2 *A) (1 - q^)u =f(x) , 



so that 



« = y r nir i i q n A n f(x) . 



£-[ro -l]![n]! tf J y ' 



§ 7. Forms of A n $( x ). 

 In this section of the paper the following forms will be obtained : — 



A^(*) = 2g W "- 1, [W + r][W + r ~ 1] ' ' ' ' i [r+3][r + 2lr+1 W ^'(Q) > . . (16a) 



^ q ¥^H~ (rjl" W ' 



A "^)= i A CJMrxT 1+ + MsF + ■■■ \« x) * 



A r = i [n] n+r - [n][n - ll»+'+ ( - lVg*"'- 1 ' r ^ ! r -,, [n-sf+ r 



[ ' [n - s\\ [s\\ 



a. ( _ 1 \»/7i(»+l)(n-2)r«l I (^~ V 



+ •••■( ^2 L "Jfn + rigw*-!)' 



subject to the conditions that ^(x) and all its derivatives are finite and continuous. 

 Theorem (16a) can be derived directly from the equation 



A nj < W J U- *(g"*)-[ttXg n - 1 «)+ (-lrgW'-^a!) 



1 v * /; ( 2 n a: - g"- 1 ^)^ 71 - 1 ^ - 2"- s a:) . . . . (qx-x) ' ' 



for, on replacing <$>(q n x), &(q n ~ 1 x), by Maclaurin series, we obtain from the 



numerator an expression 



|~3>(0) + 2 "a*'(0) + + ^W M (0) + | 



- [n] *(0) + g"-W(0) + +g__a; r $" , (0)+ j 





(_l)» 2 M«-i> _H! r#(0) + 3 "- s a!*'(0)+ +2!!LV*W(0)+ i 



[?t-sJ!|sJ!L /■'• J 



(-l)Y»(»-«("*(0)+ a*'(0) + +-y *"X°) - ]• 



TRANS. ROY. SOC. EDIN., VOL. XLVI. PART II. (NO. 11). 39 



