ON ^-FUNCTIONS AND A CERTAIN DIFFERENCE OPERATOR. 259 



§ 5. Ee verse Operation A -1 with (/-Finite Integration. 



The operative function 



(Q-i)(Q-?)(Q-? 2 ) (Q-'T 1 ) 



may be annulled by 



Q-" 



(i-Q- 1 )(i-aQ- 1 Xi-fl"Q- 1 ) ■ • ■ • (i-^Q- 1 )' 

 which can be expressed in many different forms, one of which is 



<H i + (g^i) Q -i + (g"-i)(g" +1 - 1) I 



= Q-" + HQ-"- 1 + W[«+i]^^M[« + i][" + 2] Q [ 3^ 3 + , • • (13) 



so that the reverse operations indicated by the following equations will be legitimate, 

 provided that the series obtained are finite or convergent. 



§ 6. Solution of A n w = (f>(x). 



A"w = $(a;), 

 u = A~"<£(a;) , 

 u={Q- + [n]Q- n - 1 + } {gM n -V(q - l)V*(a;)} , 



= 3 w«-*'(, i -l)»|Q-« + [„]Q-»-> + W^ + l]Q-»- 3 + . . .}.*"*(*). . . (14) 



Now, since &x m = [rri]x m ~' 1 , it is plain, that just as in indefinite integration, so here 

 we may introduce an arbitrary periodical * constant of multiplicative periodicity at 

 each performance of the operation A -1 . An example of such a function, which may 

 be termed a function of multiplicative period q, is, 



... s ^ ( /27rlogx'\ , 7 ■ /27rloga:\ ) 



satisfying <f>(x) = <f>(qx) = $(.q n %) • 

 We obtain then 



u = a + ai x+ .... +^_ 1 ^ + 2 w»- 1 '(2-ir{p*(^) + W^*(^r 1 )+ } . (15) 



The connection of these forms with integration is manifest, for on writing 1 + e for 

 q, and putting n= 1, we obtain form (15), 



u = a + ai x+ . . . +«»-^- 1 + -{ (r ^ ) ^( f fJ + (T ^^( (1 ^y 2 )+ } 



and so retaining only terms involving e and neglecting e 2 , e 3 , 



u = a + a^x+ . . . . + a^xX 11 - 1 + L ^ \ex<f>(x - ex) + ex<j>(x - 2ex) + ]. 



It is easy to verify (15) in particular cases, for consider 



u=.- (a) then Au= — r (/3) 



(l-x) (\=x)(\-qx) 



* Cf. Boole, Finite Differences, pp. 46, 47. 



