258 THE REV. F. H. JACKSON 



§ 4. Interpretation of A - ". 



If in any function of x we substitute qx for x, let the operation be denoted by a 



a ... d 



prefixed symbol Q, thus Q = q x <ix in terms of the differential operator x-=- • 



so that 



Q-tf*> -/(£)■ 



Consider now the equation 



[VI ' 

 4>(q"x)-[n]<P(q"- 1 x) + .... + ( - l)y ( '~" r i, [ iM '/'"'- 1 -) + • ■ • ■ (-l)Y"" ,_,, *(a;) 



A"*(a-) = -, - L ?, J-L M - ,,- J- . . (10) 



(q n x - (f x) (q 2 x - qx)((/x - x) 



In terms of the operating symbol Q, this may be written 



\Q n -[nW- 1 + q^k 1 ~-\ n - i - (-lJV*-*'!^) 



A' ! d>(V) = I LfJ: ' (in 



\ J ( 2 4»(»-«( 2 - 1)V " ' V ' 



The operating function 



| cy-MQ*-^ (-1)^-1,/. [»]i q»-'-+ (-i)v ( "- 1,/2 Q° | , 



may be expressed by the (/-Binomial theorem in the form 



(Q-1)(Q-<Z)(Q-? 2 ) (Q -'/'-'), 



since Q is an operative symbol obeying the laws 



Q*.Q\u = Q m+n .u=Q n .Q, m u, 



0,(11 + v) = Q.m + Q.y , 



therefore we write 



A « M _ (Q -i)(Q-tf)(Q-g 2 ) (Q -g-> ( 12) 



Now in the case of the " q " Binomial series, 



(I - a;)(l - qx)( 1 - q*x) . . . ( 1 - <f-hc) = 1 - [n]x + y^[" ~ ^x- - gsML" - ^I" " 2 ] x 3 + ( _ i f^n-m^ ( 



if we denote the product by (l - x)„, it is found that the equation 



(l-^-l-^ + jfcfcllLi- 



is valid for negative and non-integral values of n, provided (l — x) n be suitably 



interpreted from the following index relations, 



( 1 - *)„( 1 - q n x) m = (1 - x) n+m = ( 1 - *),„( 1 - q m x) n , 

 whence 



(l-^'x),, - 1 - tf-arXl - r/-" +1 a;) ... (1 - 2 _! k)(l - jf 1 *) ' 



= 1 -[-»> + f/ " "J[ " " ~ ^a 2 - ad inf., 



which result may be applied in the case of the operative function (12) as follows 



