ON 2-FUNCTIONS AND A CERTAIN DIFFERENCE OPERATOR. 255 



We proceed as follows by defining an operation A<fi(x) as the quotient of two 

 differences. 



§ 1. Forms of A n . 

 AWs)}-*^) -*<*>, (1) 



qx-x 

 We see that in the limiting case, q= 1, this equation is D {$(«)} = &'(x), provided that 

 <£(;») has a first derivative &(x). Repeating the operation symbolised by A, we obtain 



as{<*>(*)} = ^ x) ~ frjfo^ + <**<*> .... (2) 



(q 2 x - qx)(qx - x) 

 and, by induction it is easy to establish 



A«{$( a -)}= 4>( ' /a;) ~ w * ( ' / "" lx)+ • • • • ( - 1 ) r g r(r - i,/2 M ,[J_: r]! *(g"- r *) + • • • • (-i)y i, "- i,/2 «>(^ (3) 



(q n x - q n ~ 1 x)(q n ~ 1 x - q n ~-x) (q 2 x- qx)(qx - x) 



in which, as in previous papers, # [n] denotes (q n —l)/(q—\). 



In this expression the coefficients of the functions <fi(q n ~ r x) follow what may be 

 termed the g-Binomial form, that is, they are identical with the coefficients in the 

 equation 



(x+!/)(x + qy) .... (x + q"~ l y) = x n +[n\z"-hj + .... + W g *»-i>g"-y + , (4) 



By means of the operator A , the connection between the A -equations satisfied by 

 certain g-series functions of x and the linear differential equations satisfied by the 

 simple (q = 1) forms of such series is made clear, for from the equation 



*.£ 



a r/f./ \i <&(qx)-®(x) . q dx 



A{4>(a:)} = v * ' ±-' , we see x.k. — ± — 



qx — x q-\ 



also that with certain restrictions on the nature of <p(x) we may write 



i -»=x: + e- i >i ! s + <*- i >f 2 s + • • • ■ < 5 > 



d 

 and that xA is related to the differential operator x-p in a manner analogous to the 



relation between n and its g-form (q 11 - l)/(q - 1). 



§ 2. Basic Exponential and Transformation of an Arbitrary Power Series. 



The following simple example will suffice at this stage for illustration of the connec- 



d 

 tion between the ^-operator A and the differential operator -=- . 



Compare two equations 



I- W 



* Trans. Roy. Soc. Edin., vol. xli. pp. 1-28, 105-118, 399-408. 



