ON 9-FUNCTIONS AND A CERTAIN DIFFERENCE OPERATOR. 257 



has a solution 



y ~ X [2][2n-lf X + ln]{) 



analogous to Legendre's series P„(a;). 



In the case of the A operator we have 



qx 2 A 2 y-^ 2 y+{l-[n]-[-n-\]}xAy + [n][-n-l]y = 0, . . (8) 



with a solution 



y - X [2J[2^T] + 



Of course, when we put q = 1, the right side of (7) vanishes, and both equations (7) and 

 (8) become identical with Legendre's equation. The A-equation is, however, an exact 

 parallel of Legendre's differential equation, and any transformation of Legendre's 

 equation can be at once followed by similar transformations of (8). The presence of the 

 two functions on the right side of (7) prevents our doing this in the case of (7). More- 

 over, the A operation can be applied at once to product-functions, etc., which is quite 

 impossible in the case of the operations denoted D ( '°. 



§ 3. Elementary Properties of the Operator A. 



So long as we confine ourselves to the direct operations denoted by A" [n a positive 

 integer), it is clear that A as an operative symbol obeys the laws 



A m {A"$(a;)} = A"' + "{$(a;)} = A n {A'"$(a;)} 



A(m + v) = Am + Aw 



Aaw = aAw . 



Also in analogy with 



d , > dv , du , d ( u\ I da do | „ 



—(uv) = w— + v— and — - = { v— + u-— } -=- v 2 

 dx dx dx dx\ v J I dx ax ) 



we have 



AU X V X = V qx &U x + Uz&Vz 



A ^ _ vAu x - u x Av x ^ 

 v x v qx v x 



We note the following special cases : — 



A. a:" = [?<]x' !_1 , with analogies D.<" = nx n ~ x . 



A"E 9 (a.c) = a"E a (aa:) , D" Exp. (ax) = a n Exp. (ax). 



. , log a \ tm 1 



A log x = — 2_ji . _ ; JJ log x = — . 



q- 1 x x 



A(z- \)(x-q) . . . (x-q n - l ) = [n](;x-\)(x-q) . . . (x-q'" 2 ), B(x - 1)" = n(x - \f~\ 



A~E q (ax).x n = q n x n dE q (ax) + \n\x n ~^ q (ax) , De"V = ax"e ax + nx n ~ 1 e az . 



From which formulae corresponding to those of integration by parts may be found 



and (/-analogies of e n or n dx = T(n) formed, which I leave to a supplementary paper. 



