256 THE REV. F. H. JACKSON 



On substituting 2,c,xc a+r m the first equation (a), we find, since (A^'" = [m]x m_1 ), an 

 indicial equation [a] = 0, and a function 



X [a+l] + [a+l][a+2] + 



The equation [a] = 0, viz. (q a — l)/(<7 — l) = 0, has the doubly infinite system of roots 

 « = , 2 7l . , (r = 0, 1, 2, 3, . . . .), (s = 0, 1, 2 . . . .)■ 



lOg g + 2S7T« 



From the first or principal root, which is zero, we deduce a solution 



Cf OP a* 



In connection with this function, which may also be expressed as an infinite 

 product, we note that Boole's transformation of an arbitrary power series, viz. 



= e*{a + x. Aa + |p A 2 a + |- • A 3 a + J 



possesses a remarkably simple generalisation, in which E q (x) replaces e x , [2]! , 

 [3]! , . . . . replace 2! , 3! , . . . . and g-operators replace the A operator of Boole's 



formula ; thus 



A p a* = (D-l)(D-g) . . . (B-q r - 1 )a n 

 replaces the 



A\ = (D-1)\, Da„ = a n+1 , 



usual in Finite Differences. 



I think that these examples will suffice to show that for (^-functions the natural 

 analogue of the theory of differential equations is the theory of the difference quotient 



d 



A<I>(.r) = —sLJ. \-L or operator xA = ? = \xTf\ . . . (6) 



qx-x q-\ 



In the later portions of the paper several A-equations will be solved, which will 

 show clearly that the theory of A-equations is exactly parallel to that of differential 

 equations. Such solutions as Hankkl's solution of Bessel's equation have a place in 

 the A-analogue of Bessel's equation. 



In previous papers the writer made use of an operator 



^ s a^){i&r*)\ • • • ■ I d&r*) i • ■ ■ • \W)\ — id®}- • • I, 



which could, however, only be applied to such special series as ^cx lr] , and was quite 

 inapplicable to arbitrary functional forms. A comparison of the following equations 

 will show the advantage which the operator A has over D ( ' l) . 



qsS*B»y - DM? + { 1 - [n] - [ - n - \}}xDy + [n][ - n - l]y = P ln ^(x) - P[„_ 2] (^) . (7) 



