U A \(l-x)(l-qx)l 



260 THE REV. F. H. JACKSON 



If we now reverse the operator A in equation (/3), we obtain 



1 



(l-x)(l-qx) 



and (15) becomes 



,=tt,, "1f|H + #-:) + <-l4) + -'' 1 



On choosing the arbitrary constant c = 1 , and replacing the terms on the left by 



1 f 1 1 1 



series of powers of x. we find the coefficient of x r is [r~\ — { 1 + — + — +... '■ (q — 1 ), 



q' I q q rr J 



which ((/>1) is identically unity; so that a special value of the series is \+x + x 2 



+ . .... or 7— , , and the validity of the transformation is justified in this simple 

 (1-x) J J L 



case. The result may be expressed also as 



(T~^x) = ( q ~ 1 H(l-x)(l -g_x) + (q - x)(q* -x) + (g 2 - x)(q3 -x) + ' ' ' I 



It will be noticed that such an equation as Aw = — cannot be solved in this form, 



though a solution is furnished by 



, A.(ff - 1) 



u = e+ —3 ' log x , 



log? 



reducing, when q = l, to c + Wogx. 



In the same manner, if we perform the operations denoted by A H on the function 



(1 - x)(l - qx)(l - q 2 x) .... {\-q m - l x) 



we obtain 



bPn = - [m][m+l] . ■ . . [m + n-l] 



(l-x)(l-qx) .... (1 - q m +"- 2 x)(l - q m+n - l x)' 



If we solve this by the inverse method, we find in general 

 u = a + a x x + a 2 x 2 + . . . . + ^os" -1 + (q - 1 ) n qW>-v x n 



*s?[ n][n + 1] . . . [n + r- 1] • [m][m+ 1] . . . [m + rij-l] ,,g* 



one value of which is the particular expression for u given above. This value will be 

 obtained by the following choice of arbitrary constants : 



a =l , 



_[m][?« + 1] .... [m + r-2][m + r-l] 



