2G2 THE REV. F. H. JACKSON 



Arranging this series according to powers of x, we find that the term containing x r is 



[2]! 1 +( l) [n-s]\[s)\ 



C'$" (0) j .„ _ r w -[„r|H-l) + [ n ][ n - 1L .(«-2|+1 + +(_iy M - „r(n-s)+ls(*-l) 



+ + ( - l)»gW«i-i) I . 



This expression is well known as being 



?! ^V-i)('/''- i -i)('/'- 2 -i) — (f-^-i) 



for all values of r > n. In case r < n or = w, the expression is identically zero. 



Hence, taking account of the denominator of (17), A n (p(x) may be written in the 

 form 



^ MIV-11 . . . . |>-ra + ll .... 



or, since [r][r — 1] . . . . [r — n+ 1] contains a zero factor for all values of r <.n, finally 

 we write 



r=0 L J 



In the equation 



qx-x 



let us substitute 1 + e for q ; then, in case &(x + ex) is capable of expansion in the form 

 of Taylor's series, 



A*(a;) = *'(*) + —*"(») + .... +-^-^"+ l \x)+ . ... 

 2! ra+1! 



Similarly, from the equation 



it is easy to obtain 



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



A^(x) = (l± 2 -V'(.r)+|! , ( £ +2)(e+3)*'>)+ (19) 



w=» u— \,y,ll 



l(' +2 >"-w 



71 = 1 X ' 



or symbolically 



(g+iK ■ (g _ i)-i (g +i){( g +i)»-i }af-v <* Y +1 ran 



In the same manner, if we replace q by 1 4- e, in the general equation 



W ' (q n x - q n - l x)(q n - l x - q n ~ 2 x) . . . (qx-x) ' 



