ON ^-FUNCTIONS AND A CERTAIN DIFFERENCE OPERATOR. 263 



and suppose ($>{x) capable of expansion in the form of Taylor's series, we obtain 



A"<j>(x) 



= [*(*)+ + {(l+e)"-l}^ (r)(a;)+ J 



-w[^)+ + {(i+ e r i -i}v rw+ .... .J 



, Mrn-lir./ v , , {(l + e)"- 2 -l} r r . w/ * , 

 + 3 f2l! L^ ~^ ri L *y(*)+ ■ • • 



) +q in{n - 1] (q-\)"x n . 



(21a) 



+ ( _ l)» 2 W»-D^(a;) 



Collecting the terms in sets according to the orders of cj>', <p", , we find that all 



terms below the set containing </> (n) (x) vanish identically, since 



[n] m -[n][n-\] m + ql^^[n-2] m + + ( - lyv/''- 1 '"- >[n][l] m = 



for all values of m (an integer) < n. This identity is easily established if we consider 



that 



A 2 <£(a;) = c<t>"(x) + terms in <£'" , </> (iv) , , 



then replacing q by 1 + e, and operating with A , we find 



A^(a;) = ciJ Li — ^AJ + similar terms in d)"', 



ex 



= c 2 <f>'"(x) + . . . . similar terms in <£ ,iv) , </> (v) , 



and by induction A n <p(x), when expressed in terms of <p(x) and its derivatives, contains 

 neither <p(x) nor any of the derivatives below <p {n) (x), so that the coefficient of <p°' l) (x), 

 viz. 



2 -in(»-i)( 2 _l)-»r( 3 »_l)^_[ TC ]( 2 »-i_l)» + gWElzlJ( 3 »-2_i)'»- 



L L 2 J ! 



- ( - \fq^-^-\q - 1)»[»] ~\^ , 



is identically zero for all values of m < n (m, n being positive integers). 

 The coefficient of ft n+r \x) is 



|~( g n _ i)-* _ [„] (g n-i _ 1)n+ r + q [n][n- l] (gB ., _ 1)B+r _ _ _ _ 



(22) 



n + r\ 



(23) 



Hence, taking account of the denominator in expression (21a), we find 



i *)=(^J tA '(ir tj ' 



d_\ n+2 



dx 



t 



</>(*)> 



(24) 



