ON ^-FUNCTIONS AND A CERTAIN DIFFERENCE OPERATOR. 275 



in which the coefficients A , A x , . . . . are independent of m and are given by 



For example, 



[a 1 + 3][a 2 + 3] [a„ + 3] _ [a, + 2] [a, + 2] U + \] .... [a„+l] _ [qj .... [q w ] 



[3]! [2]![1]! ~ +/ [1]![2]! Z " [3]! ~' 



the ^-analogue of a well-known algebraical transformation, 

 if the A difference equation 



r=n r=m 



in which A r is defined above, 



and B,.= g(-1)^ « [? ._ s]![s]! , 

 be compared with form (73) we see that on substituting ^c,x" txr , 



<f>[m] = A + A 1 [m] + A 2 [m][m- 1]+ . . . +A„[m]. . . [m-n+1], 

 = n„[a 4- iii\ , (as defined above) . 



In the same way 



i//[m] = B [m] + B 1 [?»][?n - 1] + . . . + B„[?ra][m- 1] . , . [m-raj, 

 = [m]n n [f3 + m-l], 



so that the solutions are given by a function 



n n [a + m] +1 n w [a + OT ]n„[ a+w+l ] m+i n > 



[m+ljn„,(j8 + ni] [m+l][m + 2]n„ 1 [ J 8 + m]n„ 1 [ j 8 + 7» + l] ' ' ^ ; 



in which m is any root of 



[m]n„[/3 + m-l] = 0. 



The principal roots are 0, I— /5 1} 1— /3 2 > 1— £ 3 , , and corresponding to each 



of these n l principal roots there is doubly infinite system as in general 



m=l-& + , grT/ nj , (r,*=0,l,2 .... ad inf.) (s = 0, 1,2,3 . . . ».). 



lOg (/ + 2^7rt A/ 



The series from the principal root zero is 



, /= i + hi M • • • [ a «] , + [ a i][ a i + 1 l • • • K][q»+i] o 7r , 



7 [iMLAJ • • W WMA+i] • • • L&I&+1] ( } 



From the principal root 1 - & we shall have 



The particular case 



(^-l)( t /-l) 



' 1+ (q-l)fr-l) X+ 



is a solution of 



[a][p]y+{q«[p+l]x + qP[a]x-[y]}Ay+{q*+P+h*-qyx}*y = 0. . . (78) 



