560 PROCEEDINGS OF THE AMERICAN ACADEMY. 



and let (ji (x) be a solution of the (/-difference ecjuation of the type (54) 

 (55) g (q.v) = (x - m) (J (x) 



for ni = (ti- If one takes for new variable 



Y{x) = g,{x)..Aj,{x)Y{x), 



a new matrix equation (54) in Y{x) is obtained with Q{x) polynomial 

 in X. 



Let us write 



log g ■ 

 In terms of this new variable a solution of (55) for ni = is 



For in 9^ 0, the transformation 



qiiP-i)^ 



. log X lOgZ 



X = mx, y {x) = e" ^~^ log a wios e y (x) 



takes (55) to the normal form 



(56) y(qx) = (l-x)y(x). 



Two solutions of this equation are 



(57) 





1-J 1-i 



.r qx 



as one may verify by direct substitution. The function y^ (x) plays 

 the same role for the linear ^-difference equations as the gamma func- 

 tion does in the theory of linear difference equations. I have men- 

 tioned these functions in order to supply an example later. 



The fundamental existence theorems for linear ^-difference equa- 

 tions are essentially a consequence of the work of Grevy^° and Leau.^^ 

 The first complete treatment has been given by Carmichael,^^ and the 



30 Paris thesis, 1894. 31 Paris thesis, 1897. 



32 Am. Jour. Math., 34, 147-168 (1912). 



