ON g-FUNCTIONS AND A CERTAIN DIFFERENCE OPERATOR. 281 



The theorem analogous to this is 



U2„_l d y i_ 2)i CI \ I T m+n+r j _ m+n-T 



which in ^-functions is 



{(x 2, '->A)(z 1 --''A)} , x m+n+, J q (m,n + r) = x m+n J q (m,n). . . . (X) 



Other A equations are 



xL | s«+»J m . B J + arij- { a -+»+ 2 J,„ +1 , H+1 | = (2m + 2«) a f l +»J m> „ . . fc) 



| ce m+B Jj(m , n , x) 1 + ./-'A | a;'"+" +2 J g (m + l,n+l,a:)i= [2m + 2n]( - Y' + ' J,(m , n , gcc) . (v) 



A{a;" ,+ "J 4 (??i, w)}=a / "'+"- , </) 9 (?n ) «,a-), . . . . (p) 



<£,/?» , n , .<) + <£,(to 4 1 , n+ 1 , .'.) = ^~r n -^ g (m , n , qx) , 

 from which, of course, various series may be obtained by repetition. 



.rA 

 If 



then 



TRANS. ROY. SOC. EDIN., VOL. XL VI. PART II. (NO. 11) 42 



