MAJOR P. A. BUoMAHON: MEMOIR ON THE 



. ' 1 



+ (b+l)(c)IL( oo ; a, 6-1, r-1)} = OJL( oo ; a, b, c) ; 

 ; ) ^--l)-(a+2)IL(oc;a-l > M 

 + (c)(a+2)IL( oo ; a-1, 6, c-1)} = OJL(eo ; a, 6, c) ; 

 IL(;-l,M)-(b+l)n J (oc;a )fc -l,) 

 + (a+8)(b+l)IL( oo ; o-l, 6-1, c)} = 0*IL(ao ; a, b, <). 



I further write , N 



O.JL ( oo ; a, 6, c) 



= x -{IL( oo ; , b, r)-(a+8)IL( oo ; a-1, b, c)-(b+l)IL( co ; a, ^-1, c) 

 -(c)IL( oo ; a, 6, c-1) +(a+8)(b+l)IL( oo ; a-1, 6-1, c) 

 + (a4-2)(c)IL(oo;a-l,6, ( -l)+(b+l)(c)IL(oc;a > 6-l,o-l) 



_(a+2)(b+l)(c)IL(oo ; o-l, 6-1, c-1)} ; 

 and it is easy to establish the operator relations 



o a o t o c = i> 

 o.o^eu 0.0. = o, o 4 o e = o fc , 



OaO,, = 0,0^ = 0,0^ = 0^=1. 



Art, 12. I now operate with these operators upon the known solution of the 

 functional equation. To clear it of fractions I multiply throughout by (l)'(2). 

 Operating m times in succession with O c I obtain the result 



^ +> (b) + jJ + +4 (a-b) 

 b-l)-^ +3 (2a-2b)} (c) 

 {(1) +aj(a-b)} (c-1) (c). 

 Whence I conclude that 



P, = (l)-x a+1 (2)(b+l)-x a+!( (a+l)+x 6+!1 (b)+x a+26+4 (a- b )' 



are solutions of the functional equation. 

 I find that 



but new solutions are obtained by operating upon PI, P a , and P 3 with O a and O 6 . 



