THEORY OF THE I'AKHTIONS OF NUMBERS. 



359 



All of the processes employed above are obviously valid when applied to the 

 functional equation of order n and lead to the expression of IL( oo ; a lt a 2 , ..., a,,) as a 

 quotient of determinants. 



Art. 20. Before proce-din<;- further I collect together the chief results obtained 

 above. 



lL(oo;a,6) = 



(a+1), (b) 



.' 



1, 1 

 f, 1 



L ( oo ; a, b) = 



(2) (3). ..(+!).(!) (2). ..(/,) 



GF( oo ;,&)== 



(2) (3)... (a+1). (1) (2) ...(b)' 



IL ( oo ; a, b, c) = 



(b+1) , 



x 3 



*(a+2) , (b+1) , 

 x? x 



(c) 

 1 



(c) 



1 



1, 1, 1 



X, }, I 



a*, x, 1 



1, 1, 1 



*" 1, 1 



x 3 , x, 1 



(3) (4) ... (a+2). (2) (3). ..(b+1) T^J^Ji^eJ 

 and, not putting the denominator determinant in evidence, 



(a+1) (a+2), (b)(b+l), *(c-l)(c) 

 x(a+2) , -(b+1) , (c) 



.T 3 X 1 



(l)(2)...(a+b+c); 



GF(oo;, 6,c) = 



x(a+2)(a+3) , 

 .r 3 (a+3) , 

 r* 



, .r 3 (d-2)(d-l)(d) 



(d) 



I 



(1) ... (a+3) .(!)... (b+2) .(!)... (c+1) . (1) ... (d) 



