TIIKORY OF TIIK I'AKTITIONS OF NTMIll 353 



We may therefore take (a+1) and (b) as the two fundamental solutions, and cle;n ly 

 we may always multiply a solution by any function of x which does not involve " ->i //. 

 'I'll.' lin.il r\|irfssion of IL(oo ; it, b) which I adopt is 



(+l), (b) 1, 1 



I I. ( oo ; a, 6) = -J- 



s , 1 x, 1 



and we will find that, expressed thus as the quotient of two determinants, it is 

 generalizable. I might now, knowing d posteriori the expression for IL ( oo ; a6<-), 

 proceed in a simpler manner than what follows ; but I think it better to put l>efore 

 the reader the actual course that the investigation took. 



Art. 11. In the functional equation satisfied by IL(oo ; a, b, c), which may be 

 written 



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



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

 I write 



V, ( oo ; a, b, c) = x-"-' {IL ( oo ; a, b, e)-(b + l) IL( oo ; a, 6-1, c) 



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



and thence derive the relation 



(++) V, (; a, M 



= (a+2) V, ( oo ; -l, 6, c) + (b+l) V, (oo ; a, b- 1, <-) + (c) V, ( oo ; a, 6, c-1) 

 -(a+2) (b + 1) V, ( oo ; a-1, b- 1, r)-(a+2) (o) V, ( GO ; o- 1, b, ,-l) 

 -(b+1) (c) V, ( oo ; , 6-1, t -l) + (a+2) (b+l)(c) V, ( ; a-1, 6-1, c-1). 



Comparini; this with the functional equation it is clear that V, ( oo ; a, 6, c), as 

 defined, is a solution. 



I'r.KVfiliiii: similarly \\- tindsix solutions \\liidi I i-xliihit as .>j,.Tat ions |ifrfo|-[ii.-,l 

 upon IL ( oo ; a, 6, c) as follows : 



x- {IL( oo ; a, b, o)-(a+2)IL(oo ; a-1, 6, c)} = O.IL( oo ; , 6, c) ; 

 *' {IL( oo ; a, 6, <) -(b+1) IL( oo ; a, 6-1, c)} = OJL ( oo ; a, 6, c); 

 a:-{IL(oo;rt,ft,c)- (c) lL(oo;a,6,c-l)} = OJL ( oo ; a, 6, c) ; 



VOU CXJXI. A. 2 Z 



