ON 9-FTJNCTIONS AND A CERTAIN DIFFERENCE OPERATOR. 



279 



is independent of x and is equal to q n2 **, from which many curious results may be 

 derived. We pass on, however, to give the four principal solutions of (84) — 



>J\ = ?"%(« , x)J q -i(m , x = (fZ q {m , x)J q -i(n , x) , 



Vi = 2 m2 J«(rc . ar) J 9 -i( -m,x) = (f 2 J q ( - m , x)J q -i(n , x) , 



V 3 = (Z'" 2 J 9 ( - n . *)Jj-i( -m,x) = q n2 i q ( - m , «) J g -i( -n,x), 



y 4 = q r " 2 J tJ ( - n , x)J q -i(m , x) = q n \\ q {m , x)J q -i( -n,x), 



which may be expressed in a great variety of forms, as follows. 

 It is easy to establish a theorem for q functions analogous to 



(93) 



t / x # n I ■> , • , (2» + 3) , , , (2n + 5) ., , , ) 



>(2« + 2) 



(93a) 



which is a well-known result in the theory of Besskl's Function. 



The corresponding result for an arbitrary power series, and the q- generalisation of 

 the same, I propose to discuss in the supplementary paper. 



The (/-analogues of (93a) are 



whence 





{2«} 

 E q (ix)J q - 1 (n,x) = q*^{ 



[2][2n + 2] [3]![2n + 2] 



1 + KB + ; 



[2n4-3] 



[2][2n+2] 



J q -i(w , x) = , 8 E g -i(Kc) _ if 



J q (n , x) \\{ix) E q (ix)E q ( - ix) 



(94) 



(95) 



(96) 



and so the solution y x can be expressed in the form 



y _ lf c>+n ^ v {n,x)3 q (m,x) 

 Jl + E q (ix)E q (-ix) ' 



so that if we denote this solution of (84) by <J> 9 (m, n, x), we have the following forms — 



%{m,n,x) = q'" 2 J q (u,x)J q _ 1 (m,x), ....... (97) 



= q nll J q (m,x)J q _ 1 (n,x), 



= q"> ,+ " 2 J q (m , x)J q (?i ,x)±{ E q (ix)E q ( - ix) } , . 

 = qi^^{J q (m , x)J q (n,x)J q ^(m, x)J ^(n , x)} ! > , 



{2m + 2n + 4:r}\x m+n+2r 



= 'Z m2+ " 2 Z(- 1 ) r T2 



{2m + 2n + 2r}\ {2m + 2r}\ {2n + 2r}\ {2r}\ : 



(98) 



(99) 



(100) 



(101) 



= 2+nJ fy ( _ i y H'» + 4r}!^-"^ P ly, , , {4 W + 4r}!^^ ) >> , 



)^- V ; {4m+2r}!{2m + 2r}!2{2r}! J I ^ V ' {4rc + 2r}\ {2n+ 2r}! 2 {2r}! J ' V ; 



§ 5. Relations between various Series. 



Several interesting q series may be deduced from these relations. By equating 

 (97) and (98) we obtain 



'/" 2+ "'{J,(»M| x)J v _,(m, aOJ^n.aOJf.ttfi, *)} 



= gww ly/, v, {2m + 2n + 4r}\x'"+ n+ir ) 



* \^ y ' J2»n + 2* + 2r}!{2»t + 2»-}!J2n + 2r}!{2r}!j 



(«) 



