ON g-FUNCTIONS AND A CERTAIN DIFFERENCE OPERATOR. 277 



§ 3. Solution corresponding to Hankel's Y n . 

 Such a solution of equation (82) is 



y = C 2 ( ~ * ) '[2][4] . . . [2r] . [2 - 2»] . . . [2r - 2ref 2 '" 



t 



+ D log x"V ( - l) r i~ -,,'tn , a 1 



o ^jV / 1 2?-}! {2ra + 2r} 



(86) 



"^"^ 1[2] + [4] + • • • [2r] + [2ra + 2j + [2 M + 4] ' ' ' [2n + 2r]f {2r}\ \2n+ 2r\ 



The constants C and D not being independent, I do not propose to discuss such 

 solutions of the A -equations, but merely indicate that there are such solutions, as 

 showing the parallel nature of the theory of these equations with the theory of ordinary 

 differential equations. It is clear that Hankel's solution of Bessel's equation can be 

 derived from the above in the limiting case 3=1. 

 The solution for n = , is 



^ I 2<f 2<f 2q 2 ' \ .<■"• 



y = cJ q (0,qx)logz-c^ \ [2j + [iJ + ■ ■ ■ ■ +[27] / [2]»|4]» . . . [2sf 



I do not propose to discuss here the solutions obtained from the other roots of the 

 indicial equation, namely those corresponding to 



a=±n+ 2 ™ . ,(r = 0, 1,2,3, ....). 

 . . — — log q + 2<tti 



§ 4. Solution of ^-Difference Equation satisfied by J m . J n . 



In the case of equation (80), on substituting ^c s x a+2s for y, we obtain an indicial 

 equation 



[m + n + a][m -n + a~\[m + n - a][ - m - v + a] = , 

 and an indicial function 



J ° I [a + m + n + 2][a--m + n + 2][a + m-ii + 2][a-m-n+ 2] I ' 



and from the four principal roots of the indicial equation we derive four principal 

 solutions, which, however, are not distinct in case m and n are both zero, and which 

 become formally infinite in certain cases, so that special solutions involving log x would 

 be necessary to replace these indeterminate forms. The four principal solutions, when 

 distinct, will be denoted J q (m , n , x) , J q ( — m , n , x) , 3 q (m , — n,x),J q ( — m, —n,x), • 



J«(".".«) = Z(-D , {2BI + 2yt + 2r}!{2w + 2 7J T { 2 )> + 2r}lW l- ■ (87) 



As before {2r}! denotes [2][4] . . . [2r]. 



This function possesses many interesting properties, which will be given in § (4) 

 et seq. It can be derived from the product of two functions J q (n , x). 



TRANS. ROY. SOC. EDIN., VOL. XLVI. PART II. (NO. 11). 41 



