278 THE REV. F. H. JACKSON 



In the case of equation (84), if we substitute ^c s x a+2 * for y, we obtain an expression 



2, J [a + m +n + 2s][a -m-n + 2s][a + m-n + 2s][a - m + n + 2s]c s x a+2 » - [2a + 4s][2a + 4s - 2 ]<yc a + 2s + 2 I , 



so that the indicial equation is 



[a + m + n][a -m- n][a + m - ?i][a -m + n] = 0, 

 and the indicial function 



/■-./«.+ [2a. + 4][2a + 2] ) 



•^ °l [o + m + » + 2][a-m-n+2][a + m-ra+2][a-w+re+2] J 



Choosing c n = -7 ^r-p — ,- , we obtain from the root m + n of the indicial equation 



5 ° {2mj! {2w}! ' H 



a principal solution 



^ {2m + 2n + ±r}\x'"+ n+ * r , g 



V ^>{2m + 2n + 2r}\[2m + 2r}l{2n + 2r\\{2r}\' ' K ' 



which I have shown"" to be the product of two q generalised Bess el functions, so we 

 write 



y 1 = eJ q (m,x).J q _ 1 (n,x). ..... (89) 



The meaning of J 9 _ x will require explanation. We have in (85) 



X m [ X* xt ) 



Jq ( m '^ = {2l^}\ I 1 ~ [2m + 2][2] + [2m + 2][2m + 4][2][4] " j ' ' 



If in this series we replace q by q-i a new series is formed, viz. — 



/ _ 1 V___f n „a+-2,im+r) 



K ; {2m + 2r}!{2r}r 



which I denote Jg.^m , as). Both J g _! and J 9 have a common region of convergence. 

 In previous papers I have used the symbols J [m] (as), q m2 3m( x )> to denote these series 

 respectively, but I think it is better to use the notation J q (m , as), as being both easier 

 to print, and as showing explicitly the nature of the base. Moreover, it is apparent 

 that the functions J q , J g _i are not distinct functions, but may be derived immediately, 

 the one from the other, by inversion of the base q. 

 Nowt 



xi nt / \ mw iv {2m + 2n + 4r\\x m+n+ » r /Q1 , 



J.(n,s).J 9 -,KsH g Z(- l ) { 2 m + 2n + 2 r}n2m + 2r}l{2n + 2r}l{2 7}l- ' ' (91) 



t/ \t / \ m»V/ i\r \2m + 2n + ir}lx" l+n+ ' 2r /Q0 . 



J q (m , x)J q - l{ n , x) = <t 2( " 1 )' {2m + 2 , + 2, }!{2OT + 2 ; }!{2M + 2 , }!{2r} ! ■ < 92 > 



It is interesting to notice that 



J v (n , x)J q -i(m , x) 

 J q (m , x)J q -\{n , x) 



* Proc. Roy. Soc, vol. lxxiv. p. 67. t Ibid., pp. 67-68. 





