280 THE REV. F. H. JACKSON 



which by (87) may be written 



|y ( _l V {4m + 4r}!^-"+ 2 >- l|V(-ir < 4w+ ^V-^L \ /m 



I ^ V '{im + 2r)\{2m + 2r\\{2m+2r}\{2r}\\\^ y ' {4n + 2r)\ {2n + 2r}\ {2n + 2r}\ {2r}\ J VP; 



= I y C - 1 V {2m + 2n + ir}\ x m+n+2r ) J 



~ I ^ { { 2m + 2n + 2r} ! { 2n+2r}\ {2m +2r}\ {2r} ! J ' 



whence, equating coefficients of x 2m+2n+2 '' , 



Z{4wi + 4»-}!{4n + 4i/-4?-}! 

 (- 1 )' {4»i + 2r}!{4rc + 2v-2r}!{2m + 2r}! 2 {2n + 2v-2r}! 2 {2v- 2r}! {2r}! 



v {2m + 2w + 4v-4r}! {2m + 2rc + 4r}! 



= ^' ~ l '2m + 2n + 2v- 2r}\ {2m + 2n + 2r}\ \2n + 2v- 2r\l {2n + 2r)\ 



{2m + 2v-2r}!{2m + 2r}!{2v + 2r}!{2r}! 



in which {2?-}! = [2][4] . . [2rJ and [>■] = (q 2r - \)/(q - 1). This identity holds when 

 m and n are not integral, if \2n] ! be interpreted generally by a Basic-gamma Function. 

 When q=l, the series reduce to special hypergeometric series (x=l), with eight ele- 

 ments in the denominator and two in the numerator of each term. It would be tedious 

 to obtain further identities. I need mention only, that by considering such products as 



J q (?i , xt)J q (n , ■jrt-')J q _ l ((v , xt)3 q _i(v , xt~ l ) 



a great variety of relations can be obtained.* 



§ 6. Recurrence Relations. 



It would be tedious to give further analysis of these functions ; so, to conclude the 

 paper, I shall merely state certain recurrence and other relations satisfied by these 

 functions 



q 2n J q (m ,n-l, x 2 ) + J q (m ,n + l,x 2 ) = L^iAj^wi ,n,x 2 ), . (r?) 



x L 



q 2 '"J q (n , m - 1 , -c 2 ) + J,(» , m + 1 , x 2 ) = LJJaJ ,(m , n , x-) , 



x* 



whence we deduce 



<f" J g (m , n - 1) + J q (m , n + 1) _ [2w] _ (g 2n - 1) ($) 



q 2 '"J q (n , m - 1) + J q (n , m + 1) [2m] (q 2m - 1 ) ' ' 



q**J a (m ,n-l) + J q (m , n + 1) = ^ | J q '(m , n) + J^l3;\m , n) + x 2< ^lj q '"(m , n) + . ...'}, («) 

 whence, in case q=l, 



J m ,»-i(.'-) + Jm,n+l(*) = — J'(™ ,»>«)' 



In the theory of Bessel's Function 



I a: eta J 



* Trans. R.S.E., vol. xli., pt, ii., p. 406'. 



