ON tf-FUNCTIONS AND A CERTAIN DIFFERENCE OPERATOR. 273 



Now 



im)Mm 



_ , (m - v)x (u- v)(u - q 2 v)x 2 (u-v)(u- q*v)(u -q*v)x? m) 



" 1 + -[2T + [2p] "M4][6] 



so that the expression (64) may be written 



1 | (u -»)■>•? + (u-v)(u-q*v) r2fi+ 1 y Y x , (u - v)xt- i + ( y - «)( tt - g 8 ")^- 4 + 



[2] [2]L4] 



[2J [2][4] 



(67) 



= | + (< a + «- s )li + (^ + r ')-la+ 



in which the coefficients 3f are ? analogues of Bessel's function J n (u— v). We see that 



„ (tt - t-)(tt - g'tQ ...(«- 7 g "-^ ) / , ^ (» - »)(« ~ g 2 "") ^ (» - «)(« - g 2 »)(" - g^Xt* - q in+ 'v) , I , fi8 v 



|n [2][4]...[2»] I [2n + 2][2] [2n + 2][2n + 4][2][4] J' 1 °' 



The q, or quasi-addition theorem, for this function is, in the notation of a former 

 paper, 





Io(« , + »)= J ra (»)I m («) + ffiMhfc) + £fW«)ImW + 



in which 



J[ " ,)(m) = 2{2« + 2»-}!{2r}! = 2 I> + r + 1)I>+ l)(2) n+r (2) r ' 

 ^ lm]W ~ 2-{2m + 2r}!{2»-}! 2 



■ 3\ 



~2r(m+r) 



3f[„] is not a function distinct from J [n] , but can be derived from J [n] by inversion of the 

 base q. It will be more convenient in subsequent work to denote these functions J q (n , x) 

 and J q _ 1 (n , x) respectively, which will show explicitly that they are g-functions, and in 

 what manner they are related to one another. For a detailed discussion of their 

 properties I refer to Proc. L. M.S., series 2, vol. ii. pp. 193-220, vol. iii. pp. 1-24; 

 Trans. R.S.E., vol. xli. pp. 401-408, pp. 105-118; Proc. R.S., vol. lxxvi. A., pp. 

 127-145, vol. lxxiv. pp. 64-72. 



From (61), (64), and (67) we obtain 



H Q (x) + (t* + r i ) H . 1 (x) + (t* + t- i )^(x)+ 



Mo(|)+(^+nM 1 (|)+(^+^-v 2 (|)+ 



= $ (u,v) + (t* + t->)Uu,v)+ , . (69) 



whence we derive 



/*o(*)=/*o(-f}&> + 2 /*i(f)li + 2 ^(|)l 2 + 



ft^HZ {^(|)l» + r + /w(|)l,} (70) 



