224 ^- FUJISAWA. 



that we may here al«o restrict ourselves to the consideration of B. 

 Moreover, since in this case 



(165) B {X, k) = B (J^, k^ k'^ x'^, 

 it follows, 



(166) B,^,„, = B,,,, k'^-^'\ 



and we need only determine the first half of the coefficients i^g,,,. 

 Now B satisfies the diiferential equation 



(167) {1-(1 + A:V + AV['^.+ I [('27i^-l)Ä.-2-ll^-2(H2_l)^-2^-3}^ 



+ 2;i2Ä'(l-A-2)^+ {n\n''-l)k:'x^ + n-\ B =0, 



whence we deduce 



(168) (2w + l)(2w + 2)B2,„+,+ { {n^-é7ii') + 4:m0v'-m)k^B,r„, 



clh: 

 Substituting 



•^2»» = B2n, + -^2/», 2 A"' + • • • + -^^2/n, 2r A-'' + . . . + B.2,n^ -2,11 ^'"\ 



m ^p, 



in (168), we get 



(:2m+ l)(27;i + 2)B.,,„+2,o +in'-^m')B,,,^o = , 



B.2m,o = 0, /i<c2w<2^, 



(2w+ I),2;;i + 2)ß2«f2.2 +{5n''-^m')B,^,2 



+ 4m{n'-vi)B,^^, +{n--2m+\){n^-'2m + 2)B,,,.,,, = 0, 



