202 R- FUJISAWA. 



and, generally, 



(117) ç^^^^(^n'-\)-^Ê^ + %,^niH„ = U„, , m = q-2, q-3, ... , 



where U„, = (1 + ^-^1^' - '2(n— ^)^"-^^ + »'0^' " D'^'^.u+i 



It will be seen that equation (115) is satisfied by ç*^"; indeed 

 we might have determined Un by means of tliis equation, for if 



the operator g-^j- be denoted by ^, then the diiferential equation may 



be written 



the general solution of which is 



where C, C" arearbitrnry constants ; and, as H,j can contain only even 

 powers off, C' = 0, or 6"' = 0, according as —^r— is odd or even. 



Let us now investigate e([uation (iK))« ^or this purpose, it 

 will be convenient to distinguish two cases according as — -^— is even 



or odd. When —2— is even, ^„ = —;?(;?-!) and iJ^ = 2pf^"<""i>, further 



rJ^H rlTT 



(^ + n^ - %n^ - 1 )^~' + n\n'' - l)^m,^ 



=2^--2|(«,_2)(»,- l>i(».+ l)fä"("-^> 2+(/, -\)n{n + 1)0? + '2),-"("-i>+2[ , 

 so that equation (IKi) now assumes the form 



= 2^-2|(;^--2)(n-l);^(H+ L,-'"^"--i)-2+(/i- l);?,(^i + l)('y^ + 2;c^''(''-"+2[ . 



