200 ^^- FUJIKAWA. 



From the well known relations 



(109) A{x, k) = {- l)'^'D(Jy, äVä^ B{x, ;,) = (- 1)V^^J^, Ak'^x'p, 



which may also be written in the form 



(llU) A{a, ç)==(- \)^'d(^x, jY', D{a, ç) = (- iy^A(^a, 4^ç^ 



we see that A and 1) are of the same degree in a and that, moreo\ er, 



(111) E„,(ç) = ( - \)^H„(iy'^, H„,{ç) - ( - l)''^V«(y)>-^^. 



AVhen the multiplicatcrr // is required to be put in evidence, we 

 shall write A\_n~\ and lJ\_n'\. Xoav the terms containing the highest 

 power of a in J[3], J[o], J[7] are -2^-^«, 'A-^V/, -2''^'^y', and those 

 in D[3], D[5], l)[7] are '2\-V., -i^çi«./, •Ji^i.s^.«. j>^ ,,^^.^11^ ^f the relation 



(112) A[n + 2]A[n-2] = iyp]A%N]-(\--aç'+^')A\2]D-lnl 



which is easily deducible from the addition-equation, we conclude by 

 applying mathematical induction that, generally, the terms involving 

 the highest power of a in A^nl, i)[//] are 



(118) E,/j:' = {-\) ^ 2^ç "a", H,//.'J = 2^ç^"f/', 



where /„ + /i„ = «^— 1. 



To find À,„ /A„ we deduce from (11^^) 



/„+2+'<„-2 = 2//,,+ 2 - 2>i2_'2/„, 

 which may be written 



