214 ^- FUJISAWA. 



we need onlv detennine either A or D. Let U8 take- D and apply 



n—l 



(134). We obtain V^^-l, A=0, 1)4^,=(-1) 2 n¥", and, generally, 



(143) (2m + lXi^;» + 2)A.«+2+4w|(n^-7;0^''-m|A«. + '2^i^Atl-Ä:^)'^ 



When VI ^p, 



(1 44) D,„, = D,», ( 1 + A-'" ) + A™, 2 (A-- + A""^-) + ■ • ■ + ^2«. 2. (A'" + /.''"--n + . . . , 



the last term heing A,».™ A'"' or A,„,,»-i (A"'"' + A™^') Mccording as /» is 

 even or odd. Substituting this in (113), we obtain 



{'2m + 1 ){2m + 2]{ D,,,^,, ,{] + Ä^-+^) + !)„„,,, , (A- + A'^-^) + . . . 



+ Am4 2,2r(A'' + A-"'--^+)+...} 



+ 4//t (;r-;;^) |A,«,o (A'+ A'™'') + A».,2(A' + A:-"')+ ... 



+ A., 2r( A "■+'-+ /.-'"--'• + ')+...} 



- 4;;r I A,,,, „ ( 1 + Ä-'") + D,„, . (A - + A -'"-2) + . . + D„„, 0,. ( A -' + A-'" -"1 + , . } 

 + 2ir |I)2„,,,(*+2;;iA;-''0 + A„,,2(2A- + ('im-'2)A2''--)+ .. 



+ A», .r (2rA^'- + (2w--2y)A-'«- -"•) + . . . [ 



- 2n- { A-«, Ü i* + •2;;iA-"'--) + D,„^ , (2 A* + (2;;i - M"") +... 



+ A,«,2.i^''A''-'-+(2^«-2r)A2--2'+2) + ...} 

 + (n' - 2 ;;i. + 1 )(^r - 2m + 2) ] Do„,^., „ ( A" + A-'" ) + D,„_,, ., ( A '^ + A -™ -^ ) + . . . 



+ Am-2, 2. lA-'-^ -+ A-'"-2'-) + . . . } 

 = 0. 



Hence 



{2m + 1 )[2m + 2)D.,„, ^0.0= 4y;rD2m, , -^^2,.. n = < '. 



(2w+1X2;m + 2A«.2,2 = -4(yr-y/rA,»,2 



(2w + l)(27;i + 2) A,«+2, 4 = - 4 1 {fn - 1 )n- - ui' \ A«. 2 - 4i2;/- - //r ) Am, 4 > 



- [n''-2m + l)(;/,2- 2;« + 2)A,.-2. 2 . 

 ( 2y/i + 1 X2m + 2)D2,„+o. e = - 4 ^ {m - 2)n- - m^ \ A,„, 4 — 4(3;/- - ^/r^i).,,,,, e , 



-{}v'-2m+ \){n'-2m^2)D.„, .,,, , 



