216 R- FUJISAWA. 



Again equation (143) may be written in the form 



(147) ('2wi+ 2)(2wi + 3)Ä;2D,p_2„,_2+ '2;i2A-(l - k'f^^^ 



■\-{n'-2m){u'-'^m+l)D,,_,^^, = . 

 Now, m being less than p, 



(14o) -i^ép-Sni = -^4p-2ot, 2p— Swi ("- "T "^ ) "T" -^4p-2m, 2p— 2?»+2 ("' ^ '"' + «''"■'^ )+... 



_L n ( l.23J—2m+2r I I.2i>— 2r\ r 



T^ -^4p-2m, 2î)-2»i+2r [f^ T^ "• ,> T^ • • • > 



the last term being 



-^4J'— 2;ft, 2p-în ^ 01' •^4/)-2m,2;)— jn— 1 ("' "T k " ), 



according as m. is even or odd. 



From (147) and (148), we obtain 



(2?n + 2)(2?;i+8)D4^._2„j_2,2p-2m-2 + |''"-('^''«+ l)'}A?>-2m,2p-2.« = , 



(2;;i + 2)(27;i + 8)D^,_2^_2, 2;,-2m + I ön^ - (2;;^ + 1)^ [ D^p^s,«, 2p-2,«+2 

 + {(4w+ l);A2_(2m + l)-[D4p_2m,2p-2,» 



+ (;i2-2w)(;i2_0;;^_J_ljX>^^2«i + 2,2i,-2,»r2 = 0, 

 {1m + 2){2m + 3)D,,^2,n-2, 2,-2.+2 + I ^hl'-{:2m +\f\ D,,_2,«, 2;,-2,« + 4 



+ {{-^111—3)1^ — {2m. + If [ 1)4^.-2™, 2;>-2'«+2 



4-(»2-2w)(?i2_2^;^+l)2)4p_^,„+2,2?>-2»,+4 = 0, 



generally, 



(149) (2?^t+2)(2;;i+3)Z)4p_2,,_2,2p_2,„+2r-2 



+ I (4r +\)}v'-{2m+lf\ D,^^^^ 2j>-2m+2r 



+ { (4/M - 4/- + 5)^2 _ ('2wi + 1 )2 [ 19,^,^2,,^_ 2;^2,«+2.-2 



+ (n2-2w)(7t2_'2m+l)Z)4p_2„,+2,2?J-2m + 2r = 0, 



