46 N. F. DUPUIS ON THE SYMBOLIC 



Denoting the sum of the first by C, and of the second by S, 



C + iS = Ya + xV{a-\-6) + x'V{a+2d) + ... 



= Va ]]. ^ xV + X V- + X F' + . . . \ 



by separation of symbols, 



Va 



— \-xVti 



Realizino- the denominator by (14), and equating real parts, and also imaginary parts, 



gives, 



„ Cos a — X cos (a— 6) 



~ 1— 2a; cos 0+x^ ' 



o Sin a — XBu\ia — 6)^ 



1— 2,x cos 6+3:- 



' Ex. 4. To find the generating function of 



l+3a; sin O+Wx" pin 2^+43.r sin '68+. . . 



Denote the G. F. of the series by S, and of that of the conjugate series, in the cosines, by 



C. Then— 



+ «5 = 1 + « + 3x- F+ 11,1-° V- + 43:^= F' + . . ; 



and taking a;Fas the variable, the Gr. F. of this series is 



Thence, realizing denominators, and equating real and imaginary parts, gives — 



_ ,. 1 f 8x' sin 6 I X s,m 6 ) . 



* 1 1 -8XC0S 6»+ lfix= 1 - 2x cos ^ + x" ) ' 



, f 2 — 9,x cos 1 — .T C09 (9 I 



~ * 1 1 — 8.r cos 6 + \&x- 1 — 2a; cos 6» + .x'- J ' 



Ex. 5. To find the generating functions of 



x cos 6 —'— cos 2^ + '— cos 3^ H . • • 



And a; sin ^ — '^ sin 2(9 + ^ sin 3fl — -t- . . . 



As the generating functions of these series are not functions of the same species, the 

 series are better dealt with separately. 



Denote the G. F. s by (7 and S, as before. Then, 



1. 2(7 = X ( V + T^"' ) - 1^ ( y^- y') + -f ( y+ y~') + - • • • 



2 T7- 3 T-rS 



= a-F-^ + ^- + . . . 



I a;'-F"" , .x'F"" , 

 + a.F -_^-_+_^-+... 



= Z(l + a-F) + Kl +-«F'') 



= I {\ -\- 1x cos fl H- .x-,) by reduction. 



