Chap. XVII] AlGEBEAIC ThEORY OF RECURRING SeRIES. 409 
M. d'Ocagne^^^ considered the recurring series Ui with 
and with Uq, . . ., Up_i arbitrary; and the series u with the same law, but 
with Ui = {i = 0,. . ., p — 2), Up_i = l. Then 
Un=UoUn+p-i-\-{Ui-aiUo)Un+p-2+ ■ ■ • + ( C/p_i " ^i f7p_2 - . . . -ap_iC/o)^n- 
For each series he found the sum of any fixed number of consecutive terms 
and the Hmit of that sum. 
M. d'Ocagne^^^ treated Up+^ = Up^n-i+ ■ ■ ■ +Un' He^^^ discussed the con- 
vergents to a periodic continued fraction by use of t«„ = atW„_i + ( — l)*w„_2, 
L. Gegenbauer^^^" found the solution Pm of gnPn = 2^UnPn—i+i/nPn-2, 
where 
S. Pincherle^^^^ applied p„+i(x) = (a; — aj(x— i8„)p„(a;) to developments in 
series. 
E. Study "^^ showed how to express the general term of a recurring series 
as a sum of the general terms of simpler recurring series, exhibited explicitly 
the general term when n = 3, and applied the theory to bilinear forms. 
M. d'Ocagne^^^ considered a recurring series with the law of recurrence 
(^1, . . . , Ap) : F„+AiF„_i+ . . . -{-ApYn—p^O 
of order p and generating equation 
^{x)=x''+Aix''-'+ . . . +Ap-=0. 
Set 
Qi{x)=x'+A,x'-'+ . . . +A„ ^{x) = Yp_,+Q,{x)Yp-2+ ■ ■ • +Qp-iix)Yo. 
The existence of a conomon root a of $(x) =0, '^ (a;) =0 is a necessary and 
sufficient condition that the Y's satisfy also a law of recurrence of order 
p — 1, viz., (Qi(a),. . ., Qp_i(a)), and then the initial law of recurrence is 
said to be reducible to one of order p — 1. 
M. d'Ocagne^^'' considered the series with the law of recurrence 
U^n — ^*0 '^ n—l + a itt n—2 "T • • • + Ct Pi—lU n—Pi 
and generating equation 
(l>iix) =x^' — a\ of *~^ — ... — a*p.._i, 
"«Nouv. Ann. Math., (3), 2, 1883, 220-6; 3, 1884, 65-90; 9, 1890, 93-7; 11, 1892, 526-532 (5, 
1886, 257-272). BuU. Soc. Math. France, 12, 1883-4, 78-90 (case p = 2); 15, 1886-7, 
143-4; 19, 1890-1, 37-9 (minor applications). Nieuw Archief voor Wiskunde, 17, 1890, 
229-232 (applications to sin ma as function of sin a and cos a). 
"'Comptes Rendus Paris, 104, 1887, 419-420; errata, 534. 
"8/bid., 108, 1889, 499-501. 
"s^Sitzungsber Ak. Wiss. Wien (Math.), 97, Ila, 1888, 82-89. 
"s^Atti R. Accad. Lincei, Rendiconti, 5, 1889, I, 8-12, 323-7. 
"8cMonatshefte Math. Phys., 2, 1891, 22-54. 
"'Bull. Soc. Math. France, 20, 1892, 121-2. 
"oComptes Rendus Paris, 115, 1892, 790-2; errata, 904. 
