# Full text of "History Of The Theory Of Numbers - I"

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```CHAP, xviij       ALGEBRAIC THEOKY OF RECURRING SERIES.                 409
M. d'Ocagne116 considered the recurring series Ut with
and with UQ,..., Up_i arbitrary; and the series u with the same law, but with  w» = 0 (i = 0,. .., p-2), wp_! = l.   Then
For each series he found the sum of any fixed number of consecutive terms and the limit of that sum.
M. d'Ocagne117 treated u^ = wp+n_14- . . . +un. He118 discussed the con-vergents to a periodic continued fraction by use of wn = aJkun_1+( — l)*un_2> tt0 = 0, Wi = l.
•L. Gegenbauer118a found the solution Pm of where
P0=l,     Pi = 2x,     <7n=2*"*uln,    ^w
S. Pincherle118 b applied pn+i(z) = (x - an) (x - A»)p»(*) to developments in series.
E. Study1180 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 armliftH fh* +i^/vn,r *~ u:i: ---- *---
M. d'Ocagne119 considered
C^-i? • • • > -^-W : of order p and generating ec
\$(*)=-Set
The existence of a common root a of \$(x) = 0, ^(z) =0 is a necessary and sufficient condition that the F'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'Ocagne120 considered the series with the law of recurrence
uin = aiQUi and generating equation
118Nouv. Ann. Math., (3), 2, 1883, 220-6; 3, 1884, 65-90; 9, 1890, 93-7; 11, 1892,526-532 (5, 1886, 257-272). Bull. 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).
117Comptes Rendus Paris, 104, 1887, 419-420; errata, 534.
"*Ibid., 108, 1889, 499-501.
118aSitzungsber Ak. Wiss. Wien (Math.), 97, Ha, 1888, 82-89.
118bAtti R. Accad. Lincei, Rendiconti, 5, 1889, 1, 8-12, 323-7.
118«Monatshefte Math. Phys., 2, 1891, 22-54.
119Bull. Soc. Math. France, 20, 1892, 121-2.
»°Comptes Reudus Paris, 115, 1892, 790-2; errata, 904.```