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410                        HlSTOHY OF THE THEORY OF  NUMBERS.             [CHAP. XVII
such that, for i=0, w0=    =Wj,_2 = 0, wp_i = l.   If <t>o(x) = $i(z)*   <l>m(x),
summed for all combinations of n's for which nx + . .. -\-nm=n. Application is made to the sum of a recurring series with a variable law of recurrence.
M. d'Ocagne121 reproduced the last result, and gave a connected exposition of his earlier results and new ones.
R. Perrin122 considered a recurring series U of order p with the terms The general term of the fcth derived series of U is defined to be
un      un+1    ... un+k
If any term of the (p l)th derived series is zero, the law of recurrence of the given series U is reducible (to one of lower order). If also any term of the (p 2)th derived series is zero, continue until we get a non-vanishing determinant; then its order is the minimum order of 17. This criterion is only a more convenient form of that of d'Ocagne.119'm
E. Maillet123 noted that a necessary condition that a law of recurrence of order p be reducible to one of order pq is that $($) and *&(x) of d'Ocagne119 have q roots in common, the condition being also sufficient if <(z) = 0 has only distinct roots. He found independently a criterion analogous to that of Perrin122 and studied series with two laws of recurrence.
J. Neuberg124 considered wn = aun__i+mn-2 and found the general term of the series of Pisano.
C. A. Laisant125 treated the case F a constant of d'Ocagne's121 uk{f(u)}=F(k).
S. Latt&s126 treated un+p=f(un+.^i,..., wn), where / is an analytic function.
M. Amsler127 discussed recurring series by partial fractions.
E. Netto,127a L. E. Dickson,1276 A. Ranum,128 and T. Hayashi129 gave the general term of a recurring series. N. Traverso130 gave the general term for Qn=(n~l)(Qn_i+Qn_2) and ttn=<mn_i+6ttn_2.
Traverso131 applied the theory of combinations with repetitions to express, as a function of p, the solution of Qm = p(Qm-i+Qm-2+ ... +&>_>)._______
121 Jour, de I'gcole polyt., 64, 1894, 151-224.
122Comptes Rendus Paris, 119, 1894, 990-3.
123Mm. Acad. Sc. Toulouse, (9), 7, 1895,179-180, 182-190; Assoc. frang., 1895, 111, 233 [report
with miscellaneous Dioph. equations of order n, Vol. 11]; Nouv. Ann. Math., (3), 14,
1895, 152-7, 197-206.
124Mathesis, (2), 6, 1896, 88-92; Archive de mat., 1, 1896, 230.
iBull. Soc. Math. France, 29,1901,145-9.          ^Comptes Rendus Paris, 150,1910, 1106-9.
127Nouv. Ann. Math., (4), 10, 1910, 90-5.             127aMonatshefte Math. Phys., 6,1895, 285-290.
127&Amer. Math. Monthly, 10, 1903, 223-6. l28Bull. Amer. Math. Soc., 17,1911, 457-461. 129/bwZ., 18, 1912, 191-2. 130Periodico di Mat.. 29. 1913-4. 101-4: 145-1 fin.