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 p—q 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. 123M£m. 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. i»Bull. 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.