410 
History of the Theory of Numbers. 
[Chap. XVII 
such that, for I = 0, Uq= . . . = Wp_2 = 0, Up_i = 1 . If 4>o{x) = </)i(x)- • • </>„(x) , 
U n+p-l =2W r^+p^-l' ' "^"""w+P^-l' 
summed for all combinations of n's for which tzj + . . . -\-n^ = n. Application 
is made to the sum of a recurring series with a variable law of recurrence. 
M. d'Ocagne^-^ reproduced the last result, and gave a connected expo- 
sition of his earlier results and new ones. 
R. Perrin^" considered a recurring series U of order p ^ith the terms 
Uq,Ui,. . .. The general term of the A-th derived series of U is defined to be 
u 
(*)_ 
Un+1 
Wn+2 
Un+k 
• • 'Un+2k 
ies is zero, the law of recurrence of 
If any term of the (p — l)th derived ser 
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 U. This criterion is 
only a more convenient form of that of d'Ocagne.^^^'^^^ 
E. ]Maillet^"-^ noted that a necessarj^ condition that a law of recurrence 
of order p be reducible to one of order p—q is that ^(x) and ^{x) of 
d'Ocagne^^^ have q roots in common, the condition being also sufficient if 
$(x)=0 has only distinct roots. He found independently a criterion anal- 
ogous to that of Perrin^-- and studied series with two laws of recurrence. 
J. Neuberg^^ considered w„ = aw„_i+6u„_2 and found the general term 
of the series of Pisano. 
C. A. Laisant^^^ treated the case F a constant of d'Ocagne's^^^ 
u,{f{u)]=Fik). 
S. Lattes^^^ treated Wn+p=/(Wn+p-i>- • •> ^J» where / is an analytic 
function. 
M. .Amsler^^^ discussed recurring series by partial fractions. 
E. Netto,^-"'' L. E. Dickson,^-"' A. Ranum,^-^ and T. Hayashi'-^ gave 
the general term of a recurring series. N. Traverso^^" gave the general 
term for Q„= (n — l)(Q„_i+Q„_2) and u„= aUn-i+hu^_2. 
Traverso^^^ applied the theory of combinations with repetitions to express, 
as a function of p, the solution of Q„, = p(Qm-i+Qm-2+ • • • +Qm-n)- 
'"Jour, de l'6cole polyt., 64, 1894, 151-224. 
i=»Comptes Rendus Paris, 119, 1894, 990-3. 
i»M6m. Acad. Sc. Toulouse, (9), 7, 1895, 179-180, 182-190; Assoc, fran?., 1895, III, 233 [report 
with miscellaneous Dioph. equations of order n, Vol. II); Nouv. Ann. Math., (3), 14, 
1895, 152-7, 197-206. 
"*Mathesis, (2), 6, 1896, 88-92; Archive de mat., 1, 1896, 230. 
"*Bull. Soc. Math. France, 29, 1901, 145-9 
""Nouv. Ann. Math., (4), 10, 1910, 90-5. 
^"'-Amer. Math. Monthly, 10, 1903, 223-6. 
"'Bull. Amef. Math. Soc, 17, 1911, 457-461. 
"•/Wd., 18, 1912, 191-2. 
"oPeriodico di Mat., 29, 1913-4, 101-4; 145-160. 
"'Ibid., 31, 1915-6, 1-23, 49-70, 97-120, 145-163, 193-207 
i»>Comptes Rendus Paris, 150, 1910, 1106-9. 
"'"Monatshefte Math. Phys., 6, 1895, 285-290. 
