408 History of the Theory of Numbers. [Chap, xvii 
J. B. Fourier's"^" error in appl^dng recurring series to the solution of 
numerical equations was pointed out by R. Murphy.^^^'' 
P. Frisiani"^" applied recurring series to the solution of equations. 
E. Betti^^^'^ emploj'ed doubly recurring series to solve equations in two 
unknowns, by extending the method of Bernoulli. ^°' 
W. Scheibner^^- considered a series with a three-term recursion formula, 
deduced the linear relation between any three terms, not necessarily con- 
secutive, and applied his results to continued fractions and Gauss' h>TDer- 
geometric series. 
D. Andr^^^^ deduced the generating equation of a recurring series F» 
from that of a recurring series L'„ given a Unear homogeneous relation 
between the terms F, multiplied by constants and the terms C/„, C/„_i, . . ., 
multipUed by polynomials in n. 
D. Andr^^^^ considered a series Ui, U2, . . ., with 
where w„, X„ are given functions of n, X„ being an integer ^n — 1, while 
Ai"^ is a given function of k, n. It is proved that 
C/„ = 2 ^(n, p)u„ ^(n, p) =i:AiyA[y ...., 
p=i 
where the second summation extends over all sets of integral solutions of 
k\-\-k2+ ■ ■ ■=n-p, ni = ki+p, n< = A\+n<_i (0</:<<Xn,). 
Application is made to eight special types of series. 
D. Andr^"^ discussed the sums of the series whose general terms are 
n(/i+l) . . . {n+p - 1)' {an+j3) ! 
where w„ is the general term of any recurring series. 
G. de Longchamps"^" proved the first result by Lagrange^^^ and 
expressed y^ as a sj-mmetric function of the distinct roots a, /3, . . . . He"^* 
reduced Un = AiU„_i+ . . +A^L''„_„+/(n), where / is a polj^nomial of de- 
gree p, to the case f(n) = by making a substitution ?7„= F„-f-Xon^+ . . -\-\p. 
C. A. Laisant^^^*" studied the ratios of consecutive terms of recurring 
series, in particular for Pisano's series. 
'""Analyse des Equations, Paris, 1831. 
'»»Phil. Mag., (3), 11, 1837, 38-40. 
"'•^Effemeridi .\stronomiche di Milano, 1850, 3. 
"I'^Annali di Sc. Mat. Fis., 8, 1857, 48-61. 
"«Berichte Gesell. Wiss. Leipzig (Math.), 16, 1864, 44-68. 
i"Bull. Soc. Math. France, 6, 1877-8, 166-170. 
"«Aim. 8C. r^cole norm, sup., (2), 7, 1878, 375-408; 9, 1880, 209-226. Summary in Bull, dea 
Sc. Math., (2), 1, I, 1877, 350-5. 
"HIbmptes Rendus Paris, 86, 1878, 1017-9; 87, 1878, 973-5. 
'"^Assoc. frang., 9, 1880, 91-6. 
"»*7&id., 1885, II, 94-100. 
»"«Bull. dea Sc. Math., (2), 5, I, 1881, 218-249. 
