Chap. XVII] ALGEBRAIC THEORY OF RECURRING SERIES. 407 
E. Piccioli^^ noted that in Pisano's series 1, 1, 2, 3, . . ., 
according as A; is odd or even. 
T. A. Pierce®^ proved for the two functions HlZiil^aD of the roots ai 
of an equation with integral coefficients properties analogous to those of 
Lucas' Un, Vn. 
Algebraic Theory of Recurring Series. 
J. D. Cassini^°° and A. de Moivre^"^ treated series whose general term is 
a sum of a given number of preceding terms each multiplied by a constant. 
D. Bernoulli^°^ used such recurring series to solve algebraic equations. J. 
Stirling^°^ permitted variable multipUers. 
L. Euler^°^ studied ordinary recurring series and their application to 
solving equations. 
J. L. Lagrange^°^ made the subject depend on the integration of linear 
equations in finite differences, treating also recurring series with an additive 
term. The general term of such a series was found by V. Riccati.^°® 
P. S. Laplace^"^ made systematic use of generating functions and applied 
recurring series to questions on probability. 
J. L. Lagrange^°^ noted that if Ay^-\-Byi.^i-}- . . .-\-Nyt+n = is the 
recurring relation and if A+Bt-{- . . .+iVT = has distinct roots a, jS, . . ., 
the general term of the series is y^ = aa^+&/3'^+ • ■ • • For the case of multiple 
roots he stated a formula which G. F. Malfatti^°^ proved to be erroneous; 
the latter gave a new process explained for 2, 3 or 4 equal roots. 
Lagrange^^° had noticed independently his error and now gave the 
general term of a recurring series in the case of multiple roots by a more 
direct process than that of Malfatti. 
Pietro Paoli^^^ investigated the sum of a recurring series. 
98Periodico di Mat., 31, 1916, 284-7. 
s'Annals of Math., (2), 18, 1916, 53-64. 
""Histoire acad. roy. sc. Paris, annee 1680, 309. 
"iPhil. Trans. London, 32, 1722, 176; Miscellanea analytica, 1730, 27, 107-8; Doctrine of 
chances, ed. 2, 1738, 220-9. 
i»2Comm. Acad. Petrop., 3, ad annum 1728, 85-100. 
lO'Methodus differentialis, London, 1730, 1764. 
"^Introductio in analysin infinitorum, 1748, I, Chs. 4, 13, 17. Cf . C. F. Degen, Det K. Danske 
Vidensk. Selskabs Afhand., 1, 1824, 135; Oversigt. . .Forhand., 1818-9, 4. 
ii^Miscellanea Taurinensia, 1, 1759, Math., 33-42; Oeuvres, I, 23-36. 
"^Mem. present^s div. sav. Paris, 5, 1768, 153-174; Comm. Bonon., 5, 1767. Cf. M. Cantor, 
Geschichte Math., lY, 1908, 261. 
"^Mem. sav. etr. ac. sc. Paris, 6, annee 1771, 1774, p. 353; 7, annee 1773, 1776; Oeuvres, VIII, 
5-24, 69-197. M6m. ac. roy. sc. Paris, ann^e, 1779, 1782, 207; Oeuvres, X, 1-89 (ann^e 
1777, 99). 
"«Nouv. Mem. Ac. Berlin, annle 1775, 1777, 183-272; Oeuvres, IV, 151. 
losMem. mat. fis. soc. Ital., 3, 1786-7, 571. 
""Nouv. M6m. Ac. Sc. Berlin, ann^es 1792-3, 247; Oeuvres, V, 625-641 (p. 639 on the error). 
">Mem. Acad. Mantova, 1, 1795, 121. See Partitions in Vol. Ill of this History. 
