CHAP, xvii] ALGEBKAIC THEORY OF RECURRING SERIES. 407
E. Piccioli98 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 II-=i(l =*=<*/") of the roots at-of an equation with integral coefficients properties analogous to those of Lucas' un, vn.
ALGEBRAIC THEOBY OF RECURRING SERIES.
J. D. Cassini100 and A. de Moivre101 treated series whose general term is a sum of a given number of preceding terms each multiplied by a constant. D. Bernoulli102 used such recurring series to solve algebraic equations. J. Stirling103 permitted variable multipliers.
L. Euler104 studied ordinary recurring series and their application to solving equations.
J. L. Lagrange105 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.106
P. S. Laplace107 made systematic use of generating functions and applied recurring series to questions on probability.
J. L. Lagrange108 noted that if Ayt+Byt+i+ . . . +Nyt+n = Q is the recurring relation and if A-\-Bt+ . . . +Arr = 0 has distinct roots a, |3, . . ., the general term of the series ir- --- *'**»*• ™ •-
roots he stated a formula w^ the latter gave a new proces:
Lagrange110 had noticed mu^ general term of a recurring series A direct process than that of Malfatti.
Pietro Paoli111 investigated the sum of a recurring series.
98Periodico di Mat., 31, 1916, 284-7. "Annals of Math., (2), 18, 1916, 53-64. 100Histoire acad. roy. sc. Paris, anne"e 1680, 309. 101Phil. Trans. London, 32, 1722, 176; Miscellanea analytica, 1730, 27, 107-8; Doctrine of
chances, ed. 2, 1738, 220-9.
102Comm. Acad. Petrop., 3, ad annum 1728, 85-100. 103Methodus differentialis, London, 1730, 1764. 104Introductio in analysin mfinitorum, 1748, I, Chs. 4, 13, 17. Cf. C. F. Degen, Det K. Danske
Vidensk. Selskabs Afhand., 1, 1824, 135; Oversigt. . .Forhand., 1818-9, 4. 10BMiscellanea Taurinensia, 1, 1759, Math., 33-42; Oeuvres, I, 23-36. loflM6m. pre'sente's div. sav. Paris, 5, 1768, 153-174; Comm. Bonon., 5, 1767. Cf. M. Cantor,
Geechichte Math., IV, 1908, 261. 107M6m. sav. 6tr. ac. sc. Paris, 6, anne'e 1771, 1774, p. 353; 7, ann6e 1773, 1776; Oeuvres, VIII,
5-24, 69-197. Me"rn. ac. roy. sc. Paris, anne'e, 1779, 1782, 207; Oeuvres, X, 1-89 (anne'e
1777, 99).
108Nouv. Me"m. Ac. Berlin, anne'e 1775, 1777, 183-272; Oeuvres, IV, 151. 1(19Mem. mat. fis. soc. Ital., 3, 1786-7, 571.