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Full text of "History Of The Theory Of Numbers - I"

CHAP, xvii]       ALGEBRAIC THEOEY OF RECURRING SERIES.               411
F. Nicita132 found many relations like 2att2-6n2= ~(-l)n between the two series ai = l, 02=2,.., an=%(an+i-an-i),. . .;  bi = l, b2 = 3,...,
Reference may be made to the text by A. Vogt133 and to texts and papers on difference equations cited in Encyklopadie der Math. Wiss., 1, 2, pp. 918, 935; Encyclopedic des Sc. Math., I, 4, 47-85.
A. Weiss134 expressed the general term tk of a recurring series of order r linearly in terms of tq, ^_i , . . . , ^^r+i, where q is an integer.
W. A. Whitworth135 proved that, if Co+C!X+c2x2-|-. . . is a convergent recurring series of order r whose first 2r terms are given, its scale of relation and sum to infinity are the quotients of certain determinants.
H. F. Scherk136 started with any triangle ABC and on its sides constructed outwards squares BCED, ACFGy. ABJH. Join the end points to form the hexagon DEFGHJ. Then construct squares on the three joining lines EF} GH, JD and again join the end points to form a new hexagon, etc. If at-, bi} Ci are the lengths of the j oining lines in the tth set, an+i = 5an_i — an_3 . The nth term is found as usual.
Sylvester137 solved ux=ux^1+(x-l)(x-2)ux,2' A. Tarn138 Created recurring series connected with the approximations to V2, Vs, Vs.
V. Schlegel139 called the development of (1— x— x2— . . . — of)"1 the (n— l)th series of Lame"; each coefficient is the sum of the n preceding. For n= 2, the series is that of Pisano.
References on the connection between Pisano' s series and leaf arrangement and golden section. (Kepler, Braun, etc.) have been collected by R. C. Archibald.140
Papers by C. F. Degen,141 A. F. Svanberg,142 and J. A. V6sz143 were not available for report.
di Mat., 32, 1917, 200-210, 226-36. l33Theorie der Zahlenreihen u. der Reihengleichung, Leipzig, 1911, 133 pp. *3<Jour. fiir Math., 38, 1849, 148-157. 1350xford, Cambridge and Dublin Mess. Math., 3, 1866, 117-121; Math. Quest. Educ. Times,
3, 1865, 100-1.
188 Abh. Naturw. Vereine zu Bremen, 1, 1868, 225-236. 137Math. Quest. Educ. Times, 13, 1870, 50. 138Math. Quest, and Solutions, 1, 1916, 8-12. "•El Progreso Mat., 4, 1894, 171-4. 140Amer. Math. Monthly, 25, 1918, 232-8. 141M<§m. Acad. Sc. St. PStersbourg, 1821-2, 71. . 142Nova Acta R. Soc. Sc. Upsaliensis, 11, 1839, 1. 143jBrtekez. a Math., Magyar Tudom. Ak. (Math. Memoirs Hungarian Ac. Sc.), 3, 1875, No. 1.