Chap. XVII] ALGEBRAIC THEORY OF RECURRING SERIES. 411 
F. Nicita^^^ found many relations like 2aJ^ — hn^= — { — !)'' between 
the two series ai = l, 02 = 2,.., an = 2(an+i — «n-i), • • • ; &i = l, 62 = 3,..., 
&n = l(^n+l~-^n-l)j • • •• 
Reference may be made to the text by A. Vogt^^^ and to texts and papers 
on difference equations cited in Encyklopadie der Math. Wiss., I, 2, pp. 918, 
935; Encyclopedie des Sc. Math., I, 4, 47-85. 
A. Weiss^^^ expressed the general term 4 of a recurring series of order 
T linearly in terms of tq, tg_i , . . . , ^q-r+i, where q is an integer. 
W. A. Whitworth^^^ proved that, if Co+CiX+C2X^+ ... 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. Scherk^^^ Started with any triangle ABC and on its sides con- 
structed outwards squares BCED, ACFG, 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 tti, hi, Ci are the lengths of the joining lines in the iih. set, a„+i = 5a„_i — a„_3. 
The nth term is found as usual. 
Sylvester^" solved Uj,= u^_i-\-{x — \){x — 2)Ux_2' A._ Tarn^^^ treated 
recurring series connected with the approximations to \^2, Vs, Vs. 
V. SchlegeP^^ called the development of {1 — x—x^— . . . —x'')~'^ 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 arrange- 
ment and golden section (Kepler, Braun, etc.) have been collected by R. C. 
Archibald."^ 
Papers by C. F. Degen,^^^ A. F. Svanberg,^^^ and J. A. Vesz"^ were not 
available for report. 
"2Periodico di Mat., 32, 1917, 200-210, 226-36. 
^'^Theorie der Zahlenreihen u. der Reihengleichung, Leipzig, 1911, 133 pp. 
"4Jour. fur Math., 38, 1849, 148-157. 
"^Oxford, Cambridge and Dublin Mess. Math., 3, 1866, 117-121; Math. Quest. Educ. Times, 
3, 1865, 100-1. 
i36Abh. Naturw. Vereine zu Bremen, 1, 1868, 225-236. 
• i"Math. Quest. Educ. Times, 13, 1870, 50. 
•38Math. Quest, and Solutions, 1, 1916, 8-12. 
139E1 Progreso Mat., 4, 1894, 171-4. 
""Amer. Math. Monthly, 25, 1918, 232-8. 
i"M6m. Acad. Sc. St. Petersbourg, 1821-2, 71. 
i«2Nova Acta R. Soc. Sc. UpsaUensis, 11, 1839, 1. 
i«EIrtekez. a Math., Magyar Tudom. Ak. (Math. Memoirs Hungarian Ac. Sc), 3, 1875, No. 1. 
