1625 



Finite Summation. Recurring Series 



1625 



Recurring series, use in analysis of equations. 

 Frisiani, P. Mil. Effem. As. (1850) 3-. 



systems, generation by a linear differential 

 equation. Pincherle, S. Acta Mth. 16 (1892- 

 93) 341-. 



, 1st order and 2nd degree. Pincherle, S. 



Em. B. Ac. Line. Ed. 5 (1889) (Sem. 1) 8-, 



323-. 

 Eelation I 3 + 2 3 + ... + n 3 = (1 + 2 + ... + n) 2 . 



Gelin, E. Mathesis 14 (1894) 220-. 

 Series with alternate law of recurrence. 



Ocagne, M. d>. Par. S. Mth. Bll. 12 (1884) 



78-. 

 , doubly recurring. Betti, E. Tortolini A. 



8 (1857) 48-. 



of Fibonacci, on [u n + l=u n + u n -l]. 

 Cesaro, E. G. Teix. J. Sc. 6 (1885) 17-. 



, Fibonacci, peculiarities. Fontes, . As. 



Fr. C. B. (1894) (Pt. 2) 217-. 

 , general term found as with recurring series. 



Andre, D. Par. EC. Norm. A. 7 (1878) 



375-. 

 , geometrical, fractional number of terms. 



Choquet, C. N. A. Mth. 1 (1842) 74-. 



of quotients, Euclid's summation. Zeuthen, 

 H. G. (xn) Ts. Mth. 6 (1876) 168-, 181- ; 

 6 (1882) 97- ; 1 (1883) 145-. 



, self-repeating. Warburton, H. Camb. Ph. 



S. T. 9 (1856) (Pt. 4) 471-. 

 , whose series of differences are recurring. 



Raabe, J. L. Baumgartner Z. 6 (1829) 



307-. 

 , summation. (Dienger, Grunert Arch. 8 



213-214.) Arndt, F. Grunert Arch. 8 (1846) 



383-. 

 , of certain. Tait, P. G. Edinb. Mth. 



S. P. 3 (1885) 107-. 

 , class. Ocagne, M. d'. C. B. 115 



(1892) 790-, 904. 

 , , successive. Dantas Pereira, J. M. 



Lisb. Mm. Ac. So. 2 (1799) 168-. 

 Sum of a combinatory series, properties. 



Unferdinger, F. Grunert Arch. 26 (1856) 



227-. 



cubes = a square. Catalan, E. C. [1866] 



. Rm. At. N. Line. 20 (1867) 1-, 77-. 



of secondary numbers, theorem. 

 Barbier, E. Bll. Sc. Mth. As. 7 (1883) 42-. 



first n products of p consecutive numbers. 



Laisant, A. N. A. Mth. 6 (1867) 366-. 

 products of n quantities taken m together. 



Trowbridge, D. Des Moines Anal. 1 (1874) 



154-. 



r different terms of series. 

 Hammond, J. L. Mth. S. P. 7 (1875-76) 

 119-. 



reciprocals. Bouvier, L. C. Gergonne 



A. Mth. 15 (1824-25) 39-. 



. Blissard, J. QJ. Mth. 6 (1863) 



242-. 

 . Worontzof, M. M. QJ. Mth. 8 



(1867) 185-, 310-. 

 of first 1000 integers. Adams, J. C. 



B. A. Bp. (1877) (Sect.) 14-. 

 Summation of series, general term of which is 



na n , a being a given number. Bertrand, J. 

 N. A. Mth. 8 (1849) 354-. 



Summation of terms of recurring series, forma- 

 tion of integers by. Maillet, E. As. Fr. 

 C. B. (1896) (Pt. 2) 78-. 



Summations, successive, l m + 2 m + 3 m + . . . x m . 

 Lucas, E. [1877] Mess. Mth. 7 (1878) 



Sums of 4 algebraic series. Glaisher, J. W. L. 



[1877] Mess. Mth. 7 (1878) 119-. 

 powers of reciprocals of integers. Schlo- 



milch, O. X. Leip. Mth. Ps. B. 29 (1877) 



106-. 



products of an arithmetical progres- 

 sion, taken 2, 3, 4 ... together, calculation. 

 Lampe, E. Fschr. Mth. (1893-94) 409. 



natural numbers. Mollame, V. 



N. A. Mth. 5 (1886) 364-. 

 numbers, theorem. Lerch, . 



G. Teix. J. Sc. 8 (1887) 161-. 



series l n + 2 + . . . x n , l n - 2 n + . . . x n . 

 Glaisher, J. W. L. QJ. Mth. 30 (1899) 

 166-. 



Special series. 



Collins, E. St Pet. Ac. Sc. Mm. 8 (1817-18) 



242-. 



Schlomilch, O. Grunert Arch. 10 (1847) 342-. 

 Thacker, A. Camb. and Dubl. Mth. J. 5 (1850) 



243-. 



Mourgues, , N. A. Mth. 10 (1851) 78-. 

 Catalan, E. C. N. A. Mth. 15 (1856) 230-. 

 Siacci, F. (vm) A. Mt. 4 (1861) 46-. 

 Lucas, E. N. A. Mth. 9 (1870) 49-. 

 Amigues, E. N. A. Mth. 10 (1871) 79-. 

 Dostor, G. Arch. Mth. Ps. 54 (1872) 70-; 



57 (1875) 222-. 

 Lampe, E. [1877] Crelle J. Mth. 84 (1878) 



270-. 



Lucas, E. N. A. Mth. 16 (1877) 18-. 

 Pelndr, M. Casopis 6 (*1877) 279-; Fschr. 



Mth. (*1877) HO-. 

 Dostor, G. Arch. Mth. Ps. 63 (1879) 435-; 



64 (1879) 310-. 



Marchand, D. N. A. Mth. 20 (1881) 140-. 

 Schier, 0. Wien Ak. Sb. 85 (1882) (Ab. 2) 



503-. 

 Boije af Gennas, C. 0. Stockh. Ak. Hndl. 



Bh. 10 (1885) No. 4, 6 pp. 

 Slavik, J. Casopis 16 (1887) 121- ; Fschr. 



Mth. (1887) 240. 



Locchi, E. Mh. Mth. Ps. 4 (1893) 85-. 

 Nielsen, N. N. Ts. Mth. 4 (B) (1893) 1-; 



Fschr. Mth. (1893-94) 410-. 

 Barbette, E. Mathesis 14 (1894) 105-. 

 Jung, V. Casopis 28 (1899) 124- ; Fschr. Mth. 



(1899) 247. 



Tr 3 . 



r=l 



Genocchi, A. A. Mt. 7 (1865) 151-. 

 Cantor, M. Z. Mth. Ps. 11 (1866) 248-; 12 

 (1867) 170-. 



91 



