×

On the Somos-4 sequence. (Russian. English summary) Zbl 1411.11013

Summary: In [Bull. Lond. Math. Soc. 37, No. 2, 161–171 (2005; Zbl 1166.11333)] A. N. W. Hone has obtained an explicit formula for Somos-4 sequences in terms of the Weierstrass sigma function. The proof is valid only for sequences without zero terms. We prove that arbitrary sequence satisfying the Somos-4 equation is determined by the same formula iff it satisfies some determinant identity (analogue of the formula of addition). We give examples of sequences that satisfy the Somos-4 equation but do not satisfy the determinant identity.

MSC:

11B37 Recurrences
33E05 Elliptic functions and integrals

Citations:

Zbl 1166.11333
Full Text: MNR

References:

[1] M. Ward, “Memoir on elliptic divisibility sequences”, Amer. J. Math., 70 (1948), 31-74 · Zbl 0035.03702 · doi:10.2307/2371930
[2] R. M. Robinson, “Periodicity of Somos sequences”, Proc. AMS, 116:3 (1992), 613-619 · Zbl 0774.11009 · doi:10.1090/S0002-9939-1992-1140672-5
[3] R. Shipsey, Elliptic Divisibility Sequences, PhD Thesis, L.: Univ. London, 2000
[4] C. S. Swart, Elliptic curves and related sequences, PhD Thesis, L.: Royal Holloway, Univ. London, 2003
[5] A. N. W. Hone, “Elliptic curves and quadratic reccurence sequences”, Bull. Lond. Math. Soc., 37 (2005), 161-171 · Zbl 1166.11333 · doi:10.1112/S0024609304004163
[6] A. J. van der Poorten, “Hyperelliptic curves, continued fractions, and Somos sequences”, Dynamics \(\&\) Stochastics, Lecture Notes-Monograph Series, v. 48, ed. Denteneer, Dee and Hollander, Frank den and Verbitskiy, Evgeny, Institute of Mathematical Statistics, Beachwood, Ohio, USA, 2006, 212-224 · Zbl 1221.11015
[7] A. J. van der Poorten, C. S. Swart, “Recurrence relations for elliptic sequences: every Somos 4 is a Somos \(k\)”, Bull. Lond. Math. Soc., 38:4 (2006), 546-554 · Zbl 1169.11013 · doi:10.1112/S0024609306018534
[8] A. N. W. Hone, “Sigma function solution of the initial value problem for Somos 5 sequences”, Trans. AMS, 359:10 (2007), 5019-5034 · Zbl 1162.11011 · doi:10.1090/S0002-9947-07-04215-8
[9] A. N. Hone, C. Swart, “Integrality and the Laurent phenomenon for Somos 4 and Somos 5 sequences”, Math. Proc. Camb. Philos. Soc., 145:1 (2008), 65-85 · Zbl 1165.11018 · doi:10.1017/S030500410800114X
[10] A. N. Hone, “Analytic solutions and integrability for bilinear recurrences of order six”, Applicable Analysis: An International Journal, 89:4 (2010), 473-492 · Zbl 1185.11012 · doi:10.1080/00036810903329977
[11] X.Ma, “Magic determinants of Somos sequences and theta functions”, Discrete Math., 210:1 (2010), 1-5 · Zbl 1217.11016
[12] Y. Fedorov, A. N. Hone, “Sigma-function solution to the general Somos-6 recurrence via hyperelliptic Prym varieties”, Journal of Integrable Systems, 1:1 (2016) · Zbl 1400.37083 · doi:10.1093/integr/xyw012
[13] V. A. Bykovskii, A. V. Ustinov, “Somos-4 i ellipticheskie sistemy posledovatelnostei”, DAN, 471:1 (2016), 7-10 · Zbl 1375.11023
[14] V. A. Bykovskii, “Giperkvazimnogochleny i ikh prilozheniya”, Funkts. analiz i ego prilozheniya, 50:3 (2016), 34-46 · Zbl 1360.30023 · doi:10.4213/faa3244
[15] A. A. Illarionov, “Funktsionalnoe uravnenie i sigma-funktsiya Veiershtrassa”, Funkts. analiz i ego prilozheniya, 50:4 (2016), 43-54 · Zbl 1362.33025 · doi:10.4213/faa3253
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.