Far Eastern Mathematical Journal

To content of the issue

On the connection between hyperelliptic systems of sequences and functions

Illarionov A.A., Romanov M.A.

2017, issue 2, Ñ. 210-220

We study the connection between 1-periodic solutions of the functional equation $$ f(x+y)g(x-y)=\sum_{j=1}^N\varphi_j(x)\psi_j(y) \quad (x,y\in \CC) $$ and some sequences of special kind. As an application we solve the equation in the case when $g$ is Jacobi theta function.

functional equation, Weierstrass sigma function, Jacobi theta function, addition formula, elliptic functions, nonlinear sequences

Download the article (PDF-file)


[1] A.A. Illarionov, “Reshenie funktsional'nykh uravnenii, sviazannykh s ellipticheskimi funktsiiami”, Analiticheskaia teoriia chisel, Sbornik statei. K 80-letiiu so dnia rozhdeniia Anatoliia Alekseevicha Karatsuby, Tr. MIAN, 299, MAIK, M., 2017 (v pechati).
[2] V.A. Bykovskii, “Giperkvazimnogochleny i ikh prilozheniia”, Funkts. analiz i ego prilozheniia, 50:3, (2016), 34–46.
[3] R. Rochberg, L. Rubel, “A Functional Equation”, Indiana Univ. Math. J., 41:2, (1992), 363–376.
[4] M. Bonk, “The addition theorem of Weierstrass’s sigma function”, Math. Ann., 298:1, (1994), 591–610.
[5] P. Sinopoulos, “Generalized sine equations, I”, Aequationes Math., 48, (1994), 171–193.
[6] P. Sinopoulos, “Generalized sine equations, II”, Aequationes Math., 49, (1995), 122–152.
[7] M. Bonk, “The Characterization of Theta Functions by Functional Equations”, Abh. Math. Sem. Univ. Hamburg, 65, (1995), 29–55.
[8] P. Sinopoulos, “Generalized sine equations, III”, Aequationes Math., 51, (1996), 311–327.
[9] M. Bonk, “The addition formula for theta function”, Aequationes Math., 53:1–2, (1997), 54–72.
[10] P. Sinopoulos, “Contribution to the study of two functional equations”, Aequationes Math., 56, (1998), 91–97.
[11] A. Jarai, W. Sander, “On the characterization of Weierstrass’s sigma function”, in: Functional Equations – Results and Advances, Adv. Math., v. 3, Kluwer Acad. Publ., Dordrecht, 2002, 29–79.
[12] V. Bykovskii, Elliptic systems of sequences and functions, Torus Actions in Geometry, Topology, and Applications February 16–21, 2015, SkolTech, Moscow, Russia, http://www.skoltech.ru/app/data/uploads/sites/29/2015/02/Skolkovo Bykovskii.pdf.
[13] A.A. Illarionov, “Funktsional'noe uravnenie i sigma-funktsiia Veiershtrassa”, Funkts. analiz i ego prilozheniia, 50:4, (2016), 43–54.
[14] V.A. Bykovskii, “O range nechetnykh giperkvazimnogochlenov”, Dokl. RAN, 470:3, (2016), 255–256.
[15] M.D. Monina, “O tselochislennykh posledovatel'nostiakh Somos-8 i Somos-9”, Dal'nevost. matem. zhurn., 15:1, (2015), 70–75.
[16] M.D. Monina, “Mul'tiplikativnyi metod postroeniia tselochislennykh posledovatel'nostei Somos-8 i Somos-9”, Dal'nevost. matem. zhurn., 16:1, (2016), 62–68.
[17] M.O. Avdeeva, V.A. Bykovskii, “Giperellipticheskie sistemy posledovatel'nostei i funktsii”, Dal'nevost. matem. zhurn., 16:2, (2016), 115–122.
[18] A.O. Gel'fond, Ischislenie konechnykh raznostei, 2, Gos. izd. fiz.-mat. lit., Moskva, 1959, 400 s.

To content of the issue