×

Functional equations and Weierstrass sigma-functions. (English. Russian original) Zbl 1362.33025

Funct. Anal. Appl. 50, No. 4, 281-290 (2016); translation from Funkts. Anal. Prilozh. 50, No. 4, 43-54 (2016).
Summary: It is proved that if an entire function \(f: \mathbb{C}\to\mathbb{C}\) satisfies an equation of the form \(f(x+y)f(x-y)=\alpha_1(x)\beta_1(y) + \alpha_2(x)\beta_2(y) + \alpha_3(x)\beta_3(y)\), \(x,y\in\mathbb C\), for some \(\alpha_j,\beta_j: \mathbb{C}\to\mathbb{C}\) and there exist no \(\tilde{\alpha}_j\) and \(\tilde{\beta}_j\) for which \(f(x + y)f(x - y) = \tilde{\alpha}_1(x)\tilde{\beta}_1(y) + \tilde{\alpha}_2(x)\tilde{\beta}_2(y)\), then \(f(z) = \exp(Az^2 + Bz + C)\cdot \sigma_\Gamma(z - z_1)\cdot \sigma_\Gamma(z - z_2)\), where \(\Gamma\) is a lattice in \(\mathbb{C}\); \(\sigma_\Gamma\) is the Weierstrass sigma-function associated with \(\Gamma\); \(A,B,C,z_1,z_2\in\mathbb{C}\); and \(z_1-z_2\notin (\frac{1}{2}\Gamma)\setminus\Gamma\).

MSC:

33E05 Elliptic functions and integrals
30D05 Functional equations in the complex plane, iteration and composition of analytic functions of one complex variable
39B32 Functional equations for complex functions
Full Text: DOI

References:

[1] V. M. Bukhshtaber and D. V. Leikin, “Trilinear functional equations,” Uspekhi Mat. Nauk, 60:(2) (2005), 151-152; English transl.: Russian Math. Surveys, 60:2 (2005), 341-343. · Zbl 1093.14049 · doi:10.4213/rm1410
[2] V. M. Bukhshtaber and D. V. Leikin, “Addition laws on Jacobian varieties of plane algebraic curves,” Trudy Mat. Inst. Steklov., 251 (2005), 54-126; English transl.: Proc. Steklov Inst. Math., 251 (2005), 49-120. · Zbl 1132.14024
[3] V. M. Bukhshtaber and I. M. Krichever, “Integrable equations, addition theorems, and the Riemann-Schottky problem,” UspekhiMat. Nauk, 61:(1) (2006), 25-84; English transl.: Russian Math. Surveys, 61:(1) (2006), 19-78. · Zbl 1134.14306 · doi:10.4213/rm1715
[4] V. A. Bykovskii, “Hyperquasipolynomials and their applications,” Funkts. Anal. Prilozhen., 50:(3) (2016), 34-46; English transl.: Functional Anal. Appl., 50:3 (2016), 193-203. · Zbl 1360.30023 · doi:10.4213/faa3244
[5] S. Janson, J. Peetre, and R. Wallsten, “A new look on Hankel forms over Fock space,” Studia Math., 95:1 (1989), 33-41. · Zbl 0703.47018 · doi:10.1002/sapm198981133
[6] R. Rochberg and L. Rubel, “A Functional Equation,” Indiana Univ. Math. J., 41:(2) (1992), 363-376. · Zbl 0756.39012 · doi:10.1512/iumj.1992.41.41020
[7] M. Bonk, “The addition formula for theta function,” Aequationes Math, 53:1-2 (1997), 54-72. · Zbl 0881.39016 · doi:10.1007/BF02215965
[8] M. Bonk, “The addition theorem of Weierstrass’s sigma function,” Math. Ann., 298:(1) (1994), 591-610. · Zbl 0791.39009 · doi:10.1007/BF01459753
[9] M. Bonk, “The characterization of theta functions by functional equations,” Abh. Math. Sem. Univ. Hamburg, 65 (1995), 29-55. · Zbl 0852.39009 · doi:10.1007/BF02953312
[10] Jarai, A.; Sander, W., On the characterization of Weierstrass’s sigma function, 29-79 (2002), Dordrecht · Zbl 0996.39020
[11] C. M. Cosgrove, “Higher-order Painlevé equations in the polynomial class I. Bureau Symbol P2,” Stud. Appl. Math., 104:(1) (2000), 1-65. · Zbl 1136.34350 · doi:10.1111/1467-9590.00130
[12] S. Stoilow, Teoria functiilor de o variabilă complexă, Vol. 1: Notiuni si principii fundamentale, Editura de Stat Didactica si Pedagogica, Bucharest, 1954; Russian transl.: Inostrannaya Literatura, Moscow, 1962.
[13] G. Peano, “Sur le déterminant wronskien,” Mathesis IX, 1889, 110-112. · JFM 21.0153.01
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.