On a multilinear functional equation. (English. Russian original) Zbl 1508.39016
Math. Notes 107, No. 1, 80-92 (2020); translation from Mat. Zametki 107, No. 1, 59-73 (2020).
Summary: The following functional equation is solved:
\[f\left( {{x_1} + z} \right) \cdots f\left( {{x_2} + z} \right)f\left( {{x_1} + \cdots + {x_{s - 1}} - z} \right) = {\phi_1}\left( x \right){\psi_1}\left( z \right) + \cdots + {\phi_m}\left( x \right){\psi_m}\left( z \right),\]
where \(x =(x_1,\dots,x_{s -1})\), for the unknowns \(f,{\psi_j}:\mathbb{C} \to \mathbb{C}\) and \({\phi_j}:{\mathbb{C}^{s - 1}} \to \mathbb{C}\) for \(s \geq 3\) and \(m \leq 4s - 5\).
MSC:
| 39B32 | Functional equations for complex functions |
| 33E05 | Elliptic functions and integrals |
| 14H42 | Theta functions and curves; Schottky problem |
Keywords:
functional equation; theta function; Weierstrass sigma function; elliptic function; addition theorems; multilinear functional-differential operatorsReferences:
| [1] | Buchstaber, V. M.; Leikin, D. V., Trilinear functional equations, Uspekhi Mat. Nauk, 60, 2362, 151-152 (2005) · Zbl 1093.14049 · doi:10.4213/rm1410 |
| [2] | Buchstaber, V. M.; Leikin, D. V., Russian Math. Surveys, 60, 2, 341-343 (2005) · Zbl 1093.14049 · doi:10.1070/RM2005v060n02ABEH000829 |
| [3] | Buchstaber, V. M.; Leikin, D. V., Addition laws on Jacobian varieties of plane algebraic curves, Trudy Mat. Inst. Steklova, Vol. 251: Nonlinear Dynamics, 54-126 (2005), Moscow: MAIK Nauka/Inteperiodika, Moscow · Zbl 1132.14024 |
| [4] | Buchstaber, V. M.; Leikin, D. V., Proc. Steklov Inst. Math., 251, 49-120 (2005) · Zbl 1132.14024 |
| [5] | Buchstaber, V. M.; Krichever, I. M., Integrable equations, addition theorems, and the Riemann-Schottky problem, Uspekhi Mat. Nauk, 61, 1367, 25-84 (2006) · Zbl 1134.14306 · doi:10.4213/rm1715 |
| [6] | Buchstaber, V. M.; Krichever, I. M., Russian Math. Surveys, 61, 1, 19-78 (2006) · Zbl 1134.14306 · doi:10.1070/RM2006v061n01ABEH004298 |
| [7] | Rochberg, R.; Rubel, L. A., A functional equation, Indiana Univ. Math. J., 41, 2, 363-376 (1992) · Zbl 0756.39012 · doi:10.1512/iumj.1992.41.41020 |
| [8] | Bonk, M., The addition theorem of Weierstrass’s sigma function, Math. Ann., 298, 4, 591-610 (1994) · Zbl 0791.39009 · doi:10.1007/BF01459753 |
| [9] | Sinopoulos, P., Generalized sine equation. I, Aequationes Math., 48, 2-3, 171-193 (1994) · Zbl 0811.39003 · doi:10.1007/BF01832984 |
| [10] | Bonk, M., The characterization of theta functions by functional equations, Abh. Math. Sem. Univ. Hamburg, 65, 29-55 (1995) · Zbl 0852.39009 · doi:10.1007/BF02953312 |
| [11] | Bonk, M., The addition formula for the theta function, Aequationes Math., 53, 1-2, 54-72 (1997) · Zbl 0881.39016 · doi:10.1007/BF02215965 |
| [12] | Járai, A.; Sander, W., On the characterization Weierstrass’s sigma function, Adv. Math. (Dordr.), Vol. 3: Functional Equations—Results and Advances, 29-79 (2002), Dordrecht: Kluwer Acad. Publ., Dordrecht · Zbl 0996.39020 |
| [13] | Bykovskii, V. A., Hyperquasipolynomials and their applications, Funktsional. Anal. Prilozhen., 50, 3, 34-46 (2016) · Zbl 1360.30023 · doi:10.4213/faa3244 |
| [14] | Bykovskii, V. A., Functional Anal. Appl., 50, 3, 193-203 (2016) · Zbl 1360.30023 · doi:10.1007/s10688-016-0147-y |
| [15] | Bykovskii, V. A., On the rank of odd hyperquasipolynomials, Dokl. Akad. Nauk, 470, 3, 255-256 (2016) · Zbl 1361.30046 |
| [16] | Illarionov, A. A., Functional equations and Weierstrass sigma-functions, Funktsional. Anal. Prilozhen., 50, 4, 43-54 (2016) · Zbl 1362.33025 · doi:10.4213/faa3253 |
| [17] | Illarionov, A. A., Functional Anal. Appl., 50, 4, 281-290 (2016) · Zbl 1362.33025 · doi:10.1007/s10688-016-0159-7 |
| [18] | Illarionov, A. A., Solution of functional equations related to elliptic functions, Trudy Mat. Inst. Steklova, Vol. 299: Analytic Theory of Numbers, 105-117 (2017), Moscow: MAIK Nauka/Inteperiodika, Moscow · Zbl 1397.39015 |
| [19] | Illarionov, A. A., Proc. Steklov Inst. Math., 299, 96-108 (2017) · Zbl 1397.39015 · doi:10.1134/S0081543817080065 |
| [20] | Illarionov, A. A., Hyperelliptic systems of sequences of rank 4, Mat. Sb., 210, 9, 59-88 (2019) · Zbl 1442.11051 · doi:10.4213/sm9050 |
| [21] | Illarionov, A. A., Sb. Math., 210, 9, 1259-1287 (2019) · Zbl 1442.11051 · doi:10.1070/SM9050 |
| [22] | Illarionov, A. A.; Romanov, M. A., Hyperquasipolynomials for the theta-function, Funktsional. Anal. Prilozhen., 52, 3, 84-87 (2018) · Zbl 1423.30021 · doi:10.4213/faa3507 |
| [23] | Illarionov, A. A.; Romanov, M. A., Functional Anal. Appl., 52, 3, 228-231 (2018) · Zbl 1423.30021 · doi:10.1007/s10688-018-0232-5 |
| [24] | Illarionov, A. A., Solution of a functional equation related to trilinear differential operators, Dal’nevost. Mat. Zh., 16, 2, 169-180 (2016) · Zbl 1375.39043 |
| [25] | Whittaker, E. T.; Watson, G. N., A Course of Modern Analysis, Pt. 2: Transcendental Functions (1996), Cambridge: Cambridge Univ. Press, Cambridge · Zbl 0951.30002 |
| [26] | Mumford, D., Tata Lectures on Theta. I (1983), Inc., Boston, Mass.: Birkhäuser Boston, Inc., Boston, Mass. · Zbl 0509.14049 |
| [27] | Stoilov, S., Theory of Functions of a Complex Variable (1962), Moscow: Inostr. Lit., Moscow |
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.
