Illarionov, Andreĭ A. Hyperelliptic systems of sequences of rank 4. (English. Russian original) Zbl 1442.11051 Sb. Math. 210, No. 9, 1259-1287 (2019); translation from Mat. Sb. 210, No. 9, 59-88 (2019). The main result of the paper describes all \(1\)-periodic entire functions \(f\colon \mathbb{C} \rightarrow \mathbb{C}\) satisfying the functional equation \[f(x+y) f(x-y)=\varphi_{1}(x) \psi_{1}(y)+\cdots+\varphi_{N}(x) \psi_{N}(y)\tag{*}\] with \(N=4\). Such functions are closely connected with hyperelliptic system of sequences of finite rank, i.e. a sequences \(A, B\colon \mathbb{Z} \rightarrow \mathbb{C}\) such that the expansions \[ A(m+n) B(m-n)=\sum_{j=1}^{N_{0}} a_{j}(m) b_{j}(n) \] and \[ A(m+n+1) B(m-n)=\sum_{j=1}^{N_{1}} \tilde{a}_{j}(m) \tilde{b}_{j}(n) \] hold for some sequences \(a_j, b_j, \tilde{a}_j, \tilde{b}_j: \mathbb{Z} \rightarrow \mathbb{C}\) and \(N_0, N_1 \in\mathbb{Z}_{+}\). The paper describes different relations between hyperelliptic system of sequences and solutions of the equation \((*)\). Reviewer: Alexey Ustinov (Khabarovsk) Cited in 6 Documents MSC: 11B83 Special sequences and polynomials 39B32 Functional equations for complex functions 11B37 Recurrences 33E05 Elliptic functions and integrals Keywords:functional equations; elliptic functions; nonlinear recurrent sequences; addition theorems × Cite Format Result Cite Review PDF Full Text: DOI References: [1] M. Ward 1948 Memoir on elliptic divisibility sequences Amer. J. Math.70 31-74 · Zbl 0035.03702 · doi:10.2307/2371930 [2] A. N. W. Hone 2005 Elliptic curves and quadratic recurrence sequences Bull. London Math. Soc.37 2 161-171 · Zbl 1166.11333 · doi:10.1112/S0024609304004163 [3] A. N. W. Hone 2007 Sigma function solution of the initial value problem for Somos 5 sequences Trans. Amer. Math. Soc.359 10 5019-5034 · Zbl 1162.11011 · doi:10.1090/S0002-9947-07-04215-8 [4] Yu. N. Fedorov and A. N. W. 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