On The Solution of an Infinite System of Discrete Equations

Abstract

 In this work, we construct the transformation operator for the infinite system of the difference equations 

$a_{n-2}y_{n-2}+b_{n-1}y_{n-1}+c_{n}y_{n}+b_{n}y_{n+1}+a_{n}y_{n+2}=\lambda y_{n}$ $(n=1,2,...)$,

where $a_{n}\neq0,$ $b_{n},$ $c_{n}$ $(n=1,2,3,...)$ are given complex numbers, investigate some important properties of the special solutions of the difference system.

Keywords

• Difference equation
• Jacobi matrix
• Transformation operators
• Direct problem
• Inverse problem

References

1. Guseinov, G.Sh., {\em Determination of an infinite Jacobi matrix from scattering data}, Doklady Akademii Nauk SSSR, \textbf{227}(6)(1976), 1289--1292.
2. Guseinov, G.Sh., {\em The inverse problem of scattering theory for a second order difference equation on the whole axis}, Doklady Akademii Nauk SSSR, \textbf{17}(1976), 1684--1688.
3. Guseinov, G.Sh., {\em Determination of an infinite non-self-adjoint Jacobi matrix from its generalized spectral function}, Mathematical Notes, \textbf{23}(2)(1978), 130--136.
4. Guseinov, I.M., Khanmamedov, Ag. Kh., {\em The $t\rightarrow\infty$ asymptotic regime of the Cauchy problem solution for the Toda chain with threshold-type Initial data}, Theoretical and Mathematical Physics, \textbf{119}(1999), 739--749.
5. Khanmamedov, Ag. Kh., {\em Inverse scattering problem for a discrete Sturm-Liouville Operator on the entire line}, Doklady Akademii Nauk, \textbf{431}(1)(2010), 25--26.
6. Kishakevich, Yu.L., {\em Spectral function of Marchenko type for a difference operator of an even order}, Mathematical Notes, \textbf{11}(4)(1972), 266--271.
7. Marchenko, V.A., Sturm-Liouville Operators and Their Applications, Naukova Dumka, Kiev, 1977.
8. Zagorodnyuk, S., {\em The direct and inverse spectral problems for $\mathit{(2N+1)}$-diagonal complex transposition-antisymmetric matrices}, Methods Funct. Anal. Topology, \textbf{14}(2)(2008), 124--131.

9. Zagorodnyuk, S.M., {\em Direct and inverse spectral problems for $\mathit{(2N+1)}$-diagonal, complex, symmetric, non-Hermitian matrices}, Serdica Mathematical Journal, \textbf{30}(4)(2004), 471--482.