On zeros of an entire function coinciding with exponential typequasi-polynomials, associated with a regular third-order differential operator on an interval
Кесiндiдегi үшiншi реттi регулярлы дифференциалдық оператормен байланысқан, экспоненциалды типтегi квазикөпмүшелiктермен сәйкес келетiн бүтiн функцияның нөлдерi жайлы
О нулях целой функции, совпадающей с квазиполиномами экспоненциального типа, связанной с регулярным дифференциальным оператором третьего порядка на отрезке
Imanbaev N.S. Kurmysh Ye.
2021E.A.Buketov Karaganda State University Publish House
Bulletin of the Karaganda University. Mathematics Series
2021#103Issue 344 - 53 pp.
In this paper, we consider the question on study of zeros of an entire function of one class, which coincides with quasi-polynomials of exponential type. Eigenvalue problems for some classes of differential operators on a segment are reduced to a similar problem. In particular, the studied problem is led by the eigenvalue problem for a linear differential equation of the third order with regular boundary value conditions in the space W23(0, 1). The studied entire function is adequately characteristic determinant of the spectral problem for a third-order linear differential operator with periodic boundary value conditions. An algorithm to construct a conjugate indicator diagram of an entire function of one class is indicated, which coincides with exponential type quasi-polynomials with comparable exponents according to the monograph by A.F. Leontyev. Existence of a countable number of zeros of the studied entire function in each series is proved, which are simultaneously eigenvalues of the above-mentioned third-order differential operator with regular boundary value conditions. We determine distance between adjacent zeros of each series, which lies on the rays perpendicular to sides of the conjugate indicator diagram, that is a regular hexagon on the complex plane. In this case, zero is not an eigenvalue of the considered operator, that is, zero is a regular point of the operator. Fundamental difference of this work is finding the corresponding eigenfunctions of the operator. System of eigenfunctions of the operator corresponding in each series is found. Adjoint operator is constructed.
eigenvalues , entire function , indicator diagram , operator , quasi-polynomials , regular periodic boundary value conditions , series , system of eigenfunctions , zeros
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South Kazakhstan State Pedagogical University, Shymkent, Kazakhstan
Institute of Mathematics and Mathematical Modeling, Almaty, Kazakhstan
South Kazakhstan State Pedagogical University
Institute of Mathematics and Mathematical Modeling
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