• Bulletin of the Korean Mathematical Society
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• v.46 no.6
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• pp.1153-1158
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• 2009

Given two integral self-reciprocal polynomials having the same modulus at a point $z_0$ on the unit circle, we show that the minimal polynomial of $z_0$ is also self-reciprocal and it divides an explicit integral self-reciprocal polynomial. Moreover, for any two integral self-reciprocal polynomials, we give a sufficient condition for the existence of a point $z_0$ on the unit circle such that the two polynomials have the same modulus at $z_0$.

• Communications of the Korean Mathematical Society
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• v.12 no.3
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• pp.645-653
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• 1997

We study centrally symmetric orthogonal polynomials satisfying an admissible partial differential equation of the form $$ Au_{xx} + 2Bu_{xy} + Cu_{yy} + Du_x + Eu_y = \lambda_n y, $$ where $A, B, \cdots, E$ are polynomials independent of n and $\lambda_n$ is the eignevalue parameter depending on n. We show that they are either the product of Hermite polymials or the circle polynomials up to a complex linear change of variables. Also we give some properties of them.

• Communications of the Korean Mathematical Society
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• v.33 no.4
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• pp.1303-1308
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• 2018

In this paper, we consider both maximum modulus and minimum modulus on a circle of some polynomials. These give rise to interesting examples that are about moduli of Chebyshev polynomials and certain sums of polynomials on a circle. Moreover, we obtain some root locations of difference quotients of Chebyshev polynomials.

• Bulletin of the Korean Mathematical Society
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• v.47 no.6
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• pp.1189-1194
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• 2010

Kim and Park investigated the distribution of zeros around the unit circle of real self-reciprocal polynomials of even degrees with five terms, where the absolute value of middle coefficient equals the sum of all other coefficients. In this paper, we extend some of their results to the same kinds of polynomials with arbitrary many nonzero terms.

• Bulletin of the Korean Mathematical Society
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• v.44 no.3
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• pp.461-473
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• 2007

For integral self-reciprocal polynomials P(z) and Q(z) with all zeros lying on the unit circle, does there exist integral self-reciprocal polynomial $G_r(z)$ depending on r such that for any r, $0{\leq}r{\leq}1$, all zeros of $G_r(z)$ lie on the unit circle and $G_0(z)$ = P(z), $G_1(z)$ = Q(z)? We study this question by providing examples. An example answers some interesting questions. Another example relates to the study of convex combination of two polynomials. From this example, we deduce the study of the sum of certain two products of finite geometric series.

• Bulletin of the Korean Mathematical Society
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• v.41 no.4
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• pp.641-646
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• 2004

A convex combination of two products with same degree of finitely many finite geometric series with each having even degree does not always have all its zeros on the unit circle. However, in this paper, we show that a polynomial obtained by just adding a finite geometric series multiplied by a large constant to such a convex combination has all its zeros on the unit circle.

• Communications of the Korean Mathematical Society
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• v.27 no.1
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• pp.175-183
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• 2012

Let $\mathcal{P}_n$ be the set of all monic integral self-reciprocal poly-nomials of degree n whose all zeros lie on the unit circle. In this paper we study the following question: For P(z), Q(z)${\in}\mathcal{P}_n$, does there exist a continuous mapping $r{\rightarrow}G_r(z){\in}\mathcal{P}_n$ on [0, 1] such that $G_0$(z) = P(z) and $G_1$(z) = Q(z)?.

• The Pure and Applied Mathematics
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• v.24 no.2
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• pp.69-77
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• 2017

If q(z) is a polynomial of degree n with all zeros in the unit circle, then the self-reciprocal polynomial $q(z)+x^nq(1/z)$ has all its zeros on the unit circle. One might naturally ask: where are the zeros of $q(z)+x^nq(1/z)$ located if q(z) has different zero distribution from the unit circle? In this paper, we study this question when $q(z)=(z-1)^{n-k}(z-1-c_1){\cdots}(z-1-c_k)+(z+1)^{n-k}(z+1+c_1){\cdots}(z+1+c_k)$, where $c_j$ > 0 for each j, and q(z) is a 'zeros dragged' polynomial from $(z-1)^n+(z+1)^n$ whose all zeros lie on the imaginary axis.

• Kyungpook Mathematical Journal
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• v.57 no.4
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• pp.537-543
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• 2017

In this paper, we generalize some earlier well known results by considering polynomials of lacunary type having some zeros at origin and rest of the zeros on or outside the boundary of a prescribed disk.

• Journal of the Chungcheong Mathematical Society
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• v.22 no.4
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• pp.739-741
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• 2009

Given $\alpha$ > 1, there exist $C(1/\alpha)$-polynomials of the form $z^n-\sum_{k=0}^{n-1}\;a_kz^k$, where $\sum_{k=0}^{n-1}\;a_k=1$, $a_{n-1}>0$ and $a_k{\geq}0$ for each k. In this paper, we obtain lower bounds for $a_{n-1}$.