• 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 $z_0$.

• 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.59 no.5
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• pp.1269-1277
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• 2022

In this paper we consider a sequence of polynomials defined by some recurrence relation. They include, for instance, Poupard polynomials and Kreweras polynomials whose coefficients have some combinatorial interpretation and have been investigated before. Extending a recent result of Chapoton and Han we show that each polynomial of this sequence is a self-reciprocal polynomial with positive coefficients whose all roots are unimodular. Moreover, we prove that their arguments are uniformly distributed in the interval [0, 2𝜋).

• Bulletin of the Korean Mathematical Society
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• v.53 no.3
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• pp.833-842
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• 2016

For a certain kind of reciprocal polynomials $P(x),Q(x){\in}{\mathbb{Z}}[x]$, their sums are considered. We demonstrate that the Mahler measure of polynomials plays a role to prove the irreducibility of the sums over the field of rationals.

• 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)?.

• 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.

• Journal of the Korean Mathematical Society
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• v.45 no.3
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• pp.835-840
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• 2008

The main result of this paper shows that every reciprocal Littlewood polynomial, one with {-1, 1} coefficients, of odd degree at least 7 has at least five unimodular roots, and every reciprocal Little-wood polynomial of even degree at least 14 has at least four unimodular roots, thus improving the result of Mukunda. We also give a sketch of alternative proof of the well-known theorem characterizing Pisot numbers whose minimal polynomials are in $$A_N=\{[{X^d+ \sum\limits^{d-1}_{k=0} a_k\;X^k{\in} \mathbb{Z}[X]\;:\;a_k={\pm}N,\;0{\leqslant}k{\leqslant}d-1}\}$$ for positive integer $N{\geqslant}2$.

• Bulletin of the Korean Mathematical Society
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• v.50 no.3
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• pp.983-991
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• 2013

For each real number $n$ > 6, we prove that there is a sequence $\{pk(n,z)\}^{\infty}_{k=1}$ of fourth degree self-reciprocal polynomials such that the zeros of $p_k(n,z)$ are all simple and real, and every $p_{k+1}(n,z)$ has the largest (in modulus) zero ${\alpha}{\beta}$ where ${\alpha}$ and ${\beta}$ are the first and the second largest (in modulus) zeros of $p_k(n,z)$, respectively. One such sequence is given by $p_k(n,z)$ so that $$p_k(n,z)=z^4-q_{k-1}(n)z^3+(q_k(n)+2)z^2-q_{k-1}(n)z+1$$, where $q_0(n)=1$ and other $q_k(n)^{\prime}s$ are polynomials in n defined by the severely nonlinear recurrence $$4q_{2m-1}(n)=q^2_{2m-2}(n)-(4n+1)\prod_{j=0}^{m-2}\;q^2_{2j}(n),\\4q_{2m}(n)=q^2_{2m-1}(n)-(n-2)(n-6)\prod_{j=0}^{m-2}\;q^2_{2j+1}(n)$$ for $m{\geq}1$, with the usual empty product conventions, i.e., ${\prod}_{j=0}^{-1}\;b_j=1$.

• Journal of Integrative Natural Science
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• v.5 no.2
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• pp.131-134
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• 2012

Bae and Kim displayed a sequence of 4th degree self-reciprocal polynomials whose maximal zeros are related in a very nice and far from obvious way. Kim showed that the auxiliary polynomials in their results are related to Chebyshev polynomials. In this paper, we study two sequences of polynomials satisfying the recurrence of the auxiliary polynomials with generalized initial conditions. We obtain same results with the auxiliary polynomials from a sequence, and some interesting conjectural properties about resultants and discriminants from another sequence.

• Journal of Integrative Natural Science
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• v.4 no.3
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• pp.227-228
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• 2011

Bae and Kim displayed a sequence of 4th degree self-reciprocal polynomials whose maximal zeros are related in a very nice and far from obvious way. The auxiliary polynomials in their results that parametrize their coefficients are of significant independent interest. In this note we show that such auxiliary polynomials are related to Chebyshev polynomials.