Concerning dense subideals in commutative Banach algebras

Żelazko, W.

doi:10.7169/facm/2013.48.1.8

Cited by 8 publications

(2 citation statements)

References 8 publications

(17 reference statements)

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“…Real SR polynomials of height 1 -namely, special cases of Littlewood, Newman and Borwein polynomials -were studied by several authors, see [27][28][29][30][31][32][33][34][35] and references therein 2 . Zeros of the so-called Ramanujan Polynomials and generalizations were analyzed in [37][38][39]. Finally, the Galois theory of PSR polynomials was studied in [40] by Lindstrøm, who showed that any PSR polynomial of degree less than 10 can be solved by radicals.…”

Section: Real Self-reciprocal Polynomialsmentioning

confidence: 99%

Polynomials with Symmetric Zeros

Vieira

2019

Polynomials - Theory and Application

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Polynomials whose zeros are symmetric either to the real line or to the unit circle are very important in mathematics and physics. We can classify them into three main classes: the self-conjugate polynomials, whose zeros are symmetric to the real line; the self-inversive polynomials, whose zeros are symmetric to the unit circle; and the self-reciprocal polynomials, whose zeros are symmetric by an inversion with respect to the unit circle followed by a reflection in the real line. Real selfreciprocal polynomials are simultaneously self-conjugate and self-inversive so that their zeros are symmetric to both the real line and the unit circle. In this survey, we present a short review of these polynomials, focusing on the distribution of their zeros.

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Section: Real Self-reciprocal Polynomialsmentioning

confidence: 99%

Polynomials with Symmetric Zeros

Vieira

2019

Polynomials - Theory and Application

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“…In particular, if the coefficients of P ( z ) are close to a constant, then all its roots lie on the unit circle. For example, let

E_{j} = j! [z^{j}] {(\cosh (z))}^{- 1}

denote Euler's numbers; then the polynomial

P_{n} (z) = {(- 1)}^{n} \sum_{0 \leq j \leq n} (\begin{matrix} 2 n \\ 2 j \end{matrix}) E_{2 j} E_{2 n - 2 j} z^{j} = [w^{n}] \frac{1}{\cos (\sqrt{w}) \cos (\sqrt{w z})},

is root‐unitary (see ) with non‐negative coefficients. See also for more information and other root‐unitary polynomials.…”

Section: Applications II Non‐normal Limit Lawsmentioning

confidence: 99%

Limit distribution of the coefficients of polynomials with only unit roots

Hwang

Zacharovas

2013

Random Struct Algorithms

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We consider sequences of random variables whose probability generating functions have only roots on the unit circle, which has only been sporadically studied in the literature. We show that the random variables are asymptotically normally distributed if and only if the fourth central and normalized (by the standard deviation) moment tends to 3, in contrast to the common scenario for polynomials with only real roots for which a central limit theorem holds if and only if the variance is unbounded. We also derive a representation theorem for all possible limit laws and apply our results to many concrete examples in the literature, ranging from combinatorial structures to numerical analysis, and from probability to analysis of algorithms. © 2013 Wiley Periodicals, Inc. Random Struct. Alg., 46,707–738, 2015

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An Unpublished Manuscript of Ramanujan on Infinite Series Identities

Andrews

Berndt

2013

Ramanujan's Lost Notebook

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Concerning dense subideals in commutative Banach algebras

Cited by 8 publications

References 8 publications

Polynomials with Symmetric Zeros

Polynomials with Symmetric Zeros

Limit distribution of the coefficients of polynomials with only unit roots

An Unpublished Manuscript of Ramanujan on Infinite Series Identities

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